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+arXiv:2301.03522v1  [gr-qc]  9 Jan 2023
+A Comment on “Traversable wormhole dynamics
+on a quantum processor”
+Galina Weinstein
+Reichman University, The Efi Arazi School of Computer Science, Herzliya;
+University of Haifa, The Department of Philosophy, Haifa, Israel.
+January 10, 2023
+Abstract
+There has been a lot of buzz surrounding the latest Nature paper,
+”Traversable wormhole dynamics on a quantum processor”. The Nature
+paper discusses an experiment in which Google’s Sycamore quantum pro-
+cessor is used to simulate a sparsified version of an SYK model. It is shown
+that the simplified model preserves the key gravitational characteristics of
+the original SYK model and that it is sufficient to produce a traversable
+wormhole behavior. The experiment does not create an actual wormhole.
+Rather, the team of researchers shows an equivalence between a gravity
+picture and a quantum information picture. This paper gives an account
+of the experiment and addresses philosophical questions arising from the
+theoretical and experimental work.
+1
+Quantum chaos and scrambling
+Let us begin with the quantum butterfly effect, which is essential for the under-
+standing of the experiment. The butterfly effect implies scrambling [5]. Quan-
+tum scrambling is the quantum analog of chaotic dynamics in classical systems.
+Scrambling describes many-body dynamics which, though ultimately unitary,
+scatter initially localized quantum information across all of the system’s avail-
+able degrees of freedom. Black holes are the fastest scramblers in the universe
+and are therefore the most chaotic bodies in the cosmos [14]; [1].
+More specifically, quantum information present in a small local area of space
+spreads out, and we must search a large region to recover the information.
+This is the scrambling of the quantum information while the system evolves.
+Heisenberg’s operators evolve in a way that reminds the chaotic butterfly effect:
+they were first local, and now they are spread over many regions in space. This
+is the butterfly effect in quantum systems.
+It should be stressed that when we speak about black holes, we are not
+talking about black holes that form from gravitational collapse. Rather what
+1
+
+is meant by black holes here and thereafter is eternal black holes (two-sided
+black holes) or two anti-de Sitter space (Ads) black holes. The eternal black
+hole is dual to two copies of the original conformal field theory (CFT) in the
+thermofield double (TFD) state.
+The TFD state is an entangled pure state
+between two identical copies of the quantum system (CFT):
+1
+√
+Zβ
+|T FDβ⟩ = eβ(HL+HR) |nn⟩L,R .
+(1)
+Tracing out one of the copies HL (the SYK Hamiltonian applied to the
+left system) or HR (the SYK Hamiltonian applied to the right system) gives a
+thermal state (with Majorana fermions). In other words, tracing out either copy
+produces the thermal density matrix at inverse temperature β. The |nn⟩L,R is
+the thermofield double state at an infinite temperature.
+The left and right
+external bulk regions of the eternal black hole are joined through a wormhole
+and are thus dual to the TFD state [10].
+The models for the onset and dynamics of quantum chaos are called the
+Sachdev-Ye-Kitaev (SYK) models. The SYK models lead to scrambling and
+spreading of the information among the quantum many-body system. But the
+SYK models possess gravity duals.
+They are also a paradigm for quantum
+holographic matter and the gravitational interpretation through the holographic
+principle or duality (the AdS/CFT correspondence or gauge/gravity duality);
+the equivalence between two descriptions of the same system: quantum gravity
+in (d+1) dimensions, on the one hand, and quantum field theory in d dimensions,
+on the other.
+The above characteristics of the SYK Hamiltonian for N fermions have led
+to realizing holographic physics in the laboratory, what is called quantum gravity
+in the lab. I will further discuss quantum gravity in the lab in section 9.
+An SYK model becomes extremely chaotic at the very beginning of its devel-
+opment. In the SYK model, the out-of-time-order correlation (OTOC) functions
+are used to diagnose quantum chaos, and measure the growth of operators in
+space, unitarily evolving (in the Heisenberg interpretation of quantum mechan-
+ics) as a function of time. With chaotic time evolution, the butterfly effect will
+cause most of the OTOC functions in the average to decay exponentially [5].
+In the semi-classical limit (in quantum systems with many degrees of free-
+dom), this scrambling of information and operator growth due to chaotic behav-
+ior is exponential and is measured using the quantum Lyapunov exponent. But
+unlike the classical Lyapunov exponent, there exists a bound on the quantum
+Lyapunov exponent. This is additionally measured by the butterfly velocity, the
+very equivalent measure of the classical chaotic butterfly effect. The quantum
+Lyapunov exponent and the scrambling rate are the ones that characterize the
+beginning and appearance of quantum chaos in this system [11].
+It should be noted that an interesting characteristic of the SYK model, which
+is related to the quantum Lyapunov exponent and the OTOC, is that the model
+exhibits maximally chaotic behavior. It means that like eternal black holes, the
+SYK model is a very fast scrambler of information. There is another important
+2
+
+quantity called, Loschmidt echo, which is intimately tied to quantum chaos. The
+echo is defined as the probability that the chaotic system would return to its
+initial state.
+As said above, we characterize quantum scrambling and quantum chaos by
+measuring the OTOC function. However, OTOCs do not generally discriminate
+between quantum scrambling and the effects of both ordinary quantum deco-
+herence and experimental noise: quantum scrambling and classical noise lead
+the OTOC to decay exponentially with time. It is a major problem if quantum
+scrambling is indistinguishable from quantum decoherence and noise, where the
+information in a system is lost to the environment [9];[15].
+Isolated systems are idealized models but unfortunately, realistic systems
+are open systems and are in interaction with the environment. Suppose there
+is a system of n qubits.
+This system is not an isolated and closed system.
+The n qubits are interacting with many interfering particles in the complex
+environment. It is almost impossible to follow the dynamics of each particle,
+so what we have here is a system that is many-body system, and decoherence
+induced by the environment. As the system evolves, the n qubits get entangled
+with the many-body system of the environment, and there are more disturbances
+and perturbations and more degrees of freedom. Decoherence happens naturally
+to quantum computers since like scrambling, qubits can’t be perfectly isolated
+from the environment.
+It was found that a quantum teleportation protocol enables one to differen-
+tiate between scrambling and decoherence. Thus using teleportation one can
+verify scrambling behavior even in the face of decoherence and experimental
+imperfection [1];[15].
+2
+SYK models and holography
+The SYK Hamiltonian is a model for quantum chaos and holography. That
+is, there is correspondence between the SYK model and scrambling/quantum
+chaotic behavior on the one hand, and eternal black holes, on the other. This
+dual possibility led a team of researchers to the realization that they might
+be able to create a model of teleportation through a traversable wormhole.
+They discovered that a process called unscrambling comes after scrambling in a
+wormhole. The discovery of a process of scrambling followed by unscrambling
+has boosted the possibility of realizing a quantum mechanism called size wind-
+ing in the lab. This process completely goes against everything we know from
+classical chaos and irreversibility. The size-winding mechanism is reminiscent
+of Poincare’s Recurrence Theorem of classical physics. But in the dual gravi-
+tational interpretation, size-winding leads to the interesting conclusion that a
+particle can pass through a wormhole (a holographic wormhole).
+The protocol is the following: on the left side of the wormhole, the infor-
+mation is scrambled. Since the two sides, right and left of the wormhole are
+connected (coupled), the information, i.e., qubits, is unscrambled and pops up
+on the right side. Two essential things enable traversability: the two sides of
+3
+
+the wormhole must be entangled before sending the information and the two
+sides must be coupled after sending the message.
+It was thought that it was possible to study the dynamics of a wormhole,
+through which a qubit can pass, by simulating the SYK model of N Majorana
+fermions. It was suggested that realizing the holographic SYK model on the
+Google Sycamore chip might open a window to an understanding of the quantum
+gravity of holographic traversable wormholes.
+The SYK models of a quantum many-body system simulate the scrambling-
+unscrambling method.
+According to the holographic principle, systems that
+are not gravitational but are entangled will exhibit properties that are identical
+to quantum gravity. Hence, reasoned the team of researchers, an experiment
+implementing the entanglement of qubits can be performed in the laboratory
+to test theories of quantum gravity. This experiment consists of two entangled
+systems of n qubits, on the right and n qubits on the left. In this protocol
+obviously, the dynamics of the system are chaotic (quantum mechanically) and
+is described by the SYK model.
+One inserts a qubit (the message) on the left side of the system (L subsys-
+tem), and it evolves in time. The qubit is entangled with one of the qubits on
+the L subsystem. It means that the qubit begins to spread among the n qubits
+(a small number of qubits) on the left side, and in all parts of the subsystem.
+After a certain time, the qubit is entangled with the qubits of the L subsystem.
+But then the qubit suddenly reappears, is unscrambled, and recoheres on the
+other side, the right side (R), very far from the L side, where it was scrambled.
+There is something that caused the original qubit, which entered on the
+far-left side, to suddenly be focused on the far-right side at a future time, even
+though it was completely mixed up on the left side.
+It is bizarre from the quantum mechanical point of view, says the team of
+researchers, but what makes things less weird is that it may be explained or in-
+terpreted using the paradigm of quantum gravity and the holographic principle:
+a traversable wormhole protocol is equivalent to the above quantum information
+protocol [2].
+We start from two separated black holes, the so-called scramblers. The tele-
+ported signal reappears on the right side when the two black holes are connected
+by a wormhole. In the quantum system, we speak of the method of scrambling-
+unscrambling. But with respect to a wormhole, the above mechanism is called
+teleportation-by-size, a protocol of quantum teleportation through the worm-
+hole, i.e., information transmission is dependent on operator-size growth.
+So, argues the team of researchers, if we imagine that the two sides of the
+system represent two sides of the eternal black holes (L and R) that are con-
+nected by a wormhole, then the explanation for the phenomenon is simpler. A
+teleported message is sent through an emergent wormhole: it is injected into L
+and arrives at R later due to a coupling operator [2].
+4
+
+3
+Perfect size winding
+The above scenario requires perfect size winding. The team of researchers first
+describes size-winding purely from the boundary point of view and then applies
+it to the traversable wormholes (in the bulk).
+In the Heisenberg picture, near the scrambling time (just before the onset of
+the chaotic behavior) for the SYK model, a thermal operator P is inserted at a
+negative time into the left boundary (the left side L). Recall that the growth
+of the size of an operator is a basic manifestation of quantum chaos and com-
+plexity of the system. The operator-size distribution is winding in the clockwise
+direction. A coupling is applied between the two subsystems L and R. The
+LR coupling unwinds the complex winding of the operator size distribution, it
+winds the size distribution in the opposite direction, accurately reversing the
+winding direction. The thermal operator P from the left side will be exactly
+mapped to its right side. We obtain a counterclockwise size distribution corre-
+sponding to a thermal operator P inserted on the other boundary (the right side
+R) at a positive time [2]. The team of researchers stresses: ”We explicitly show
+size-winding of thermal operators near the scrambling time for the SYK model,
+and we conjecture that the phenomenon can also be found in other holographic
+systems” [13].
+Perfect size winding provides a necessary condition for traversable wormhole
+behavior. It occurs in the ground state, the state of lowest possible energy where
+the temperature is zero (low temperature through the wormhole).
+The team of researchers expects systems with a holographic dual to exhibit
+perfect size winding [13]. In other words, the SYK model is dual to a traversable
+wormhole only in the low-temperature regime, and it exhibits perfect size wind-
+ing in the low-temperature limit. But this applies to large N Majorana fermions
+interacting with large q other Majorana fermions (teleportation of q fermions).
+The team of researchers then pondered: What is the most simplified Hamil-
+tonian that preserves the gravitational physics of the original SYK model? How
+many qubits do we need to simulate this Hamiltonian on a quantum device? It
+was shown that N = 10 was sufficient to produce the traversable wormhole be-
+havior. The team employed learning techniques to construct a sparsified version
+of the SYK model. Sparsification reduces the complexity of the system.
+A simplified learned Hamiltonian was constructed. Its ground state was close
+to a TFD state. Techniques from machine learning (and a kind of approximation
+called Trotterization) were applied to optimize the procedure. The techniques
+were performed on a classical computer. The sparsification procedure reduced
+the SYK model to a sparse N = 10 SYK model. ”We choose q = 4” fermions
+interacting with N other fermions of the simplified version of the SYK Hamil-
+tonian, ”and demonstrate gravitational physics at sufficiently small N”, where
+N = 7.
+Since ”The wormhole teleportation protocol also introduces a pair
+of entangled qubits, i.e., a reference qubit that is entangled with the injected
+qubit”, then ”the total circuit has 9 qubits”. Hence, the sparsified SYK model
+was experimentally realized with N = 9 qubits [6].
+It should be mentioned that at about the same time, Leonard Susskind and
+5
+
+a team of researchers were working on what seems like a bigger project, a sparse
+SYK model that recovers the global physics of ordinary SYK models. In par-
+ticular, at low temperatures, their model exhibits a gravitational sector that is
+maximally chaotic. The sparsity of the model, so writes the team, ”consider-
+ably reduces the cost of quantum simulation algorithms”. This, so claims the
+team, makes their sparse SYK model ”the most efficient currently known route
+to simulate a holographic model of quantum gravity”. The team of researchers
+add: ”On a practical level, sparse systems typically admit much more efficient
+computer simulations—both classical and quantum. By significantly reducing
+the resources needed to simulate black holes in holographic models of quantum
+gravity, these results bring us closer to the goal of studying ’quantum gravity
+in the lab’” [16].
+4
+Majorana fermions versus transmons
+The sparse Hamiltonian is doubled to give left HL and right HR Hamiltonians
+with N Majorana fermions on each side. Each side is a simulation of the SYK
+model, the learned Hamiltonian.
+The wormhole experiment was realized with superconducting qubits on the
+Google Sycamore. I would like to emphasize that I am not speaking now of
+claims related to quantum supremacy. The Sycamore consists of an array of
+54 superconducting qubits called transmons (transmission-line shunted plasma
+oscillation qubits).
+The transmon is closely related to the charge qubits or
+Cooper–Pair–Box (CPB) (Cooper pairs that are tunneling in a Josephson junc-
+tion). The transmon fixes the weakness of the CPB and as compared to the
+CPB, it greatly reduces charge noise sensitivity in the qubit [8].
+That said, the team of researchers is speaking of the Majorana SYK model
+with N fermions with which they produce evidence of gravitational physics
+in the sparsified SYK system: ”To encode 7 Majorana fermions on the left
+system and 7 Majorana fermions on the right system, we require 7 qubits (two
+fermions per qubit)” [6]. The team of researchers also writes: ”we assume that
+the total number of qubits (or fermions) on each side is n, and the number
+of message qubits (or fermions) that are transmitted by the state transfer or
+operator transfer protocols is m” [13].
+This is problematic because it is not at all clear whether one superconducting
+transmon qubit represents two Majorana fermions or rather, one transmon qubit
+represents one Majorana fermion.
+5
+The quantum information picture
+The practical steps of the teleportation protocol (step-by-step) in the quantum
+information picture (without gravity) are as follows (based on [2]):
+1) Two identical copies of the quantum system are prepared: a system of 7
+qubits on the left (side L) and a system of 7 qubits on the right (side R). The
+6
+
+two subsystems are entangled in the TFD state; that is, we have entangled Bell
+pairs shared between L and R.
+2) We evolve all the qubits on the side L “backward in time” by acting with
+the inverse of the time-evolution operator (exp+iHt).
+3) A qubit Q (the message) is injected into L at a certain time (swapped into
+side L: a SWAP gate). Now we evolve subsystem L “forward in time” using the
+time-evolution operator (exp−iHt). As a result, Q is entangled with a reference
+qubit P; Q is then scrambled with P and among the 7 qubits on the subsystem
+L (the carrier qubits).
+4) We now weakly couple side L to side R (at t = 0), applying a coupling
+operator (expiµV , where V is the interaction term and µ represents the coupling
+interaction). The coupling is applied suddenly: All the 7 qubits on side L are
+now coupled to the 7 qubits on the side R.
+5) We now evolve side R “forward in time” using the time-evolution oper-
+ator (exp−iHt). Side R is subsequently measured. The qubit Q (the message)
+reappears unscrambled, it arrives unscathed at R and there is no need to de-
+code it (a final SWAP gate: extract qubit Q from R). The message has been
+teleported while being first scrambled and then unscrambled. The teleported
+qubit is highly error-protected [4].
+The team of researchers distinguishes between two mechanisms of transmis-
+sion with the wormhole circuit [2]:
+1) The low-temperature teleportation: If µ < 0, the qubit Q experiences a
+time advance and is rescued on the side R. This is wormhole teleportation.
+2) On the other hand, when µ > 0 the qubit is entangled with the qubits of
+side L but is not unscrambled and its destiny is oblivion.
+6
+The gravity picture
+According to Occam’s razor, the simplest explanation for the above mecha-
+nism is teleportation-by-size, i.e., holographic teleportation. Thus in the grav-
+ity picture, a message has been teleported through a semi-classical holographic
+traversable wormhole [2]. Holographically, the above coupled LR quantum sys-
+tem is dual to a wormhole that connects the two sides of the eternal black hole.
+The LR coupling renders the wormhole traversable; if µ < 0, the coupling oper-
+ator generates a negative energy shockwave in the bulk, modifying the geometry
+of the wormhole and allowing traversability. When µ > 0 the coupling generates
+a positive energy shockwave and the qubit falls into the singularity. The team
+of researchers writes: ”we observe increased teleportation when the interaction
+introduces a negative energy shockwave rather than a positive one. The asym-
+metric signature is consistent with the physical interpretation that the qubit
+underwent teleportation through the wormhole” [6].
+The point is that for very low temperatures, the information does not vanish
+and the original entanglement between Q and T does not get destroyed by
+chaotic perturbations.
+How is this possible?
+Although there is scrambling
+and quantum chaotic behavior, the weak coupling interaction between L and
+7
+
+R entangles L and R, and the qubit Q is unscrambled. This is perfect size
+winding which causes teleportation around the scrambling time. In the perfect
+size winding protocol of scrambling followed by unscrambling the teleported
+qubit is highly error-protected [4]. I further discuss this issue in section 9.
+7
+Why should we believe the gravity picture?
+In the new experiment performed with the Sycamore chip, the team of re-
+searchers shows that their coarse-grained SYK model preserves key properties
+of the traversable wormhole physics: perfect size winding, coupling interac-
+tion on either side of the wormhole that is consistent with a negative energy
+shock wave, a Shapiro time delay, causal time-order of signals emerging from
+the wormhole, and scrambling [6].
+Besides sending a single qubit from left to right, another qubit is inserted
+from right to left. The result is time-ordered teleportation, which is interpreted
+as a demonstration of gravitational teleportation. At time −t0, a qubit Q is
+swapped into L. Simultaneously, a qubit R is swapped into R. At the time
+t1, the team of researchers performs a measurement and compares the two pro-
+cesses.
+They found that the presence of R delayed the arrival of the signal traveling
+left to right, and interpreted this delay observed in the learned Hamiltonian as
+due to a Shapiro time delay. It is also demonstrated that in the high-temperature
+regime, non-gravitational teleportation occurs, and there is no size winding [6].
+Working with collaborators from Caltech, Fermilab, and Harvard, the quan-
+tum system was subjected to numerous tests to determine if it showed quantum
+gravitational behavior. The above signatures were verified on classical com-
+puters, so claims the team of researchers, confirming that the dynamics of the
+quantum system were consistent with a quantum gravity interpretation and the
+holographic principle [17].
+8
+Scientific explanation is not truth
+We usually proceed from the success of an experiment to the conclusion that
+our explanation is likely to be approximately true, or true. We think that if an
+explanation is the best among the competing explanations of the experiment,
+then it is probably true. But it should be stressed that the fit between the
+simplified SYK model and the explanation in terms of an emergent wormhole
+does not mean that the latter explanation is literally true. Neither does it mean
+that holographic wormholes exist or that they are real. What is meant by saying
+that this explanation is the simplest among the other hypotheses is mainly that,
+it is the best fit for the experimental setup, and that holographic teleportation
+fits the teleportation mechanism at the basis of the said experiment.
+The point is that according to the ER = EPR hypothesis, the gravity picture
+is equivalent to the quantum information picture, and ”The traversable worm-
+8
+
+hole expressed as a quantum circuit, equivalent to the gravitational picture in
+the semiclassical limit of an infinite number of qubits” [6]. But although the
+analogy between the experimental setup and the emergent geometry is sugges-
+tive, it does not follow from the experiment that the wormhole gravitational
+picture is real. We can only say that teleportation-by-size is the hypothesis
+that explains the experiment best. This is so even if it explains the evidence.
+”Truth” requires a step beyond the judgment that the holographic wormhole
+hypothesis fits the experimental setup and the data and is better than all of its
+rivals.
+9
+Quantum gravity in the lab
+Advocates of the ”quantum gravity in the Lab” program argue: ”The ‘quan-
+tum gravity in the lab’ program does not need to wait for large error-corrected
+quantum computers. Progress can be made even in the Noisy Intermediate-Scale
+Quantum (NISQ) era” [13]. There is a problem with this statement.
+As is well known, quantum computers are prone to many errors and the
+Sycamore quantum device has a large error rate [7]. In this state of affairs, ”If,
+at any point in time, a small error occurs, the chaotic dynamics will not undo
+themselves, and the particle will not make it through the wormhole” [17].
+At large times, a small perturbation can destroy the correlations between the
+two sides L and R of the quantum system that would otherwise exist without
+the perturbation. Although the qubits of the Sycamore processor are cooled
+down to cryogenic temperatures and are held in an ultra-high vacuum chamber,
+the entangled qubits can decohere quickly due to interaction (entanglement)
+with the environment (incoherent errors). The team of researchers writes: ”In
+general, errors can include coherent errors [crosstalk errors and qubit phase] and
+incoherent sources of noise; in simulations, we assume fully incoherent errors and
+observe agreement with experimental data” [6].
+A team of researchers trained a quantum neural network (in a quantum
+machine learning context). An appropriate ansatz can mitigate coherent errors
+for only a small number of qubits (18) on the Sycamore quantum device [12].
+Proponents of the ”quantum gravity in the Lab” program show that ”with some
+caveats we can use a finite fraction of the fermions” [13]. So in order to reduce
+the coherent errors, ”the total circuit has 9 qubits” [6]. Recall that in practice,
+only 7 qubits were used to simulate a ”wormhole-like teleportation”. The other
+two qubits served as the teleported qubits [6] (see sections 3 and 5).
+Using
+machine learning, the team of researchers was able to make the quantum model
+simple enough to preserve the key gravitational properties, so that it could be
+realized with a circuit with 164 two-qubit gates [6]. A more complex model
+would increase the number of gates, and consequently also the error rate.
+In the Caltech press release, it is said that the team of researchers found a
+quantum system, a “baby” SYK-like model, prepared to preserve the key prop-
+erties of a gravitational wormhole. To achieve this, the team had to first reduce
+the SYK model to a simplified form, a feat they achieved using machine learn-
+9
+
+ing tools on conventional computers. They employed learning techniques to find
+and prepare a simple SYK-like quantum system that could be encoded in the
+Sycamore quantum architecture, and that would preserve the key gravitational
+property: the negative energy shockwave. The greatest achievement was sim-
+plifying the microscopic description of the SYK quantum system and studying
+the resulting effective model that the team found on the Sycamore quantum
+processor. The team of researchers found it “curious and surprising how the
+optimization on one characteristic of the model [the negative energy shockwave
+or LR coupling] preserved the other characteristics” [3].
+Reducing the SYK model to a simplified form is an achievement that is to
+be celebrated. But this demonstrates the amazing capabilities of conventional
+computers. It is important to stress that no wormhole was created in the lab,
+and moreover, no one has ever observed or found any evidence of any wormhole.
+Acknowledgement
+This work is supported by ERC advanced grant number 834735.
+References
+[1]
+M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D.
+Dahlen, A. Morvan, B. Yoshida, N. Y. Yao and I. Siddiqi (2021). ”Quantum
+Information Scrambling in a Superconducting Qutrit Processor.” Physical
+Review X 10, pp. 021010-1- 021010-21.
+[2]
+A. R. Brown, H. Gharibyan, S. Leichenauer, H. W. Lin, S. Nezami, G.
+Salton, L. Susskind, B. Swingle and M. Walter (2019). ”Quantum Gravity
+in the Lab: Teleportation by Size and Traversable Wormholes.” arXiv:
+1911.06314v2 [quant-ph]
+[3]
+Clavin, W. (2022). ”Physicists observe wormhole dynamics using a quan-
+tum computer.” Caltech.
+[4]
+P. Gao and D. L. Jafferis (2021). ”A traversable wormhole teleportation
+protocol in the SYK model.” Journal of High Energy Physics 2021, pp.
+1-43.
+[5]
+P. Hosur, X-L. Qi, D. A. Roberts, and B. Yoshida, (2016). ”Chaos in Quan-
+tum Channels,” Journal of High Energy Physics 2016, pp. 1-48.
+[6]
+D. Jafferis, A. Zlokapa, J. D. Lykken, D. K. Kolchmeyer, S. I. Davis, N.
+Lauk, H. Neven, and M. Spiropulu (2022). ”Traversable wormhole dynamics
+on a quantum processor.” Nature 612, pp. 51–55.
+[7]
+G.
+Kalai,
+Y.
+Rinott,
+T.
+Shoham
+(2022).
+”Google’s
+2019
+’Quan-
+tum
+Supremacy’
+Claims:
+Data,
+Documentation,
+and
+Discussion.”
+arXiv:2210.12753v2 [quant-ph], pp. 1-34.
+10
+
+[8]
+J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A.
+Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf (2007). ”Charge-
+insensitive qubit design derived from the Cooper pair box.” Physical Review
+A 76, pp. 042319-1-042319-19.
+[9]
+K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y.
+Yao, and C. Monroe (2019). ”Verified quantum information scrambling.”
+Nature 567, pp. 61–65.
+[10] J.
+Maldacena
+and
+X.-L.
+Qi
+(2018)
+Eternal
+traversable
+wormhole,
+arXiv:1804.00491v3 [hep-th], pp. 1-74.
+[11] J. Maldacena, S. H. Shenker and D. Stanford, (2016). ”A Bound on Chaos.”
+Journal of High Energy Physics (2016), pp. 1-16.
+[12] M. Y. Niu, A. Zlokapa, M. Broughton, S. Boixo, M. Mohseni, V. Smelyan-
+skyi, and H. Neven (2022). ”Entangling Quantum Generative Adversarial
+Networks.”Physical Review Letters 128.220505.
+[13] S. Nezami, H. W. Lin, A. R. Brown, H. Gharibyan, S. Leichenauer, G.
+Salton, L. Susskind, B. Swingle and M. Walter (2022). ”Quantum Gravity
+in the Lab: Teleportation by Size and Traversable Wormholes, Part II.”
+arXiv: 2102.01064v1 [quant-ph]
+[14] Y. Sekino and L. Susskind (2008). ”Fast scramblers.” Journal of High En-
+ergy Physics 10, pp. 1-14.
+[15] B. Yoshida and N. Y. Yao (2019). ”Disentangling Scrambling and Deco-
+herence via Quantum Teleportation.” Physical Review X 9, pp. 011006-1-
+011006-17.
+[16] S. Xu, L. Susskind, Y. Su, B. Swingle (2020). ”A Sparse Model of Quantum
+Holography.” arXiv:2008.02303v1 [cond-mat.str-el], pp. 1-55.
+[17] Zlokapa, A. (2022). ”Making a Traversable Wormhole with a Quantum
+Computer.” Google Research.
+11
+

49E1T4oBgHgl3EQf6QXo/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf,len=382
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='03522v1  [gr-qc]  9 Jan 2023  A Comment on “Traversable wormhole dynamics  on a quantum processor”  Galina Weinstein  Reichman University, The Efi Arazi School of Computer Science, Herzliya;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  University of Haifa, The Department of Philosophy, Haifa, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  January 10, 2023  Abstract  There has been a lot of buzz surrounding the latest Nature paper,  ”Traversable wormhole dynamics on a quantum processor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The Nature  paper discusses an experiment in which Google’s Sycamore quantum pro-  cessor is used to simulate a sparsified version of an SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It is shown  that the simplified model preserves the key gravitational characteristics of  the original SYK model and that it is sufficient to produce a traversable  wormhole behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The experiment does not create an actual wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Rather, the team of researchers shows an equivalence between a gravity  picture and a quantum information picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This paper gives an account  of the experiment and addresses philosophical questions arising from the  theoretical and experimental work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  1  Quantum chaos and scrambling  Let us begin with the quantum butterfly effect, which is essential for the under-  standing of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The butterfly effect implies scrambling [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Quan-  tum scrambling is the quantum analog of chaotic dynamics in classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Scrambling describes many-body dynamics which, though ultimately unitary,  scatter initially localized quantum information across all of the system’s avail-  able degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Black holes are the fastest scramblers in the universe  and are therefore the most chaotic bodies in the cosmos [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  More specifically, quantum information present in a small local area of space  spreads out, and we must search a large region to recover the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  This is the scrambling of the quantum information while the system evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Heisenberg’s operators evolve in a way that reminds the chaotic butterfly effect:  they were first local, and now they are spread over many regions in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This  is the butterfly effect in quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It should be stressed that when we speak about black holes, we are not  talking about black holes that form from gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Rather what  1  is meant by black holes here and thereafter is eternal black holes (two-sided  black holes) or two anti-de Sitter space (Ads) black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The eternal black  hole is dual to two copies of the original conformal field theory (CFT) in the  thermofield double (TFD) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The TFD state is an entangled pure state  between two identical copies of the quantum system (CFT):  1  √  Zβ  |T FDβ⟩ = eβ(HL+HR) |nn⟩L,R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  (1)  Tracing out one of the copies HL (the SYK Hamiltonian applied to the  left system) or HR (the SYK Hamiltonian applied to the right system) gives a  thermal state (with Majorana fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In other words, tracing out either copy  produces the thermal density matrix at inverse temperature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The |nn⟩L,R is  the thermofield double state at an infinite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The left and right  external bulk regions of the eternal black hole are joined through a wormhole  and are thus dual to the TFD state [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The models for the onset and dynamics of quantum chaos are called the  Sachdev-Ye-Kitaev (SYK) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The SYK models lead to scrambling and  spreading of the information among the quantum many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But the  SYK models possess gravity duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  They are also a paradigm for quantum  holographic matter and the gravitational interpretation through the holographic  principle or duality (the AdS/CFT correspondence or gauge/gravity duality);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  the equivalence between two descriptions of the same system: quantum gravity  in (d+1) dimensions, on the one hand, and quantum field theory in d dimensions,  on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The above characteristics of the SYK Hamiltonian for N fermions have led  to realizing holographic physics in the laboratory, what is called quantum gravity  in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' I will further discuss quantum gravity in the lab in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  An SYK model becomes extremely chaotic at the very beginning of its devel-  opment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In the SYK model, the out-of-time-order correlation (OTOC) functions  are used to diagnose quantum chaos, and measure the growth of operators in  space, unitarily evolving (in the Heisenberg interpretation of quantum mechan-  ics) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' With chaotic time evolution, the butterfly effect will  cause most of the OTOC functions in the average to decay exponentially [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  In the semi-classical limit (in quantum systems with many degrees of free-  dom), this scrambling of information and operator growth due to chaotic behav-  ior is exponential and is measured using the quantum Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But  unlike the classical Lyapunov exponent, there exists a bound on the quantum  Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This is additionally measured by the butterfly velocity, the  very equivalent measure of the classical chaotic butterfly effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The quantum  Lyapunov exponent and the scrambling rate are the ones that characterize the  beginning and appearance of quantum chaos in this system [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It should be noted that an interesting characteristic of the SYK model, which  is related to the quantum Lyapunov exponent and the OTOC, is that the model  exhibits maximally chaotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It means that like eternal black holes, the  SYK model is a very fast scrambler of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' There is another important  2  quantity called, Loschmidt echo, which is intimately tied to quantum chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The  echo is defined as the probability that the chaotic system would return to its  initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  As said above, we characterize quantum scrambling and quantum chaos by  measuring the OTOC function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' However, OTOCs do not generally discriminate  between quantum scrambling and the effects of both ordinary quantum deco-  herence and experimental noise: quantum scrambling and classical noise lead  the OTOC to decay exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It is a major problem if quantum  scrambling is indistinguishable from quantum decoherence and noise, where the  information in a system is lost to the environment [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Isolated systems are idealized models but unfortunately, realistic systems  are open systems and are in interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Suppose there  is a system of n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  This system is not an isolated and closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The n qubits are interacting with many interfering particles in the complex  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It is almost impossible to follow the dynamics of each particle,  so what we have here is a system that is many-body system, and decoherence  induced by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' As the system evolves, the n qubits get entangled  with the many-body system of the environment, and there are more disturbances  and perturbations and more degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Decoherence happens naturally  to quantum computers since like scrambling, qubits can’t be perfectly isolated  from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It was found that a quantum teleportation protocol enables one to differen-  tiate between scrambling and decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Thus using teleportation one can  verify scrambling behavior even in the face of decoherence and experimental  imperfection [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  2  SYK models and holography  The SYK Hamiltonian is a model for quantum chaos and holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' That  is, there is correspondence between the SYK model and scrambling/quantum  chaotic behavior on the one hand, and eternal black holes, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This  dual possibility led a team of researchers to the realization that they might  be able to create a model of teleportation through a traversable wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  They discovered that a process called unscrambling comes after scrambling in a  wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The discovery of a process of scrambling followed by unscrambling  has boosted the possibility of realizing a quantum mechanism called size wind-  ing in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This process completely goes against everything we know from  classical chaos and irreversibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The size-winding mechanism is reminiscent  of Poincare’s Recurrence Theorem of classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But in the dual gravi-  tational interpretation, size-winding leads to the interesting conclusion that a  particle can pass through a wormhole (a holographic wormhole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The protocol is the following: on the left side of the wormhole, the infor-  mation is scrambled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Since the two sides, right and left of the wormhole are  connected (coupled), the information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=', qubits, is unscrambled and pops up  on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Two essential things enable traversability: the two sides of  3  the wormhole must be entangled before sending the information and the two  sides must be coupled after sending the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It was thought that it was possible to study the dynamics of a wormhole,  through which a qubit can pass, by simulating the SYK model of N Majorana  fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It was suggested that realizing the holographic SYK model on the  Google Sycamore chip might open a window to an understanding of the quantum  gravity of holographic traversable wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The SYK models of a quantum many-body system simulate the scrambling-  unscrambling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  According to the holographic principle, systems that  are not gravitational but are entangled will exhibit properties that are identical  to quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Hence, reasoned the team of researchers, an experiment  implementing the entanglement of qubits can be performed in the laboratory  to test theories of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This experiment consists of two entangled  systems of n qubits, on the right and n qubits on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In this protocol  obviously, the dynamics of the system are chaotic (quantum mechanically) and  is described by the SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  One inserts a qubit (the message) on the left side of the system (L subsys-  tem), and it evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The qubit is entangled with one of the qubits on  the L subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It means that the qubit begins to spread among the n qubits  (a small number of qubits) on the left side, and in all parts of the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  After a certain time, the qubit is entangled with the qubits of the L subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  But then the qubit suddenly reappears, is unscrambled, and recoheres on the  other side, the right side (R), very far from the L side, where it was scrambled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  There is something that caused the original qubit, which entered on the  far-left side, to suddenly be focused on the far-right side at a future time, even  though it was completely mixed up on the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It is bizarre from the quantum mechanical point of view, says the team of  researchers, but what makes things less weird is that it may be explained or in-  terpreted using the paradigm of quantum gravity and the holographic principle:  a traversable wormhole protocol is equivalent to the above quantum information  protocol [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  We start from two separated black holes, the so-called scramblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The tele-  ported signal reappears on the right side when the two black holes are connected  by a wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In the quantum system, we speak of the method of scrambling-  unscrambling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But with respect to a wormhole, the above mechanism is called  teleportation-by-size, a protocol of quantum teleportation through the worm-  hole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=', information transmission is dependent on operator-size growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  So, argues the team of researchers, if we imagine that the two sides of the  system represent two sides of the eternal black holes (L and R) that are con-  nected by a wormhole, then the explanation for the phenomenon is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A  teleported message is sent through an emergent wormhole: it is injected into L  and arrives at R later due to a coupling operator [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  4  3  Perfect size winding  The above scenario requires perfect size winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers first  describes size-winding purely from the boundary point of view and then applies  it to the traversable wormholes (in the bulk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  In the Heisenberg picture, near the scrambling time (just before the onset of  the chaotic behavior) for the SYK model, a thermal operator P is inserted at a  negative time into the left boundary (the left side L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Recall that the growth  of the size of an operator is a basic manifestation of quantum chaos and com-  plexity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The operator-size distribution is winding in the clockwise  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A coupling is applied between the two subsystems L and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The  LR coupling unwinds the complex winding of the operator size distribution, it  winds the size distribution in the opposite direction, accurately reversing the  winding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The thermal operator P from the left side will be exactly  mapped to its right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' We obtain a counterclockwise size distribution corre-  sponding to a thermal operator P inserted on the other boundary (the right side  R) at a positive time [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers stresses: ”We explicitly show  size-winding of thermal operators near the scrambling time for the SYK model,  and we conjecture that the phenomenon can also be found in other holographic  systems” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Perfect size winding provides a necessary condition for traversable wormhole  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It occurs in the ground state, the state of lowest possible energy where  the temperature is zero (low temperature through the wormhole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The team of researchers expects systems with a holographic dual to exhibit  perfect size winding [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In other words, the SYK model is dual to a traversable  wormhole only in the low-temperature regime, and it exhibits perfect size wind-  ing in the low-temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But this applies to large N Majorana fermions  interacting with large q other Majorana fermions (teleportation of q fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The team of researchers then pondered: What is the most simplified Hamil-  tonian that preserves the gravitational physics of the original SYK model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' How  many qubits do we need to simulate this Hamiltonian on a quantum device?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It  was shown that N = 10 was sufficient to produce the traversable wormhole be-  havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team employed learning techniques to construct a sparsified version  of the SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Sparsification reduces the complexity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  A simplified learned Hamiltonian was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Its ground state was close  to a TFD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Techniques from machine learning (and a kind of approximation  called Trotterization) were applied to optimize the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The techniques  were performed on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The sparsification procedure reduced  the SYK model to a sparse N = 10 SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”We choose q = 4” fermions  interacting with N other fermions of the simplified version of the SYK Hamil-  tonian, ”and demonstrate gravitational physics at sufficiently small N”, where  N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Since ”The wormhole teleportation protocol also introduces a pair  of entangled qubits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=', a reference qubit that is entangled with the injected  qubit”, then ”the total circuit has 9 qubits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Hence, the sparsified SYK model  was experimentally realized with N = 9 qubits [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  It should be mentioned that at about the same time, Leonard Susskind and  5  a team of researchers were working on what seems like a bigger project, a sparse  SYK model that recovers the global physics of ordinary SYK models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In par-  ticular, at low temperatures, their model exhibits a gravitational sector that is  maximally chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The sparsity of the model, so writes the team, ”consider-  ably reduces the cost of quantum simulation algorithms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This, so claims the  team, makes their sparse SYK model ”the most efficient currently known route  to simulate a holographic model of quantum gravity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers  add: ”On a practical level, sparse systems typically admit much more efficient  computer simulations—both classical and quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' By significantly reducing  the resources needed to simulate black holes in holographic models of quantum  gravity, these results bring us closer to the goal of studying ’quantum gravity  in the lab’” [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  4  Majorana fermions versus transmons  The sparse Hamiltonian is doubled to give left HL and right HR Hamiltonians  with N Majorana fermions on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Each side is a simulation of the SYK  model, the learned Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The wormhole experiment was realized with superconducting qubits on the  Google Sycamore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' I would like to emphasize that I am not speaking now of  claims related to quantum supremacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The Sycamore consists of an array of  54 superconducting qubits called transmons (transmission-line shunted plasma  oscillation qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The transmon is closely related to the charge qubits or  Cooper–Pair–Box (CPB) (Cooper pairs that are tunneling in a Josephson junc-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The transmon fixes the weakness of the CPB and as compared to the  CPB, it greatly reduces charge noise sensitivity in the qubit [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  That said, the team of researchers is speaking of the Majorana SYK model  with N fermions with which they produce evidence of gravitational physics  in the sparsified SYK system: ”To encode 7 Majorana fermions on the left  system and 7 Majorana fermions on the right system, we require 7 qubits (two  fermions per qubit)” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers also writes: ”we assume that  the total number of qubits (or fermions) on each side is n, and the number  of message qubits (or fermions) that are transmitted by the state transfer or  operator transfer protocols is m” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  This is problematic because it is not at all clear whether one superconducting  transmon qubit represents two Majorana fermions or rather, one transmon qubit  represents one Majorana fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  5  The quantum information picture  The practical steps of the teleportation protocol (step-by-step) in the quantum  information picture (without gravity) are as follows (based on [2]):  1) Two identical copies of the quantum system are prepared: a system of 7  qubits on the left (side L) and a system of 7 qubits on the right (side R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The  6  two subsystems are entangled in the TFD state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' that is, we have entangled Bell  pairs shared between L and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  2) We evolve all the qubits on the side L “backward in time” by acting with  the inverse of the time-evolution operator (exp+iHt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  3) A qubit Q (the message) is injected into L at a certain time (swapped into  side L: a SWAP gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Now we evolve subsystem L “forward in time” using the  time-evolution operator (exp−iHt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' As a result, Q is entangled with a reference  qubit P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Q is then scrambled with P and among the 7 qubits on the subsystem  L (the carrier qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  4) We now weakly couple side L to side R (at t = 0), applying a coupling  operator (expiµV , where V is the interaction term and µ represents the coupling  interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The coupling is applied suddenly: All the 7 qubits on side L are  now coupled to the 7 qubits on the side R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  5) We now evolve side R “forward in time” using the time-evolution oper-  ator (exp−iHt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Side R is subsequently measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The qubit Q (the message)  reappears unscrambled, it arrives unscathed at R and there is no need to de-  code it (a final SWAP gate: extract qubit Q from R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The message has been  teleported while being first scrambled and then unscrambled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The teleported  qubit is highly error-protected [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The team of researchers distinguishes between two mechanisms of transmis-  sion with the wormhole circuit [2]:  1) The low-temperature teleportation: If µ < 0, the qubit Q experiences a  time advance and is rescued on the side R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This is wormhole teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  2) On the other hand, when µ > 0 the qubit is entangled with the qubits of  side L but is not unscrambled and its destiny is oblivion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  6  The gravity picture  According to Occam’s razor, the simplest explanation for the above mecha-  nism is teleportation-by-size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=', holographic teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Thus in the grav-  ity picture, a message has been teleported through a semi-classical holographic  traversable wormhole [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Holographically, the above coupled LR quantum sys-  tem is dual to a wormhole that connects the two sides of the eternal black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The LR coupling renders the wormhole traversable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' if µ < 0, the coupling oper-  ator generates a negative energy shockwave in the bulk, modifying the geometry  of the wormhole and allowing traversability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' When µ > 0 the coupling generates  a positive energy shockwave and the qubit falls into the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team  of researchers writes: ”we observe increased teleportation when the interaction  introduces a negative energy shockwave rather than a positive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The asym-  metric signature is consistent with the physical interpretation that the qubit  underwent teleportation through the wormhole” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The point is that for very low temperatures, the information does not vanish  and the original entanglement between Q and T does not get destroyed by  chaotic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  How is this possible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Although there is scrambling  and quantum chaotic behavior, the weak coupling interaction between L and  7  R entangles L and R, and the qubit Q is unscrambled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This is perfect size  winding which causes teleportation around the scrambling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In the perfect  size winding protocol of scrambling followed by unscrambling the teleported  qubit is highly error-protected [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' I further discuss this issue in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  7  Why should we believe the gravity picture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  In the new experiment performed with the Sycamore chip, the team of re-  searchers shows that their coarse-grained SYK model preserves key properties  of the traversable wormhole physics: perfect size winding, coupling interac-  tion on either side of the wormhole that is consistent with a negative energy  shock wave, a Shapiro time delay, causal time-order of signals emerging from  the wormhole, and scrambling [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Besides sending a single qubit from left to right, another qubit is inserted  from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The result is time-ordered teleportation, which is interpreted  as a demonstration of gravitational teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' At time −t0, a qubit Q is  swapped into L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Simultaneously, a qubit R is swapped into R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' At the time  t1, the team of researchers performs a measurement and compares the two pro-  cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  They found that the presence of R delayed the arrival of the signal traveling  left to right, and interpreted this delay observed in the learned Hamiltonian as  due to a Shapiro time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It is also demonstrated that in the high-temperature  regime, non-gravitational teleportation occurs, and there is no size winding [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Working with collaborators from Caltech, Fermilab, and Harvard, the quan-  tum system was subjected to numerous tests to determine if it showed quantum  gravitational behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The above signatures were verified on classical com-  puters, so claims the team of researchers, confirming that the dynamics of the  quantum system were consistent with a quantum gravity interpretation and the  holographic principle [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  8  Scientific explanation is not truth  We usually proceed from the success of an experiment to the conclusion that  our explanation is likely to be approximately true, or true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' We think that if an  explanation is the best among the competing explanations of the experiment,  then it is probably true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But it should be stressed that the fit between the  simplified SYK model and the explanation in terms of an emergent wormhole  does not mean that the latter explanation is literally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Neither does it mean  that holographic wormholes exist or that they are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' What is meant by saying  that this explanation is the simplest among the other hypotheses is mainly that,  it is the best fit for the experimental setup, and that holographic teleportation  fits the teleportation mechanism at the basis of the said experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  The point is that according to the ER = EPR hypothesis, the gravity picture  is equivalent to the quantum information picture, and ”The traversable worm-  8  hole expressed as a quantum circuit, equivalent to the gravitational picture in  the semiclassical limit of an infinite number of qubits” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But although the  analogy between the experimental setup and the emergent geometry is sugges-  tive, it does not follow from the experiment that the wormhole gravitational  picture is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' We can only say that teleportation-by-size is the hypothesis  that explains the experiment best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' This is so even if it explains the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  ”Truth” requires a step beyond the judgment that the holographic wormhole  hypothesis fits the experimental setup and the data and is better than all of its  rivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  9  Quantum gravity in the lab  Advocates of the ”quantum gravity in the Lab” program argue: ”The ‘quan-  tum gravity in the lab’ program does not need to wait for large error-corrected  quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Progress can be made even in the Noisy Intermediate-Scale  Quantum (NISQ) era” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' There is a problem with this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  As is well known, quantum computers are prone to many errors and the  Sycamore quantum device has a large error rate [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' In this state of affairs, ”If,  at any point in time, a small error occurs, the chaotic dynamics will not undo  themselves, and the particle will not make it through the wormhole” [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  At large times, a small perturbation can destroy the correlations between the  two sides L and R of the quantum system that would otherwise exist without  the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Although the qubits of the Sycamore processor are cooled  down to cryogenic temperatures and are held in an ultra-high vacuum chamber,  the entangled qubits can decohere quickly due to interaction (entanglement)  with the environment (incoherent errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers writes: ”In  general, errors can include coherent errors [crosstalk errors and qubit phase] and  incoherent sources of noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' in simulations, we assume fully incoherent errors and  observe agreement with experimental data” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  A team of researchers trained a quantum neural network (in a quantum  machine learning context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' An appropriate ansatz can mitigate coherent errors  for only a small number of qubits (18) on the Sycamore quantum device [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Proponents of the ”quantum gravity in the Lab” program show that ”with some  caveats we can use a finite fraction of the fermions” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' So in order to reduce  the coherent errors, ”the total circuit has 9 qubits” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Recall that in practice,  only 7 qubits were used to simulate a ”wormhole-like teleportation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The other  two qubits served as the teleported qubits [6] (see sections 3 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Using  machine learning, the team of researchers was able to make the quantum model  simple enough to preserve the key gravitational properties, so that it could be  realized with a circuit with 164 two-qubit gates [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A more complex model  would increase the number of gates, and consequently also the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  In the Caltech press release, it is said that the team of researchers found a  quantum system, a “baby” SYK-like model, prepared to preserve the key prop-  erties of a gravitational wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' To achieve this, the team had to first reduce  the SYK model to a simplified form, a feat they achieved using machine learn-  9  ing tools on conventional computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' They employed learning techniques to find  and prepare a simple SYK-like quantum system that could be encoded in the  Sycamore quantum architecture, and that would preserve the key gravitational  property: the negative energy shockwave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The greatest achievement was sim-  plifying the microscopic description of the SYK quantum system and studying  the resulting effective model that the team found on the Sycamore quantum  processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' The team of researchers found it “curious and surprising how the  optimization on one characteristic of the model [the negative energy shockwave  or LR coupling] preserved the other characteristics” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Reducing the SYK model to a simplified form is an achievement that is to  be celebrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' But this demonstrates the amazing capabilities of conventional  computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' It is important to stress that no wormhole was created in the lab,  and moreover, no one has ever observed or found any evidence of any wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Acknowledgement  This work is supported by ERC advanced grant number 834735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  References  [1]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Blok, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Ramasesh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Schuster, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' O’Brien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Kreikebaum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Dahlen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Morvan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yoshida, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yao and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Siddiqi (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Quantum  Information Scrambling in a Superconducting Qutrit Processor.” Physical  Review X 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 021010-1- 021010-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [2]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Brown, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Gharibyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Leichenauer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Nezami, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Salton, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Susskind, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Swingle and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Walter (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Quantum Gravity  in the Lab: Teleportation by Size and Traversable Wormholes.” arXiv:  1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='06314v2 [quant-ph]  [3]  Clavin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Physicists observe wormhole dynamics using a quan-  tum computer.” Caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [4]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Gao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Jafferis (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”A traversable wormhole teleportation  protocol in the SYK model.” Journal of High Energy Physics 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  1-43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [5]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Hosur, X-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Roberts, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yoshida, (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Chaos in Quan-  tum Channels,” Journal of High Energy Physics 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [6]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Jafferis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Zlokapa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Lykken, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Kolchmeyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Davis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Lauk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Neven, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Spiropulu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Traversable wormhole dynamics  on a quantum processor.” Nature 612, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 51–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [7]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Kalai,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Rinott,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Shoham  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  ”Google’s  2019  ’Quan-  tum  Supremacy’  Claims:  Data,  Documentation,  and  Discussion.”  arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='12753v2 [quant-ph], pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  10  [8]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Koch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Gambetta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Houck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Schuster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Majer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Blais, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Devoret, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Girvin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Schoelkopf (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Charge-  insensitive qubit design derived from the Cooper pair box.” Physical Review  A 76, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 042319-1-042319-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [9]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Landsman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Figgatt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Schuster, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Linke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yoshida, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Yao, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Monroe (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Verified quantum information scrambling.”  Nature 567, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 61–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Maldacena  and  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Qi  (2018)  Eternal  traversable  wormhole,  arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='00491v3 [hep-th], pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Maldacena, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Shenker and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Stanford, (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”A Bound on Chaos.”  Journal of High Energy Physics (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Niu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Zlokapa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Broughton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Boixo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Mohseni, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Smelyan-  skyi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Neven (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Entangling Quantum Generative Adversarial  Networks.”Physical Review Letters 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='220505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Nezami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Brown, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Gharibyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Leichenauer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  Salton, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Susskind, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Swingle and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Walter (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Quantum Gravity  in the Lab: Teleportation by Size and Traversable Wormholes, Part II.”  arXiv: 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='01064v1 [quant-ph]  [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Sekino and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Susskind (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Fast scramblers.” Journal of High En-  ergy Physics 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yoshida and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Yao (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Disentangling Scrambling and Deco-  herence via Quantum Teleportation.” Physical Review X 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 011006-1-  011006-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Susskind, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Su, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' Swingle (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”A Sparse Model of Quantum  Holography.” arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='02303v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='str-el], pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' 1-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  [17] Zlokapa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content=' ”Making a Traversable Wormhole with a Quantum  Computer.” Google Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf'}

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+MNRAS 000, 1–15 (0000)
+Preprint 12 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The comptonizing medium of the black-hole X-ray binary
+MAXI J1535−571 through type-C quasi-periodic oscillations
+Divya Rawat1⋆, Mariano M´endez2, Federico Garc´ıa2,3,4, Diego Altamirano5, Konstantinos Karpouzas2,5,
+Liang Zhang5, Kevin Alabarta2,5, Tomaso M. Belloni6, Pankaj Jain7, Candela Bellavita4
+1Inter-University Center for Astronomy and Astrophysics, Ganeshkhind, Pune 411007, India
+2Kapteyn Astronomical Institute, University of Groningen, PO BOX 800, Groningen NL-9700 AV, the Netherlands
+3Instituto Argentino de Radioastronom´ıa (CCT La Plata, CONICET; CICPBA; UNLP), C.C.5, (1894) Villa Elisa, Buenos Aires, Argentina
+4Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, Paseo del Bosque, B1900FWA La Plata, Argentina
+5School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
+6INAF-Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807, Merate, Italy
+7Department of physics, IIT Kanpur, Kanpur, Uttar Pradesh 208016, India
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present a detailed spectral and temporal analysis of the black-hole candidate MAXI J1535−571 using NICER
+observations in September and October 2017. We focus specifically on observations in the hard-intermediate state
+when the source shows type-C quasi-periodic oscillations (QPOs). We fitted the time-averaged spectrum of the source
+and the rms and phase-lag spectra of the QPO with a one-component time-dependent Comptonization model. We
+found that the corona contracts from ∼ 104 to ∼ 3 × 103 km as the QPO frequency increases from ∼ 1.8 Hz to
+∼ 9.0 Hz. The fits suggest that the system would consists of two coronas, a small one that dominates the time-
+averaged spectrum and a larger one, possibly the jet, that dominates the rms and lag spectra of the QPO. We found
+a significant break in the relation of the spectral parameters of the source and the properties of the QPO, including
+its lag spectra, with QPO frequency. The change in the relations happens when the QPO frequency crosses a critical
+frequency νc ≈ 3.0 Hz. Interestingly, the QPO reaches this critical frequency simultaneously as the radio emission
+from the jet in this source is quenched.
+Key words: accretion, accretion discs — black hole physics — X-rays: binaries — X-rays: individual: MAXI J1535−571
+1 INTRODUCTION
+In the outburst, the transient black-hole X-ray binary
+(BHXB) system shows substantial X-ray variability (Belloni
+& Stella 2014). These systems spend long periods in qui-
+escence, with sporadic outbursts lasting weeks to months,
+during which the X-ray flux increases by up to three orders
+of magnitude compared to the quiescent phase (Remillard
+& McClintock 2006). During an outburst, transient BHXBs
+initially appear in the low-hard state (LHS) and, as the
+outburst progresses, move to the high-soft state (HSS) via
+the hard-intermediate (HIMS) and soft-intermediate state
+(SIMS) (Belloni et al. 2005, 2011, and references within).
+Finally, before returning to the quiescent state, BHXBs
+transition from the HSS to the LHS. In the LHS, a hard
+component due to Comptonization from an electron plasma
+with temperature 50 − 100 keV appears in the X-ray spec-
+trum as a power law with photon index 1.5–2.0 (Gilfanov
+2010). In contrast, the HSS spectrum is dominated by an
+optically thick thermal component generally modelled with a
+⋆ E-mail: [email protected] (DR)
+multi-temperature disc blackbody, occasionally accompanied
+by a soft power-law-like component with Γ ≥2 (M´endez
+& van der Klis 1997; Done et al. 2007). The evolution of
+the outburst of a BHXB can be best characterised in a
+hardness-intensity diagram (HID), where typically systems
+trace a well-defined path often shaped as a “q” (Fender et al.
+2004, Belloni et al. 2005).
+These systems show complex fast-time variability, which
+is strongly state-dependent. This variability takes the form
+of broadband noise components on top of which, in specific
+states, quasi-periodic oscillations (QPOs) can be observed
+(e.g. Chen et al. 1997; Takizawa et al. 1997; Psaltis et al.
+1999; Nowak 2000; Casella et al. 2004, 2005; Belloni et al.
+2005). The QPOs appear in the power density spectrum
+(PDS; van der Klis & Jansen 1985) as relatively narrow
+peaks. The QPOs have been broadly divided into three
+categories, the mHz QPO with QPO frequency ranging from
+few mHz to Hz (e.g., Dewangan et al. 2006, Koljonen et al.
+2011, Altamirano & Strohmayer 2012, Pasham et al. 2013),
+low-frequency QPOs (LFQPOs) with frequencies ranging
+from just below 1 Hz up to 20 Hz (e.g., Motta et al. 2015),
+© 0000 The Authors
+arXiv:2301.04418v1  [astro-ph.HE]  11 Jan 2023
+
+2
+Rawat et. al.
+and
+high-frequency
+QPOs
+(HFQPOs)
+with
+frequencies
+above 100 Hz and up to ∼500 Hz (e.g., Miller et al. 2001,
+Strohmayer 2001, Belloni et al. 2012, M´endez et al. 2013,
+Belloni & Stella 2014). LFQPOs appear in different spectral
+states and have been further classified as type A, B, and C
+(Wijnands et al. 1999, Homan et al. 2001, Remillard et al.
+2002, Casella et al. 2004). Among the three types, type-C
+is the one that is most often observed, showing a high rms
+amplitude, between 1% and 20%, and a quality factor1
+usually larger than 6.0 (Wijnands et al. 1999; Casella et al.
+2004; Belloni & Stella 2014, see Ingram & Motta 2019, for a
+review).
+MAXI J1535−571 (hereafter MAXI J1535) is a galactic
+transient, initially detected by MAXI/GSC (Negoro et al.
+2017a) and SWIFT/BAT (Kennea et al. 2017, Markwardt
+et al. 2017) on September 2, 2017. The X-ray variability
+(Negoro et al. 2017b), optical (Scaringi & ASTR211 Stu-
+dents 2017) and near-infrared (Din¸cer 2017) properties of the
+source suggest that MAXI J1535 is a low-mass X-ray binary
+(LMXB) source. Radio observations with the Australia
+Telescope Compact Array (ATCA) show a signature of a
+compact radio jet (Russell et al. 2017); this and the observed
+luminosity suggest that this system harbours a black hole
+(Negoro et al. 2017b). Study of radio (Chauhan et al. 2019)
+and X-ray (Sridhar et al. 2019) observations suggest that the
+distance to the source is 4–6 kpc, and the jet inclination angle
+is constrained to ≤ 45◦ (Russell et al. 2019). X-ray spectral
+studies suggest that the system harbours a near-maximally
+spinning black hole (Gendreau et al. 2017, Xu et al. 2018,
+Miller et al. 2018). There are some conflicting estimates of
+the mass of the black hole in the system (Sreehari et al.
+2019, Sridhar et al. 2019), but they are all based on fits to
+the X-ray spectrum and are therefore model dependent. No
+dynamical mass measurement from optical observations is
+available.
+A state transition study of MAXI J1535 during outburst,
+from September 2017 to April 2018 (Nakahira et al. 2018)
+shows that the source behaved like other BHXB systems
+tracing a q-shape in the HID (Tao et al. 2018). In the LHS
+and HIMS, starting from September 9-18, 2017, MAXI J1535
+showed a type-C QPO with a centroid frequency in the
+0.2-3.4 Hz range (Gendreau et al. 2017, Mereminskiy et al.
+2018, Stiele & Kong 2018, Huang et al. 2018, Bhargava et al.
+2019). The source transitioned to the SIMS and then to
+the HSS from September 19-26, 2017. The stable and weak
+type A/B LFQPO appears in the SIMS (Stiele & Kong
+2018, Stevens et al. 2018, Huang et al. 2018). In the HIMS
+and LHS, the type-C QPO reappears from September 26
+to October 9, 2017. After the end of the main outburst
+in mid-May 2018, five re-brightening events were reported
+by Parikh et al. (2019). A state transition during these
+re-flares was reported by C´uneo et al. (2020) using NICER
+observations.
+Kumar & Misra (2014) proposed a model to study the
+Comptonisation medium of neutron-star X-ray binary sys-
+1 Quality factor=QPO frequency/QPO width
+tems, which was later extended by Karpouzas et al. (2020).
+This model was originally developed for high-frequency
+QPOs in accreting neutron-star systems. Still, it has been
+recently extended by Bellavita et al. (2022) to LFQPOs
+in BHXBs and was applied to the type-C QPO in GRS
+1915+105 by Karpouzas et al. (2021) and M´endez et al.
+(2022), and the type-B QPO in MAXI J1348−630 (Garc´ıa
+et al. 2021; Bellavita et al. 2022). Zhang et al. (2022) has
+applied the same model using Insight-HXMT observations
+of the type-C QPO in MAXI J1535 up to 150 keV. The
+rationale behind applying this model to type-C in BHXB
+is that the fractional rms amplitude of these QPOs can be
+as large as ∼ 15% up to ∼200 keV (Ma et al. 2021). At
+those energies, Comptonization dominates the emission in
+these systems (e.g., the disc and the reflection component
+peak at, respectively, ∼1−3 keV and ∼ 20−25 keV and both
+drop quickly above that), and hence Comptonization is most
+likely responsible for the rms amplitude and lags of the QPO.
+In this paper, we report the results of the spectro-temporal
+analysis of MAXI J1535 using NICER observations. To study
+the Comptonization medium of the source, we fit the rms and
+phase-lag spectra of the QPO with a one-component time-
+dependent Comptonization model, vkompthdk (Karpouzas
+et al. 2020; Bellavita et al. 2022). In Section 2, we describe the
+observations and data analysis techniques, and in Section 3
+we present the results of our analysis and the fits of the model
+to the rms and lag spectra of the type-C QPO. Finally, we
+discuss our findings in Section 4 and summarise our results
+in Section 5.
+2 OBSERVATION AND DATA ANALYSIS
+We used observations of MAXI J1535 obtained in September
+and October 2017 with the Neutron Star Interior Composi-
+tion Explorer (NICER Gendreau et al. 2012). The observa-
+tions ID’s used are 1050360101-1050360120 & 1130360101-
+1130360114. NICER’s XTI (X-ray Timing Instrument Gen-
+dreau et al. 2016) covers the 0.2-12.0 keV band and has an
+effective area of >2000 cm2 at 1.5 keV. The energy and time
+resolutions are 85 eV at 1 keV and 4 ×10−8 s (hereafter
+∆tnicer), respectively. We used the nicerl22 task to process
+each observation applying the standard calibration process
+and screening. We used only those intervals for which the
+exposure time was > 100 s after running the nicerl2 task.
+For some intervals, we found that the source flux was vary-
+ing significantly. To make sure we are not averaging features
+of two spectrally and temporally different states, we divided
+a single observation into segments with a more or less con-
+stant source count rate and studied the temporal and spectral
+properties of each segment independently. The details of each
+observation and segment are given in Table 1.
+2.1 Timing analysis
+We extracted the fractional rms amplitude (root-mean
+square) normalised (Belloni & Hasinger 1990) PDS for each
+2 https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_
+threads/nicerl2/
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+3
+Figure 1. Left panel: NICER light curve of MAXI J1535−571 in the 0.5-10.0 keV band. The shaded area represents the approximate time
+when the radio emission was quenched (Russell et al. 2019). Right panel: Hardness intensity diagram (HID) using NICER observations.
+In the HID, the line shows the general movement of the source in this diagram as the outburst progressed, with the start and end points
+of the outburst at, (HR = 0.27, Intensity = 8000) and (HR = 0.22, Intensity = 8000), respectively. In both panels, each point corresponds
+to 100 sec, and the colour scale panels indicate the frequency of the QPO.
+Table 1. Observation log of MAXI J1535, including timing parameters. The columns are the observation number, the NICER ObsID, the
+start and end time of the observation, the 0.5-10.0 keV count rate, the standard deviation of the count rate, σcount, the hardness ratio,
+HR, the standard deviation of the hardness ratio, σHR, the QPO centroid frequency and the QPO fractional rms amplitude. The errors
+are at 1σ. The observations with an asterisk are those for which the QPO was insignificant in the lowest energy bands.
+Obs no.
+ObsID
+Tstart
+Tstop
+count rate
+σcount
+HR
+σHR
+QPO frequency
+QPO Fractional
+(M.J.D)
+(M.J.D)
+(0.5-10.0 keV)
+(5−10keV)
+(0.5−2.0keV)
+(Hz)
+rms (%)
+1
+1050360105
+58008.988
+58009.126
+8140 ± 5
+48
+0.272
+0.002
+2.74 ± 0.01
+7.0 ± 0.2
+2
+1050360105
+58009.165
+58009.193
+7847 ± 4
+36
+0.280
+0.002
+2.44 ± 0.01
+6.5 ± 0.2
+3
+1050360105
+58009.229
+58009.301
+7676 ± 6
+30
+0.285
+0.004
+2.32 ± 0.01
+6.7 ± 0.2
+4
+1050360105
+58009.807
+58009.945
+7327 ± 4
+65
+0.307
+0.003
+1.83 ± 0.01
+7.3 ± 0.2
+5
+1050360106
+58010.001
+58010.525
+7364 ± 1
+138
+0.311
+0.005
+1.81 ± 0.00
+7.2 ± 0.1
+6
+1050360107
+58011.865
+58011.940
+8654 ± 7
+47
+0.299
+0.002
+2.15 ± 0.01
+6.9 ± 0.2
+7
+1050360108
+58012.187
+58012.258
+9134 ± 3
+130
+0.294
+0.006
+2.41 ± 0.01
+7.4 ± 0.2
+8
+1050360108
+58012.316
+58012.583
+9492 ± 2
+320
+0.285
+0.002
+2.77 ± 0.01
+7.3 ± 0.2
+9
+1050360109
+58013.216
+58013.222
+10088 ± 1
+4
+0.285
+0.004
+2.75 ± 0.02
+7.0 ± 0.2
+10
+1050360109
+58013.281
+58013.410
+10922 ± 4
+191
+0.275
+0.008
+3.27 ± 0.02
+7.0 ± 0.3
+11
+1050360109
+58013.481
+58013.740
+11290 ± 2
+227
+0.282
+0.005
+3.19 ± 0.03
+6.7 ± 0.3
+12
+1050360109
+58013.988
+58013.998
+10461 ± 5
+71
+0.288
+0.001
+2.72 ± 0.01
+6.7 ± 0.2
+13
+1050360110
+58014.053
+58014.063
+10744 ± 1
+5
+0.286
+0.002
+2.84 ± 0.01
+7.5 ± 0.2
+14
+1050360110
+58014.824
+58014.835
+13795 ± 1
+5
+0.269
+0.003
+4.75 ± 0.01
+5.7 ± 0.1
+15
+1050360111
+58015.276
+58015.669
+16992 ± 3
+161
+0.257
+0.005
+9.01 ± 0.04
+1.7 ± 0.1
+16
+∗1050360112
+58016.240
+58016.957
+17040 ± 9
+31
+0.256
+0.010
+7.55 ± 0.06
+2.6 ± 0.2
+17
+1050360113
+58017.011
+58017.858
+16995 ± 1
+7
+0.244
+0.017
+7.45 ± 0.03
+2.9 ± 0.1
+18
+∗1130360103
+58026.726
+58026.814
+14304 ± 2
+445
+0.235
+0.002
+7.09 ± 0.03
+2.4 ± 0.1
+19
+1130360104
+58027.755
+58027.779
+12363 ± 3
+105
+0.240
+0.002
+5.42 ± 0.01
+4.7 ± 0.1
+20
+1130360105
+58028.720
+58028.872
+12321 ± 2
+213
+0.237
+0.002
+5.73 ± 0.01
+4.5 ± 0.0
+21
+∗1130360106
+58029.749
+58029.836
+12527 ± 2
+151
+0.229
+0.002
+6.77 ± 0.02
+3.5 ± 0.1
+22
+1130360107
+58030.715
+58030.865
+10831 ± 2
+381
+0.238
+0.004
+4.57 ± 0.01
+4.6 ± 0.1
+23
+1130360108
+58031.361
+58031.894
+11163 ± 2
+370
+0.234
+0.006
+4.82 ± 0.01
+3.5 ± 0.0
+24
+1130360113
+58036.498
+58036.695
+9747 ± 10
+19
+0.206
+0.007
+5.19 ± 0.03
+3.0 ± 0.2
+25
+1130360114
+58037.032
+58037.677
+8767 ± 4
+183
+0.224
+0.004
+4.50 ± 0.01
+5.0 ± 0.1
+segment using the General High-energy Aperiodic Timing
+Software (GHATS)3 version 2.1.0. The 0.2-10.0 keV data
+were re-binned in time by a factor of 62500, such that the
+time resolution was 0.0025 s, corresponding to a Nyquist
+frequency of 200 Hz, and PDS were produced from intervals
+3 http://www.brera.inaf.it/utenti/belloni/GHATS_Package/
+Home.html
+of 8192 points (20.48 s). For each segment, the PDS for
+the intervals were averaged. We fitted the PDS in the
+frequency 100-200 Hz, where the source shows no intrinsic
+variability, with a constant representing the Poisson noise,
+which we then subtracted. We ended up with an averaged,
+Poisson-noise subtracted PDS for each segment that we
+re-binned logarithmically such that each frequency bin is
+larger than the previous one by a factor exp(1/100). We
+MNRAS 000, 1–15 (0000)
+
+• without type-C QPOs
+·with type-C QPOs18000
+9
+•without type-C QPOs
+·with type-C QPOs
+16000
+8
+14000
+7
+[zH]
+Intensity [counts s'
+12000
+6
+5
+10000
+4
+8000
+3
+2
+0.18
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+HR [(5-10 keV)/(0.5-2 keV)4
+Rawat et. al.
+Table 2. Time-averaged spectra and corona model parameters of MAXI J1535. The columns are the observation number, the hydrogen
+column density, NH, the power-law photon index of nthcomp, Γ, the inner disc temperature, kTin, the seed photon temperature of
+vkompthdk, kTs, the size of the corona, L, the fraction of the flux of the seed-photon source due to feedback from the corona, η, and
+the amplitude of the variability of the external heating rate, δ ˙Hext. The errors are at 1σ. The observations with an asterisk are those for
+which the QPO was insignificant in the lowest energy bands.
+Obs no.
+NH
+Γ
+kTin
+kTs
+L
+η
+δ ˙Hext
+χ2
+ν(dof)
+1022 cm−2
+(keV)
+(keV)
+( 103 km)
+%
+1
+2.19 ± 0.01
+2.43 ± 0.02
+0.68 ± 0.01
+0.35 ± 0.05
+5.1 ± 1.0
+0.62 ± 0.05
+12.2 ± 0.6
+231.4 (243)
+2
+2.19 ± 0.01
+2.29 ± 0.01
+0.62 ± 0.01
+0.29 ± 0.03
+8.3 ± 1.1
+0.75 ± 0.09
+12.0 ± 0.5
+191.9 (242)
+3
+2.18 ± 0.01
+2.26 ± 0.01
+0.61 ± 0.01
+0.23 ± 0.04
+8.7 ± 1.1
+0.82+0.18
+−0.38
+11.3 ± 1.1
+240.5 (243)
+4
+2.17 ± 0.01
+2.12 ± 0.01
+0.55 ± 0.01
+0.14 ± 0.01
+12.6 ± 0.5
+1.00 − 0.04
+11.1 ± 0.4
+219.8 (243)
+5
+2.16 ± 0.01
+2.11 ± 0.00
+0.55 ± 0.01
+0.15 ± 0.01
+13.2 ± 0.4
+1.00 − 0.45
+11.5 ± 0.3
+242.3 (243)
+6
+2.15 ± 0.01
+2.18 ± 0.01
+0.60 ± 0.01
+0.24 ± 0.03
+9.1 ± 1.0
+0.79 ± 0.12
+12.2 ± 0.7
+177.9 (243)
+7
+2.17 ± 0.01
+2.27 ± 0.01
+0.64 ± 0.01
+0.36 ± 0.05
+6.6 ± 1.2
+0.64 ± 0.07
+15.0 ± 0.7
+173.2 (243)
+8
+2.15 ± 0.01
+2.67 ± 0.04
+0.79 ± 0.01
+0.33 ± 0.04
+5.7 ± 0.9
+0.76 ± 0.07
+11.2 ± 0.7
+234.8 (243)
+9
+2.19 ± 0.01
+2.34 ± 0.02
+0.68 ± 0.01
+0.47 ± 0.07
+4.8 ± 0.9
+0.59 ± 0.05
+14.4 ± 0.9
+169.3 (243)
+10
+2.21 ± 0.01
+2.48 ± 0.02
+0.74 ± 0.01
+0.39 ± 0.07
+4.4 ± 1.2
+0.55 ± 0.07
+13.5 ± 1.0
+155.1 (243)
+11
+2.19 ± 0.01
+2.85 ± 0.11
+0.85 ± 0.02
+0.36 ± 0.05
+5.5 ± 1.2
+0.77 ± 0.11
+10.5 ± 0.9
+192.2 (222)
+12
+2.20 ± 0.01
+2.33 ± 0.01
+0.67 ± 0.01
+0.37 ± 0.04
+6.5 ± 0.8
+0.66 ± 0.05
+13.3 ± 0.5
+152.4 (243)
+13
+2.20 ± 0.01
+2.36 ± 0.01
+0.69 ± 0.01
+0.37 ± 0.04
+6.4 ± 0.8
+0.69 ± 0.06
+14.3 ± 0.5
+176.3 (243)
+14
+2.23 ± 0.01
+2.61 ± 0.06
+0.98 ± 0.02
+0.43 ± 0.05
+3.8 ± 0.5
+0.73 ± 0.06
+14.1 ± 0.8
+168.4 (242)
+15
+2.30 ± 0.01
+2.49 ± 0.17
+1.18 ± 0.01
+0.56 ± 0.04
+4.0 ± 0.5
+1.00 − 0.11
+17.3 ± 2.0
+195.8 (239)
+16
+2.29 ± 0.01
+2.60 ± 0.14
+1.13 ± 0.02
+0.52 ± 0.07
+3.7 ± 1.0
+0.69 ± 0.18
+15.6 ± 2.4
+165.0 (236)
+17
+2.29 ± 0.00
+2.40 ± 0.24
+1.19 ± 0.01
+0.39 ± 0.04
+3.4 ± 0.2
+0.88 ± 0.04
+20.6 ± 1.2
+177.0 (242)
+18
+2.27 ± 0.01
+2.71 ± 0.10
+1.05 ± 0.02
+0.55 ± 0.04
+3.0 ± 0.3
+0.60 ± 0.06
+11.7 ± 0.8
+244.4 (229)
+19
+2.24 ± 0.00
+2.61 ± 0.08
+0.95 ± 0.02
+0.46 ± 0.04
+3.8 ± 0.5
+0.65 ± 0.05
+14.3 ± 1.1
+203.6 (242)
+20
+2.30 ± 0.01
+3.03 ± 0.02
+0.85 ± 0.01
+0.63 ± 0.02
+4.0 ± 0.2
+0.57 ± 0.03
+14.1 ± 0.4
+204.5 (240)
+21
+2.31 ± 0.01
+3.38 ± 0.05
+0.92 ± 0.01
+0.76 ± 0.04
+2.7 ± 0.2
+0.44 ± 0.03
+13.5 ± 0.8
+198.2 (215)
+22
+2.31 ± 0.02
+2.66 ± 0.03
+0.71 ± 0.02
+0.57 ± 0.03
+4.8 ± 0.4
+0.51 ± 0.04
+13.3 ± 0.6
+258.0 (219)
+23
+2.28 ± 0.01
+3.07 ± 0.04
+0.85 ± 0.01
+0.55 ± 0.03
+4.5 ± 0.4
+0.66 ± 0.05
+9.1 ± 0.5
+223.1 (238)
+24
+2.21 ± 0.00
+2.69 ± 0.08
+0.96 ± 0.02
+0.40 ± 0.05
+6.2 ± 1.5
+1.00 − 0.33
+12.8 ± 1.9
+191.4 (241)
+25
+2.23 ± 0.00
+2.58 ± 0.03
+0.82 ± 0.02
+0.43 ± 0.03
+5.5 ± 0.7
+0.67 ± 0.08
+15.9 ± 0.5
+174.9 (242)
+fitted all the PDS with a model consisting of up to five
+Lorentzians to represent the broadband noise component
+and the QPOs. Each Lorentzian has three parameters: the
+centroid frequency, ν0, the full-width at half-maximum,
+FWHM, and the total power, equal to the integral of the
+Lorentzian function over the full frequency range. We only
+included a Lorentzian in the model if its total power was at
+least 3σ different from zero, given the error of this parameter.
+We visually inspected the PDS from all segments and used
+only those with a clear type-C QPO.
+Next, we extracted PDS in 10 energy bands, 1.0–1.5, 1.5–
+1.9, 1.9–2.3, 2.3–3.0, 3.0–3.5, 3.5–4.0, 4.0–5.0, 5.0–6.0, 6.0–
+8.0, and 8.0–12.0 keV that we normalised to fractional rms for
+each band. To extract phase/time lags, we computed FFTs
+from the data in the ten energy bands and measured the lags
+using the phases of the cross-spectra with the 2.0–3.0 keV
+band as a reference, following the procedure of Nowak et al.
+(1999b). To calculate the lags of the QPO, we averaged the
+cross spectra within one full-width half-maximum around the
+centroid frequency of the QPO for each segment in which we
+detected a significant QPO. For 4 segments, marked with an
+asterisk in Table 1, the QPO was insignificant in the lowest
+energy bands. We merged some low-energy bands in those
+cases and extracted the rms and lag spectra for 7 energy
+bands (1.0–2.3, 2.3–3.5, 3.5–4.0, 4.0–5.0, 5.0–6.0, 6.0–8.0, and
+8.0–12.0 keV).
+2.2 Spectral analysis
+We produced the spectra and background files using the
+NICER background estimator tool 3C 50 RGv54. The
+background-subtracted
+spectrum
+for
+each
+segment
+was
+re-binned using grppha such that each spectral bin had
+at least 30 counts and the bins over-sampled the spectral
+resolution of the detector by a factor 3. We used Heasoft
+version 6.30 and CALDB version 20210707 to create the re-
+sponse (rmf) and ancillary response (arf) files. We fitted the
+time-averaged spectrum of the source in the 1.0 − 10.0 keV
+band using the model tbabs*(diskbb+gauss+nthcomp)
+in xspec. The Tbabs models the interstellar absorption.
+We used the cross-section tables of Verner et al. (1996)
+and the abundances of Wilms et al. (2000) and left the
+hydrogen column density as a free parameter. The diskbb
+component models the thermal emission from an optically
+thick and geometrically thin accretion disc (Mitsuda et al.
+1984, Makishima et al. 1986) while nthcomp (Zdziarski
+et al. 1996, ˙Zycki et al. 1999) models the Comptonised
+emission from the X-ray corona. We kept both the diskbb
+parameters, the temperature at inner disk radius, kTin, and
+the normalisation free. The nthcomp model parameters
+are the power-law photon index, Γ, electron temperature,
+kTe, seed photon temperature, kTbb, and normalization.
+4 https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_
+bkg_est_tools.html
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+5
+The seed-photon temperature kTbb was tied to kTin of
+the diskbb component. We have fitted a relatively broad
+iron line present in the residuals with a Gaussian, gauss
+in xspec. In addition to the broad line, the spectra show
+narrow residuals at ∼6.4 keV. We have added one more
+gauss component to account for the narrow line (if required).
+We fit the rms with the model vkompthdk*dilution5
+(Karpouzas et al. 2020; Bellavita et al. 2022) and the lag
+spectra with the model vkompthdk at the QPO frequency.
+vkompthdk can compute both the time-dependent and
+the time-averaged spectrum. The time-dependent version of
+vkompthdk is the one that fits the rms and lags. The time-
+averaged version of vkompthdk is the same as nthcomp.
+The parameters of vkompthdk are hence the temperature of
+the seed photon source, kTs, the temperature of the corona,
+kTe, the power-law index, Γ (all of them identical to kTbb,
+kTe and Γ of nthcomp), plus the size of the corona, L, the
+feedback fraction, η (between 0 to 1), the amplitude of the
+variability of the external heating rate, δ ˙Hext, and the lag of
+the model in the 2–3 keV energy band, reflag. These param-
+eters can be used to compute the fraction of the corona flux,
+ηint, that returns to the disc (see Karpouzas et al. 2020 for
+details). The parameters L, η, δ ˙Hext, and reflag are only rel-
+evant for the fits to the rms and lag spectra and do not affect
+the time-averaged version of the vkompthdk. The parame-
+ter reflag is an additive normalisation that allows the model
+to match the data, given that the observer is free to choose
+the reference energy band of the lags. We froze the electron
+temperature of nthcomp and vkompthdk at kTe = 21 keV
+(Sridhar et al. 2019) because the 1.0-10.0 keV energy band
+is not suitable to constrain it. The dilution component is
+a function of energy (E). It accounts for the fact that the
+rms amplitude we observe is diluted by the emission of the
+other components that we assume do not vary. The dilution
+component is therefore defined as;
+dilution(E) =
+nthcomp(E)
+diskbb(E) + gauss(E) + nthcomp(E)
+(See details in Bellavita et al. (2022).) Because NH towards
+the source is high, any emission below 1 keV could be at-
+tributed to calibration artefacts; therefore, we have decided
+to exclude data below 1.0 keV in our fits. Using HXMT data
+in the 2–100 keV range, Zhang et al. (2022) reported a hydro-
+gen column density, NH=5.6*1022 cm−2, that is higher than
+the value we have obtained here using NICER in the 1–10
+keV range.
+3 RESULTS
+The left panel of Figure 1 shows the NICER light curve
+of MAXI J1535 during its 2017 outburst. While the right
+panel of Figure 1 shows the evolution of the source in the
+HID. Here intensity is defined as the source count rate in
+the 0.5–10.0 keV band, and hardness ratio (HR) is the ratio
+of the source intensity in the 5.0–10.0 keV and 0.5–2.0 keV
+bands. The colour scale shown at the right of both Figures
+represents the QPO frequency range 1.8–9.0 Hz, with red
+5 https://github.com/candebellavita/vkompth
+being the lowest and navy blue being the highest end of the
+QPO frequency range. The source’s X-ray count rate and
+HR and their respective standard deviation values for each
+segment are given in Table 1.
+3.1 Spectral fits
+From the fits to the time-averaged spectrum, the rms and
+phase-lag spectra of the QPO for each segment, we find that
+during the first two days of our observations, the inner disc
+temperature, kTin, and the photon index, Γ, of the Comp-
+tonised component first drop (Figure 2) as the source moves
+to the right in the HID (Figure 1 right panel), from hardness
+ratio ∼ 0.27 to hardness ratio ∼ 0.31. Between MJD 58010
+and MJD 58012, the source intensity increases, and the spec-
+trum softens again. The source starts to move up and to the
+left in the HID, and kTin and Γ increase very quickly for
+about five days. At the end of this period, the source reaches
+the highest intensity in our observations. The accretion disc
+is the hottest, kTin ≈ 1.1 − 1.2 keV, and the Comptonised
+component is described with Γ ≈ 2.7 − 2.8. At this point, the
+source enters the HSS and the PDS show no QPOs. When
+the source transitions back to the SIMS and the HIMS, at
+around MJD 58025, kTin and Γ are approximately correlated
+with the X-ray flux (see Figures 1 and 2). We give each seg-
+ment’s spectral parameters and goodness of fit in Table 2. In
+a few segments the reduced χ2 is less than 1 (last column of
+Table 2). The low χ2 values come from the fit to the steady-
+state spectra (SSS). We provide the χ2 and the number of
+channels for the fits to the individual spectra and the total
+χ2 and the number of degree of freedom in Table A.1. Un-
+less otherwise specified, the errors represent the 1σ confidence
+(68%) interval for the corresponding parameter.
+3.2 Power Density Spectra
+Following Belloni et al. (2002), we fit the PDS with a
+0-centred Lorentzian to represent the broadband noise
+component and three separate Lorentzians to fit the narrow
+QPO, its harmonic component, and the high-frequency
+noise. The features in the PDS have a frequency in the
+ratio of 1:2, and we, therefore, identify the strongest peak
+as the fundamental and the other as the second harmonic.
+The PDS also shows a low-frequency noise component when
+the strongest QPO peak was at a frequency above 4.0 Hz
+(Figure 3). Therefore, we used an additional Lorentzian to
+fit the low-frequency noise component whenever required.
+We have studied the QPO fractional rms amplitude in the
+0.5–10.0 keV energy band as a function of QPO frequency
+(left panel of Figure 4) and confirmed that the QPO we
+have identified as fundamental followed a similar relation to
+the one found for GRS 1915+105 (Zhang et al. 2020). The
+type-C QPO appears in the LHS and HIMS as a narrow
+peak with high rms amplitude in the PDS. The properties
+of the observed broadband noise and the QPO justify the
+identification of the QPO as type-C (Casella et al. 2004).
+We fitted the PDS for three different energy bands (0.5–2.0
+KeV, 2.0–4.0 keV, 4.0–10.0 keV) when the type-C QPO was
+at 1.8 Hz, 4.5 Hz, and 7.0 Hz. We show the fitted PDS and
+their respective frequency lag spectra in Figure 3. The lag
+MNRAS 000, 1–15 (0000)
+
+6
+Rawat et. al.
+Figure 2. The evolution of Γ of the corona (left panel) and kTin of the disc (right panel) of MAXI J1535−571. The values of Γ and kTin
+are obtained from the fits to the time-averaged spectra, the rms and phase-lag spectra of the QPO.
+and rms values at the QPO frequency are given in Appendix
+Table A.2. When the QPO frequency is higher than 7.0
+Hz, the QPO fractional rms amplitude decreases, and the
+harmonic component becomes insignificant.
+The evolution of the QPO centroid frequency is shown in
+the right panel of Figure 4. The QPO frequency first decreases
+from 2.7 to 1.8 Hz and then increases to its maximum value of
+9.0 Hz. After that, the QPO frequency varies in the 4.5 − 7.5
+Hz range. The QPO frequency and fractional rms amplitude
+in the 0.5 − 10.0 keV band for each observation are given in
+Table 1. We have plotted Γ and kTin as a function of QPO
+frequency as shown in Figure 5. We found that both Γ and
+kTin increase with QPO frequency.
+To extract the rms spectrum, we fit the PDS in 10 energy
+bands, fixing the QPO centroid frequency and FWHM to the
+best-fitting values in the 2.0–10.0 keV PDS. The rms and
+phase lag spectra when the QPO frequency was 1.8 Hz, 4.5
+Hz, and 7.0 Hz are shown in the top and bottom panels of Ap-
+pendix Figure A1. While the fractional rms amplitude of the
+QPO increases with photon energy for all QPO frequencies,
+the rms spectrum steepens as the QPO frequency increases
+from 1.8 Hz to 7.0 Hz (see upper panels in Appendix Fig-
+ure A1). The change of the slope of the rms spectrum of the
+QPO is driven by a factor ∼ 3 drop of the rms amplitude at
+the lowest energies when the QPO is at low frequencies. In
+contrast, the rms amplitude at the highest energies remains
+more or less constant as the QPO frequency changes by a
+factor of ∼ 4. Although, in general, the low-energy photons
+at the QPO frequency lag behind the high-energy photons for
+all QPO frequencies, the lag spectrum of the QPO changes
+with QPO frequency. When the QPO frequency is between
+1.8 Hz and 2.4 Hz, the lag spectrum shows a minimum at ∼ 4
+keV, with the photons at low and high energies lagging the
+4–5 keV photons by 0.1 − 0.3 rad. As the QPO frequency in-
+creases, the minimum of the lag spectrum of the QPO moves
+to higher energies, with the minimum reaching ∼ 9 − 10 keV
+at the highest QPO frequency, and the low-energy photons
+lag the high-energy ones by up to ∼ 0.8 rad. The rms and
+phase-lag spectra of the QPO in MAXI J1535 in these obser-
+vations with NICER are consistent with the pattern observed
+for the type-C QPO by Rawat et al. (2019) in GRS 1915+105
+and Garg et al. (2022) in MAXI J1535 with AstroSat, over
+the common energy range of both instruments.
+3.3 One component time-dependent Comptonization model
+To understand the changes observed in the rms and lag spec-
+tra of the QPO (see Section 3.2), we fitted the rms and
+lag spectra of the QPO at each QPO frequency with the
+vkompthdk model. During the fits we linked kTe and Γ of
+nthcomp to kTe and Γ of vkompthdk. We first linked kTs
+of vkompthdk to kTin of diskbb, and we found large resid-
+uals in the fits of the phase-lag spectra (Figure 6) because
+vkompthdk fails to reproduce the minimum of the lags. We
+subsequently let kTin and kTs vary independently, and the
+fits improve significantly (Figure 6). The simultaneous fitted
+time-averaged spectra, rms spectra and lag spectra when the
+QPO frequency was ∼1.8 Hz and the residuals of the best-
+fitting model are shown in Figure 7 (The peak in the residuals
+of the time-averaged spectra at 1.84 keV corresponds to the
+absorption edge features of silicon.). We show a similar plot
+for the QPO frequencies 4.5 Hz and 7.0 Hz (for which we
+show a PDS in Figure 3) in the Appendix Figures A2 and
+A3. We discuss the implication of letting kTin and kTs free
+in Section 4.3. The best-fitting parameters and χ2 of the fits
+are given in Table 2.
+We plotted the model parameters as a function of QPO fre-
+quency in Figure 8. The size of the corona decreases from
+∼ 104 km (which corresponds to 670 Rg for a 10 M⊙ black
+hole ) to ∼ 3×103 km (201 Rg) while the temperature of the
+seed photon source, kTs, increases from ∼ 0.1 keV to ∼ 0.4
+keV as the QPO frequency increases from 1.8 Hz to ∼3.0 Hz.
+At QPO frequencies ≥3.0 Hz, the size of the corona and the
+temperature of the seed photon source remain more or less
+constant at respectively ∼ 3 − 6 × 103 km and 0.5 keV. The
+error bars on η are large, and it is hard to follow any trend if
+present, although η appears to decrease from ∼0.8 to ∼0.6 as
+the QPO frequency increases as shown in Appendix Figure
+A4. The best-fitting values of η imply that ηint is in the range
+of 10−25%. Comparing the trends in Figures 5 and 8, it is
+apparent that there is a sudden change of the properties of
+the source when the QPO frequency is below and above ∼ 3.0
+Hz. The change of behaviour of all the quantities appears to
+occur at the same QPO frequency, which we call critical fre-
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+7
+0.0010
+0.0100
+P(f)*f
+QPO= 1.824 ±  0.004 Hz
+0.5-2.0 keV
+0.0010
+0.0100
+QPO= 1.820 ±  0.004 Hz
+2.0-4.0 keV
+0.0010
+0.0100
+0.1000
+QPO= 1.824 ±  0.004 Hz
+4.0-10.0 keV
+0.1
+1.0
+10.0
+frequency (Hz)
+− 0.2
+0.0
+0.2
+Phase lag (rad)
+0.1
+1.0
+10.0
+frequenc (Hz)
+− 0.2
+0.0
+0.2
+0.1
+1.0
+10.0
+frequenc (Hz)
+− 0.2
+0.0
+0.2
+0.0010
+0.0100
+P(f)*f
+QPO= 4.43 ±  0.03 Hz
+0.5-2.0 keV
+0.0010
+0.0100
+QPO= 4.48 ±  0.02 Hz
+2.0-4.0 keV
+0.0010
+0.0100
+QPO= 4.51 ±  0.02 Hz
+4.0-10.0 keV
+0.1
+1.0
+10.0
+frequency (Hz)
+− 0.25
+0.00
+0.25
+Phase lag (rad)
+0.1
+1.0
+10.0
+freq ency (Hz)
+− 0.25
+0.00
+0.25
+0.1
+1.0
+10.0
+freq ency (Hz)
+− 0.25
+0.00
+0.25
+0.0000
+0.0001
+0.0010
+0.0100
+P(f)*f
+QPO= 6.9 ±  0.1 Hz
+0.5-2.0 keV
+0.0000
+0.0001
+0.0010
+0.0100
+QPO= 7.06 ±  0.04 Hz
+2.0-4.0 keV
+0.0000
+0.0001
+0.0010
+0.0100
+QPO= 7.13 ±  0.03 Hz
+4.0-10.0 keV
+0.1
+1.0
+10.0
+frequency (Hz)
+− 0.25
+0.00
+0.25
+Phase lag ( ad)
+0.1
+1.0
+10.0
+f equency (Hz)
+− 0.25
+0.00
+0.25
+0.1
+1.0
+10.0
+f equency (Hz)
+− 0.25
+0.00
+0.25
+Figure 3. The top panels show the power density spectra (power multiplied by frequency) of MAXI J1535−571 for three QPO frequencies,
+1.8 Hz, 4.5 Hz, and 7.0 Hz, and three different energy bands. The PDS is fitted with three to five Lorentzians. The bottom panels show
+the frequency phase-lag spectra. The reference energy band is 0.5-10.0 keV here. The vertical dashed lines indicate the ranges over which
+the QPO fundamental lags we measured (ν ± FWHM/2).
+MNRAS 000, 1–15 (0000)
+
+8
+Rawat et. al.
+2
+3
+4
+5
+6
+7
+8
+9
+QPO frequency (Hz)
+2
+3
+4
+5
+6
+7
+QPO fractional rm  (0.5-10.0 keV)
+10
+15
+20
+25
+30
+35
+time (days since MJD=58000)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+QPO frequency (Hz)
+Figure 4. Left panel: QPO fractional rms amplitude in the 0.5–10.0 keV energy band as a function of QPO frequency for MAXI J1535−571.
+Right panel: Evolution of the QPO frequency of MAXI 1535-571. The shaded area represents the radio jet quenching interval (Russell
+et al. 2019).
+2
+3
+4
+5
+6
+7
+8
+9
+QPO frequency (Hz)
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.4
+3.6
+Γ
+2
+3
+4
+5
+6
+7
+8
+9
+QPO frequency (Hz)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+kTin (keV)
+Figure 5. The dependence of Γ (left panel) and kTin (right panel) upon QPO frequency in MAXI J1535−571. The values of Γ and kTin
+are obtained from the fits to the time-averaged spectra, the rms and phase-lag spectra of the QPO.
+quency, νc.
+To estimate the critical frequency, we assume that the break
+in the relation of the disc and corona model parameters, and
+time lags as a function of QPO frequency, happens at the
+same QPO frequency, i.e., νc. In Figure 8 we show fits with a
+power-law (red) and broken power-law (blue) to the relation
+of L, kTs, time lag, kTin with QPO frequency. The parame-
+ters of the broken power law are the power-law indices α1 and
+α2 below and above the break frequency νc and a normalisa-
+tion parameter. We have calculated the F-test probability for
+the fits with a power law and a broken power-law and found
+that the probability ranges from (0.2−1)×10−4, which indi-
+cates that a broken power-law in general fits the data better
+than a power law. (To account for the dispersion of the data
+points around the model was larger than the statistical errors,
+we have added a systematic of 6%.) The break for each indi-
+vidual fit is in the range 2.7–2.8 Hz, and the break appears
+to be at the same QPO frequency in all cases. Since there is
+a hint of a break in the relationship of the time lags and kTin
+with QPO frequency, we fitted all the four relations (L, kTs,
+time lag, kTin) together with a broken power law model as
+shown in Figure 8, with the critical frequency tied. We got
+Table 3. Broken power-law best-fitting parameters to the relations
+of L, kTs, time lags of the QPO and kTin vs. QPO frequency
+shown in Figure 8. The parameters α1 and α2 are the power-law
+indices for νQP O ≤ νc and νQP O > νc, respectively.
+Parameter
+α1
+α2
+bknpower norm
+L (km)
+1.8 ± 0.4
+0.5 ± 0.2
+(3.8 ± 1.3) × 104
+kTs (keV)
+-2.2 ± 0.5
+-0.3 ± 0.2
+0.04 ± 0.01
+kTin (keV)
+-0.6 ± 0.2
+-0.4 ± 0.1
+0.7 ± 0.1
+time lag (m sec)
+0.6 ± 0.4
+1.2 ± 0.2
+0.007 ± 0.002
+Note: The best-fitting parameters values shown above are for the
+joint fits of all the parameter vs. QPO frequency plot with νc tied.
+νc = 3.0±0.4 Hz. If we let νc vary separately for each fit, the
+χ2 changes from 141.84 (dof=88) to 133.38 (dof=85) with an
+F-test probability of ∼ 0.15. This confirms that the best fit
+does not improve significantly if we let νc free. We conclude
+that the break is consistent with being at the same frequency
+in all relations plotted in Figures 5 and 8. The details of the
+best-fitting parameters are given in Table 3.
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+9
+0.0
+0.1
+0.2
+0.3
+ Phase lag (rad)
+kTin  and kTs free
+kTs=kTin
+1.0
+10.0
+Energy (keV)
+−5
+0
+5
+(data-model)/error
+Figure 6. The phase-lag spectra of the QPO of MAXI J1535−571
+fitted with the vkompthdk model keeping kTin and kTs tied to
+each other (red), and free (black). The bottom panel shows the
+respective residuals of the fits. The data corresponds to obs ID
+1050360105 with QPO frequency∼1.8 Hz
+4 DISCUSSION
+We have analysed NICER observations of MAXI J1535−571
+during the initial phase of the outburst in September and
+October 2017. The rms and lag spectrum of the type-C
+QPO, the spectral parameters deduced from fits to the
+time-averaged energy spectra of the source (the temperature
+of the accretion disc, kTin), and the parameters from fits
+to the rms and lag spectra of the QPO (the size of the
+corona, L, the temperature of the source that provides the
+seed photons that inverse-Compton scatter in the corona,
+kTs, all change in a similar manner as the frequency of the
+type-C QPO increases from 1.8 Hz to 9 Hz. While some of
+these quantities increase (kTin, kTs, phase lags) and others
+decrease (rms amplitude of the QPO, L ) with increasing
+QPO frequency, we find that all these quantities show a sig-
+nificant break in the relation at a QPO frequency νc ∼ 3.0 Hz.
+At low QPO frequencies, the lag spectrum of the type-C
+QPO in MAXI J1535 increases at low and high energies
+and is minimum at ∼ 4 keV. This is similar to what is
+observed for the type-B QPO in the black hole candidate
+MAXI J1348−630 (Belloni et al. 2020, Garc´ıa et al. 2021). In
+the case of MAXI J1348−630, Belloni et al. (2020) proposed
+that the fact that photons at energies below ∼ 3 keV lag
+behind photons at ∼ 3 keV is due to down scattering of
+the photons emitted by the disc in the corona, that they
+assume is the jet. To reach these conclusions, instead of a
+black body-like seed spectrum, Belloni et al. (2020) assumed
+a simplified seed-source spectrum that is flat between 2 and
+3 keV and does not emit at other energies. Such a spectrum,
+however, neglects the dilution of the lags caused by black
+body photons emitted below 2 keV that escape without
+being up-/down-scattered in the corona. If one considers
+a more realistic (a black body or a disc) seed spectrum of
+equivalent temperature, the lags turn out to be flat below
+∼ 2 − 3 keV, different from what is observed (Kylafis et al.
+2021). On the other hand, using the model of Karpouzas
+et al. (2020), Garc´ıa et al. (2021) showed that the shape
+of the lag spectrum (and the rms spectrum as well) of
+MAXI J1348−630 can be explained by corona photons that
+impinge back onto the accretion disc and emerge later and at
+energies below those of the photons that were up-scattered in
+the corona. This feedback loop between the corona and the
+disc is the reason for the positive lags between the photons
+with energies below ∼ 2 − 3 keV and those with energies of
+∼ 2 − 3 keV. At the same time, inverse Compton scattering
+in the corona explains that photons with energies above
+∼ 2 − 3 keV lag behind the 2 − 3 keV photons. Our fits to
+the rms and lag spectra of the QPO in MAXI J1535 here
+show the same.
+4.1 Connection of critical frequency with radio jet quenching
+Using AstroSat, and swift observation of the period MJD
+58008 − 58013 and 58004 − 58017, Mereminskiy et al. (2018)
+and Bhargava et al. (2019) found a tight correlation between
+the QPO frequency and the power-law index that models
+the hard component in the energy spectrum. Using nicer
+observation of the period MJD 58008.99 − 58037.68, we, on
+the other hand, found a significant break in the spectral and
+corona parameters as a function of QPO frequency. The rms
+and lag spectra of the QPO below and above νc are also
+significantly different. The break in the relation between the
+QPO lags and QPO frequency at νc ∼3.0 Hz in MAXI J1535
+is similar to the break found by Zhang et al. (2020) in GRS
+1915+105 when the QPO frequency is ∼2 Hz, and to the
+one in GX 339-4 (Zhang et al. 2017) at a QPO frequency of
+∼1.7 Hz.
+Interestingly, the frequency of the QPO in MAXI J1535
+crosses the value of 3.0 Hz on September 17 2017 (MJD
+58013; see Figure 4 and Table 1). This date coincides
+with the time at which the radio emission from the jet
+in this source is quenched (Russell et al. 2019), which we
+marked by the shaded area in Figure 4. Indeed, the radio
+emission of the jet in MAXI J1535 quenches in the period
+MJD 58013.60 − 58014.18; after that, in the period MJD
+58014.18 − 58015.37 (Table 1 Russell et al. 2019) the source
+makes a transition from the hard intermediate to the soft
+intermediate state. A similar behaviour has been observed
+by M´endez et al. (2022) for GRS 1915+105, i.e., a low radio
+emission at or above a QPO frequency of ∼2.0 Hz, and
+increased radio emission below that QPO frequency, the
+QPO frequency at which Zhang et al. (2020) found that the
+lags of the QPO change from soft to hard.
+4.2 Size of the corona
+From fits to the rms and lag spectra of the QPO with
+the vkompthdk, here we find that the size of the corona
+decreases very rapidly from ∼ 104 km to ∼ 4000 − 5000 km
+MNRAS 000, 1–15 (0000)
+
+10
+Rawat et. al.
+1
+10
+100
+counts s−1 keV−1
+0.1
+0.05
+Fractional rms
+1
+10
+2
+5
+−0.2
+0
+0.2
+Phase lags (rad)
+Energy (keV)
+−2
+0
+2
+(data−model)/error
+−2
+−1
+0
+1
+2
+(data−model)/error
+1
+10
+2
+5
+−1
+0
+1
+(data−model)/error
+Energy (keV)
+Figure 7. Fits of the vkompthdk model to the data of MAXI J1535—571. From top to bottom, the left panel shows the time-averaged
+spectrum of the source fitted with the model tbabs*(diskbb+gauss+nthcomp), the rms spectrum of the QPO fitted with the model
+vkompthdk*dilution, and the phase-lag spectrum of the QPO fitted with the model vkompthdk when the QPO frequency was at ∼1.8
+Hz. The right panels show the respective residuals of the best-fitting model to the data. The 2.0–3.0 keV band is the reference band for
+the phase lag spectra.
+when the QPO frequency increases from ∼ 1.8 Hz to ∼ 3.2
+Hz; from that point on the corona size remains more or less
+constant or decreases slightly from ∼ 4000 − 5000 km down
+to ∼ 3000 km as the QPO frequency increases from ∼ 3.2 Hz
+up to ∼ 9 Hz. Figure 4 shows that the QPO frequency does
+not increase monotonically during these observations. In
+contrast, from Figures 4 and 8, it is apparent that the size of
+the corona first increases from ∼ 2000 km to ∼ 104 km, and
+it then decreases back to ∼ 3000 km (first 10 points in the
+right panel of Figure 4). At this time, coincident with the
+time that the radio emission from the jet is quenched (Russell
+et al. 2019), the size of the corona continues decreasing but
+at a lower rate than before. Assuming that MAXI J1535
+harbours a 10-solar mass black hole, the maximum and mini-
+mum size of the corona are, respectively, ∼ 670 and ∼ 201 Rg.
+At low QPO frequency, the trends of the corona size and
+feedback fraction as a function of QPO frequency reported
+in this work are similar to those in Zhang et al. (2022), and
+both in their work and ours the relation between the size
+of the corona and the frequency of the QPO shows a break
+at νQP O ≈ 3 − 4 Hz. The difference between their and our
+corona sizes in the common range of QPO frequency comes
+from the coverage down to lower energies with NICER in our
+case than in Zhang et al. (2022) with HXMT: The magnitude
+of the lags of the QPO increases as energy decreases, and the
+size of the corona in the vkompth model is driven by the
+magnitude of the lags. Since we go to lower QPO frequencies
+than Zhang et al. (2022), we find that the size of the corona
+continues increasing as the QPO frequency decreases below
+∼ 2 Hz, where they do not have data. At QPO frequencies
+above ∼ 4 Hz Zhang et al. (2022) find an increase of the
+corona size, whereas here we find that the size continues
+decreasing with QPO frequency, albeit at a slower rate than
+below ∼ 3 − 4 Hz. We note that Zhang et al. (2022) did not
+include the effect of dilution of the non-variable components
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+11
+2
+3
+4
+6
+9
+QPO frequency (H )
+104
+3 × 103
+4 × 103
+6 × 103
+L (km)
+broken power-law
+power-law
+2
+3
+4
+6
+9
+QPO frequency (Hz)
+10−1
+100
+2  10−1
+3  10−1
+4  10−1
+6  10−1
+kTs (keV)
+broken power-law
+power-law
+2
+3
+4
+6
+9
+QPO freq ency (Hz)
+10−3
+6 × 10−4
+2 × 10−3
+3 × 10−3
+4 × 10−3
+time lag (secs)
+broken power-law
+power-law
+2
+3
+4
+6
+9
+QPO frequency (Hz)
+100
+6  10−1
+kTin (keV)
+broken power-law
+power-law
+Figure 8. Dependence of L, kTs, time lags of the QPO and kTin upon QPO frequency in MAXI J1535 −571. The red and blue dotted
+lines show the best-fitting power law and a broken power-law to the data. The best-fitting parameters for each relation are given in Table
+3. The time lags are between photons in the 1.0–12.0 keV and 2.0–6.0 keV bands at the QPO frequency. The vertical dotted dashed line
+represents the best-fitting break frequency, νc = 3.0 Hz.
+the rms amplitude of the QPO in their model, and that
+dilution is more important at high QPO frequency, where
+the contribution of the accretion disc to the total emission
+increases.
+Our result is similar to previous findings in other BHXBs
+(e.g. Kara et al. 2019, Karpouzas et al. 2021). In contrast to
+Kara et al. (2019) where a change of the vertical size of the
+corona is proposed to explain the shorter reverberation lags
+for MAXI J1820+070, De Marco et al. (2021) infer a change
+in the inner accretion disc radius leading to smaller coronal
+size than reported in this work. Using the JED-SAD model
+for the same source, Marino et al. (2021) reported that the
+size of the jet emitting region, which plays the corona role
+in their model, of 30-60 Rg. Axelsson & Veledina (2021)
+showed that the variability of the iron line feature could
+not be explained using the lamp-post geometry assumed
+by Kara et al. (2019) and, instead, a truncated inner hot
+flow geometry is required. Using a spectral-timing model
+based on propagating fluctuations and incorporating the
+reverberation from the variable Comptonisation components,
+Kawamura et al. (2022) further supported a truncated inner
+hot flow geometry. However, we note that the mass accretion
+rate propagation fluctuation mechanism used by Kawamura
+et al. (2022) can only explain the hard lags, and a separate
+mechanism is required to explain to soft lags in MAXI
+J1820+070 and in the QPO of MAXI J1535−571 and other
+sources.
+The trend of the size of the corona vs QPO frequency is
+similar in MAXI J1535−571 and GRS 1915+105 (see Figure
+8, and the supplementary Figure 4 in M´endez et al. 2022
+and figure 5 in Garc´ıa et al. 2022). Using a reverberation
+model for the lags of the broadband noise component in the
+power spectrum, Wang et al. (2021) found a corona that
+is ≳300 Rg in the hard to soft state transition of MAXI
+J1820+070. Similarly, using polarimetry measurements with
+PoGO+, Chauvin et al. (2018) found that the corona in
+Cyg X-1 is ≳100 Rg, while they exclude a corona of ∼6 Rg
+obtained from the lamp post model. The sizes reported in
+this work are consistent with the values published by Kylafis
+& Reig (2019), Kylafis et al. (2021), Reig & Kylafis (2021),
+who used Monte Carlo simulations of Comptonization in
+a jet. The Comptonization model used in this work has
+some simplifications; for instance, the corona is spherically
+symmetric with constant temperature and optical depth.
+MNRAS 000, 1–15 (0000)
+
+12
+Rawat et. al.
+This was discussed in Karpouzas et al. (2021), and Garc´ıa
+et al. (2021) and, as explained in M´endez et al. (2022), since
+the actual geometry of the corona is likely different, the
+values given by the model should be considered as a char-
+acteristic size of the corona rather than the actual radius of
+a spherical corona (see M´endez et al. 2022; Garc´ıa et al. 2022).
+The size of the corona that we infer from our model is
+larger than the values obtained from fits to the energy spec-
+tra of black-hole systems with models that consider reflection
+off the accretion disc from a corona that is assumed to be a
+lamppost emitter (e.g., Vincent et al. 2016). These spectral
+fits yield corona sizes of 1−20 Rg (Fabian et al. 2012). Using
+the average soft lags over a broad frequency range in the
+power spectrum and light travel-time arguments, Wang et al.
+(2022) found that corona sizes in a dozen black-hole systems
+in the hard-intermediate state, during the transition from
+the low-hard to the soft-intermediate state, are comparable,
+within a factor of a few, to the ones we infer here (see also
+Wang et al. 2021). Suppose the assumption that the lags of
+the broadband noise reflect the light travel time from the
+corona to the disc is correct. In that case, the corona sizes in
+Wang et al. (2022) are, in fact, lower limits for two reasons:
+(i) Wang et al. (2022) estimate the corona sizes based on
+the average time lag over a broad frequency range, whereas
+the magnitudes of the soft lags are larger than the average
+over a large range of QPO frequencies (see, for instance,
+their Fig. 3, panel h). (ii) Wang et al. (2022) measured the
+lags between the bands 0.5 − 1 and 2 − 5 keV. Suppose the
+lags are minimum at around ∼ 2 keV and increase both at
+energies below and above that (see their Fig. 3, panel g). In
+that case, the magnitude of the time lags between photons
+at ∼ 2 and ∼ 0.5 keV, and hence the light travel distance
+from the corona to the disc will be larger than what they
+report. Notice, however, that in Kara et al. 2019, Wang
+et al. 2021 and Wang et al. 2022, the authors estimate the
+characteristic height of the lamppost corona above the disc.
+Notice that it is not straightforward to infer sizes from
+simple light travel-time arguments applied to the time
+lags of the broadband noise components because: (i) The
+broadband noise component in the power spectrum of
+accreting black-hole and neutron-star systems is, in fact,
+the combination of multiple Lorentzians (e.g., Psaltis et al.
+1999, Nowak 2000). Since the properties of these Lorentzians
+are correlated with each other (e.g., frequency-frequency
+correlations in Psaltis et al. 1999) and with the source
+spectral parameters (e.g., Vignarca et al. 2003; Mereminskiy
+et al. 2018; Agrawal 2006 and references therein), therefore,
+most likely, these Lorentzians are not just an empirical
+description of the power spectrum, but each of them rep-
+resents a relatively well-defined, over a limited frequency
+range, variability component of the physical properties of
+the accretion flow. Suppose this decomposition is correct (as
+suggested by the works cited above). In that case, a more
+logical and accurate way is to compute the phase lag that
+results from the combined cross spectra of these Lorentzians
+in the Fourier real and imaginary space. The phase-lag
+calculated like that can be different from computed from the
+average of the cross-spectrum over a broad frequency range
+(as has been done in many works before, see, e.g. Nowak
+et al. 1999a; Reig et al. 2000; Altamirano & M´endez 2015;
+Wang et al. 2022). If the lags calculated from the Lorentzian
+decomposition, as suggested above, were due to light travel
+time, the magnitude of time lags (see, for instance, Fig. 6)
+imply large corona sizes. So even combining the lags of the
+Lorentzians in Fourier space will lead to big corona sizes.
+ii) It needs to be clarified how to convert time lags into
+distances using simple light travel-time arguments because
+the lags depend strongly upon Fourier frequency (e.g., Fig.
+3 panel h of Wang et al. 2022). Therefore, there is no single
+Fourier frequency at which the time lag would represent the
+correct light travel time that should be used to infer the
+corona size. (We note that models like RELTRANS, Ingram
+et al. (2019) calculate the full variability self consistently
+instead of using simple light travel-time arguments.)
+Given the typical magnitudes of the lags of the QPO (this
+paper; Karpouzas et al. 2020; Garc´ıa et al. 2021; Karpouzas
+et al. 2021; Bellavita et al. 2022) or of the broadband noise
+component (Wang et al. 2022; but see above for the caveats
+of these measurements) in these systems, any variability
+model that interprets the observed lags as delays of photons
+travelling through a medium around a compact object would
+necessarily yield large corona sizes since time lags of a few
+hundredths to a few tenths of seconds translate into light
+travel distances of a few thousand to a few 10,000 km.
+While propagation of accretion-rate fluctuations (Ar´evalo &
+Uttley 2006) would yield smaller sizes of the comptonizing
+region because, in this case, the viscous time scale is at play,
+propagation of accretion-rate fluctuations only account for
+hard lags. In contrast, the broadband noise component and
+the QPOs often show soft lags.
+Our results are not necessarily inconsistent with the QPO
+frequency being due to Lense-Thirring Precession (LTP,
+Stella & Vietri 1998; but see Mastichiadis et al. 2022). For
+instance, Ingram et al. (2016) fitted the energy spectra of
+the BHXRB H1743−322 over the cycle of a ∼4–5 QPO
+and concluded that the results are consistent with LTP of
+an inner hot torus in this source. However, as explained by
+Ingram et al. (2016), their data could be reproduced equally
+well if the torus was fixed and it was the disc the one that
+processed at the Lense–Thirring precession frequency. Their
+choice of one geometry over the other was based on the fact
+that the rms spectrum of the QPO is hard, and hence the
+emission at the QPO frequency could not come from the disc.
+In the model of Karpouzas et al. (2020), the rms spectrum of
+the QPO is a consequence of inverse-Compton scattering of
+soft disc photons in the corona (the torus in the scenario of
+Ingram et al. 2016), such that the high rms amplitude values
+of the QPO at high energies may reflect the variability of the
+soft disc emission at the Lense–Thirring precession frequency
+that is inverse-Compton scattered in the corona. This, plus
+the feedback from the corona to the disc, naturally explain
+the variability of the iron line discussed by Ingram et al.
+(2016) and the rms spectrum of the QPO. The LTP model
+and the reverberation model for the lags of the QPO in GRS
+1915+105 (Nathan et al. 2022) also yield a large corona
+(unless one considers an extra lag due to thermalisation;
+see Nathan et al. 2022). Therefore, the LTP model needs to
+explain how a large corona, which should necessarily extend
+beyond the disc’s inner truncation radius, can precess as
+a solid body. However, whether the QPO frequency is due
+to LTP is a matter of debate that needs to be addressed
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+13
+by general relativistic magneto-hydrodynamic (GRMHD)
+simulations, which is beyond the scope of this paper.
+4.3 A Dual Corona
+When we tied the inner-disc temperature of the time-
+averaged spectra, kTin, to the seed-photon temperature of
+the vkompthdk model, kTs, our fits could not reproduce
+the shape of the lag spectrum. Letting these two parameters
+free yields a significant improvement in the fit statistics
+(see Section 3.3 and Figure 6). We speculate that this
+difference between the seed photon temperature of nthcomp
+and vkompthdk is due to a more complex structure of
+the comptonizing region than that described by a uniform
+corona. Sridhar et al. (2019), Bhargava et al. (2019) & Garg
+et al. (2022) used AstroSat observations of MAXI J1535 that
+coincide with the first few days of the NICER observations
+reported in this work. They modelled the combined SXT and
+LAXPC spectra and reported a lower inner disc temperature
+(kTin=0.20–0.35 keV) than we found in this work. It should
+be noted that Bhargava et al. (2019) and Garg et al. (2022)
+modelled the spectra in the 1-30 keV energy range. Also, the
+source is highly absorbed, and the spectrum drops at low
+energies, so the reported inner disc temperature may not
+be accurate. Sreehari et al. (2019) used the same AstroSat
+observation and modelled the broadband spectra in the
+0.3-80.0 keV band and reported electron temperatures with
+nthcomp in the range 21-63 keV. Using the same AstroSat
+observation, Sridhar et al. (2019) reported an electron
+temperature of ∼21 keV. As the 0.8-10.0 keV spectra of
+NICER could not constrain the electron temperature, we
+chose to fix it to the values reported by Sreehari et al.
+(2019) and Sridhar et al. (2019). The electron temperature
+(∼90–108 keV) reported by Garg et al. 2022 is higher than
+the value (∼21 keV) we have used in this work. It should be
+noted that in Garg et al. (2022), they are fixed the optical
+depth of the corona, which together with Γ gives kTe.
+Using a dual-component comptonization model for type-
+B QPOs, Garc´ıa et al. (2021) and Peirano et al. (2022) ar-
+gued that the comptonizing medium of the BHXB sources,
+MAXI J1348−630 and GX 339−4 consist of two coronas. A
+relatively small corona of ∼300 km, close to the black hole
+dominates the time-averaged spectra, and a large corona of
+∼18000 km, possibly the jet, dominates the lag spectrum
+(Peirano et al. 2022). Their best-fitting results yield a lower
+seed photon temperature of the large corona compared to the
+small corona, with the seed photon temperature of the small
+corona linked to kTbb of nthcomp. Peirano et al. (2022) pro-
+posed that this difference is due to the fact that the seed pho-
+tons for the small corona come from the inner, hotter parts,
+of the disc whereas the seed photons for the large corona
+come from the outer, cooler parts, of the disc. A similar
+dual-corona geometry could explain the difference between
+kTin of the diskbb (linked to kTbb of nthcomp) and kTs of
+vkompthdk in our fits. Since we find that kTbb > kTs, also in
+MAX J1535−571 the small corona would dominate the emis-
+sion of the time-averaged spectra, whereas the big corona
+would dominate the lags. We found that the rms spectra do
+not change much between the two fits (kTs=kTin or kTs free),
+so we conclude that the rms amplitude is not affected much
+by the size of the corona. The fraction of the corona flux that
+returns to the disc is ηint 10–25 % in all the cases. This and
+the large corona size further indicate that the large corona is
+the jet.
+5 SUMMARY AND CONCLUSIONS
+We
+have
+analysed
+all
+NICER
+observation
+of
+MAXI
+J1535−571 taken on September and October 2017. We fit
+the energy spectra of the source and the rms and lag spectra
+of the type-C QPO in this source with the one-component
+time dependent Comptonization model vkompthdk. Below
+we summarize our results:
+• The size of the corona of MAXI J1535−571 decreases
+from 104 km when the QPO frequency is ≥2 Hz to ∼3000
+km when the QPO frequency is ∼9.0 Hz.
+• The behaviour of all the spectral parameters and the rms
+and lag spectra of the QPO changes above and below a critical
+QPO frequency, νc =3.0±0.4 Hz. Interestingly, the time at
+which this critical frequency happens coincide with the period
+when the radio jet emission quenches for this source.
+• Comparing our results with those in previous work, the
+data are consistent with a dual corona: a small corona lying
+close to the black hole and a larger one, possibly the jet.
+ACKNOWLEDGEMENTS
+This research is part of a project proposed for the COSPAR
+PCB fellowship program. We would like to thank the ref-
+eree for constructive comments that helped improve this
+paper. DR would like to thank COSPAR, ISRO and Pro-
+fessor Diego Altamirano for jointly funding the academic
+visit to the University of Southampton. MM, FG and KK
+acknowledge support from the research programme Athena
+with project number 184.034.002, which is (partly) financed
+by the Dutch Research Council (NWO). FG acknowledges
+support from PIP 0102 and PIP 0113 (CONICET). FG is a
+CONICET researcher. This work received financial support
+from PICT-2017-2865 (ANPCyT). KA acknowledges support
+from a UGC-UKIERI Phase 3 Thematic Partnership (UGC-
+UKIERI-2017-18-006; PI: P. Gandhi). TMB acknowledges fi-
+nancial contribution from PRIN INAF 2019 n.15. CB is a
+fellow of Consejo Interuniversitario Nacional (CIN).
+DATA AVAILABILITY
+The NICER XTI observations used in this work are available
+at NICER Archive6.
+REFERENCES
+Agrawal P. C., 2006, Advances in Space Research, 38, 2989
+Altamirano D., M´endez M., 2015, MNRAS, 449, 4027
+Altamirano D., Strohmayer T., 2012, ApJ, 754, L23
+Ar´evalo P., Uttley P., 2006, MNRAS, 367, 801
+6 https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_archive.
+html
+MNRAS 000, 1–15 (0000)
+
+14
+Rawat et. al.
+Axelsson M., Veledina A., 2021, MNRAS, 507, 2744
+Bellavita C., Garc´ıa F., M´endez M., Karpouzas K., 2022, MNRAS,
+515, 2099
+Belloni T., Hasinger G., 1990, A&A, 230, 103
+Belloni T. M., Stella L., 2014, Space Sci. Rev., 183, 43
+Belloni T., Psaltis D., van der Klis M., 2002, ApJ, 572, 392
+Belloni T., Homan J., Casella P., van der Klis M., Nespoli E.,
+Lewin W. H. G., Miller J. M., M´endez M., 2005, A&A, 440,
+207
+Belloni T. M., Motta S. E., Mu˜noz-Darias T., 2011, Bulletin of the
+Astronomical Society of India, 39, 409
+Belloni T. M., Sanna A., M´endez M., 2012, MNRAS, 426, 1701
+Belloni T. M., Zhang L., Kylafis N. D., Reig P., Altamirano D.,
+2020, MNRAS, 496, 4366
+Bhargava Y., Belloni T., Bhattacharya D., Misra R., 2019, MN-
+RAS, 488, 720
+Casella P., Belloni T., Homan J., Stella L., 2004, A&A, 426, 587
+Casella P., Belloni T., Stella L., 2005, ApJ, 629, 403
+Chauhan J., et al., 2019, MNRAS, 488, L129
+Chauvin M., et al., 2018, Nature Astronomy, 2, 652
+Chen X., Swank J. H., Taam R. E., 1997, ApJ, 477, L41
+C´uneo V. A., et al., 2020, MNRAS, 496, 1001
+De Marco B., Zdziarski A. A., Ponti G., Migliori G., Belloni T. M.,
+Segovia Otero A., Dzie�lak M. A., Lai E. V., 2021, A&A, 654,
+A14
+Dewangan G. C., Titarchuk L., Griffiths R. E., 2006, ApJ, 637,
+L21
+Din¸cer T., 2017, The Astronomer’s Telegram, 10716, 1
+Done C., Gierli´nski M., Kubota A., 2007, A&A Rev., 15, 1
+Fabian A. C., et al., 2012, MNRAS, 424, 217
+Fender R. P., Belloni T. M., Gallo E., 2004, MNRAS, 355, 1105
+Garc´ıa F., M´endez M., Karpouzas K., Belloni T., Zhang L., Al-
+tamirano D., 2021, MNRAS, 501, 3173
+Garc´ıa F., Karpouzas K., M´endez M., Zhang L., Zhang Y., Belloni
+T., Altamirano D., 2022, MNRAS, 513, 4196
+Garg A., Misra R., Sen S., 2022, MNRAS, 514, 3285
+Gendreau K. C., Arzoumanian Z., Okajima T., 2012, in Takahashi
+T., Murray S. S., den Herder J.-W. A., eds, Society of Photo-
+Optical Instrumentation Engineers (SPIE) Conference Series
+Vol. 8443, Space Telescopes and Instrumentation 2012: Ultra-
+violet to Gamma Ray. p. 844313, doi:10.1117/12.926396
+Gendreau K. C., et al., 2016, in den Herder J.-W. A., Takahashi T.,
+Bautz M., eds, Society of Photo-Optical Instrumentation Engi-
+neers (SPIE) Conference Series Vol. 9905, Space Telescopes and
+Instrumentation 2016: Ultraviolet to Gamma Ray. p. 99051H,
+doi:10.1117/12.2231304
+Gendreau K., et al., 2017, The Astronomer’s Telegram, 10768, 1
+Gilfanov M., 2010, in Belloni T., ed., , Vol. 794, Lecture Notes
+in Physics, Berlin Springer Verlag. Springer Berlin Heidelberg,
+p. 17, doi:10.1007/978-3-540-76937-8 2
+Homan J., Wijnands R., van der Klis M., Belloni T., van Paradijs
+J., Klein-Wolt M., Fender R., M´endez M., 2001, ApJS, 132,
+377
+Huang Y., et al., 2018, ApJ, 866, 122
+Ingram A. R., Motta S. E., 2019, New A Rev., 85, 101524
+Ingram A., van der Klis M., Middleton M., Done C., Altamirano
+D., Heil L., Uttley P., Axelsson M., 2016, MNRAS, 461, 1967
+Kara E., et al., 2019, Nature, 565, 198
+Karpouzas K., M´endez M., Ribeiro E. M., Altamirano D., Blaes
+O., Garc´ıa F., 2020, MNRAS, 492, 1399
+Karpouzas K., M´endez M., Garc´ıa F., Zhang L., Altamirano D.,
+Belloni T., Zhang Y., 2021, MNRAS, 503, 5522
+Kawamura T., Done C., Axelsson M., Takahashi T., 2022, arXiv
+e-prints, p. arXiv:2209.14492
+Kennea J. A., Evans P. A., Beardmore A. P., Krimm H. A., Ro-
+mano P., Yamaoka K., Serino M., Negoro H., 2017, The As-
+tronomer’s Telegram, 10700, 1
+Koljonen K. I. I., Hannikainen D. C., McCollough M. L., 2011,
+MNRAS, 416, L84
+Kumar N., Misra R., 2014, MNRAS, 445, 2818
+Kylafis N. D., Reig P., 2019, in High Energy Phenomena in Rela-
+tivistic Outflows VII. p. 37 (arXiv:1911.11975)
+Kylafis N. D., Tr¨umper J. E., Loudas N. A., 2021, A&A, 655, A39
+Ma X., et al., 2021, Nature Astronomy, 5, 94
+Makishima K., Maejima Y., Mitsuda K., Bradt H. V., Remillard
+R. A., Tuohy I. R., Hoshi R., Nakagawa M., 1986, ApJ, 308,
+635
+Marino A., et al., 2021, A&A, 656, A63
+Markwardt C. B., Burrows D. N., Cummings J. R., Kennea J. A.,
+Marshall F. E., Page K. L., Palmer D. M., Siegel M. H., 2017,
+GRB Coordinates Network, 21788, 1
+Mastichiadis A., Petropoulou M., Kylafis N. D., 2022, A&A, 662,
+A118
+M´endez M., van der Klis M., 1997, ApJ, 479, 926
+M´endez M., Altamirano D., Belloni T., Sanna A., 2013, MNRAS,
+435, 2132
+M´endez M., Karpouzas K., Garc´ıa F., Zhang L., Zhang Y., Belloni
+T. M., Altamirano D., 2022, Nature Astronomy, 6, 577
+Mereminskiy I. A., Grebenev S. A., Prosvetov A. V., Semena A. N.,
+2018, Astronomy Letters, 44, 378
+Miller J. M., et al., 2001, ApJ, 563, 928
+Miller J. M., et al., 2018, ApJ, 860, L28
+Mitsuda K., et al., 1984, PASJ, 36, 741
+Motta S. E., Casella P., Henze M., Mu˜noz-Darias T., Sanna A.,
+Fender R., Belloni T., 2015, MNRAS, 447, 2059
+Nakahira S., et al., 2018, PASJ, 70, 95
+Nathan E., et al., 2022, MNRAS, 511, 255
+Negoro H., et al., 2017a, The Astronomer’s Telegram, 10699, 1
+Negoro H., et al., 2017b, The Astronomer’s Telegram, 10708, 1
+Nowak M. A., 2000, MNRAS, 318, 361
+Nowak M. A., Dove J. B., Vaughan B. A., Wilms J., Begelman
+M. C., 1999a, Nuclear Physics B Proceedings Supplements, 69,
+302
+Nowak M. A., Vaughan B. A., Wilms J., Dove J. B., Begelman
+M. C., 1999b, ApJ, 510, 874
+Parikh A. S., Russell T. D., Wijnands R., Miller-Jones J. C. A.,
+Sivakoff G. R., Tetarenko A. J., 2019, ApJ, 878, L28
+Pasham D. R., Strohmayer T. E., Mushotzky R. F., 2013, ApJ,
+771, L44
+Peirano V., M´endez M., Garc´ıa F., Belloni T., 2022, arXiv e-prints,
+p. arXiv:2212.00062
+Psaltis D., Belloni T., van der Klis M., 1999, ApJ, 520, 262
+Rawat D., et al., 2019, ApJ, 870, 4
+Reig P., Kylafis N. D., 2021, A&A, 646, A112
+Reig P., Belloni T., van der Klis M., M´endez M., Kylafis N. D.,
+Ford E. C., 2000, ApJ, 541, 883
+Remillard R. A., McClintock J. E., 2006, ARA&A, 44, 49
+Remillard R. A., Sobczak G. J., Muno M. P., McClintock J. E.,
+2002, ApJ, 564, 962
+Russell T. D., Miller-Jones J. C. A., Sivakoff G. R., Tetarenko A. J.,
+Jacpot Xrb Collaboration 2017, The Astronomer’s Telegram,
+10711, 1
+Russell T. D., et al., 2019, ApJ, 883, 198
+Scaringi S., ASTR211 Students 2017, The Astronomer’s Telegram,
+10702, 1
+Sreehari H., Ravishankar B. T., Iyer N., Agrawal V. K., Katoch
+T. B., Mandal S., Nandi A., 2019, MNRAS, 487, 928
+Sridhar N., Bhattacharyya S., Chandra S., Antia H. M., 2019, MN-
+RAS, 487, 4221
+Stella L., Vietri M., 1998, ApJ, 492, L59
+Stevens A. L., et al., 2018, ApJ, 865, L15
+Stiele H., Kong A. K. H., 2018, ApJ, 868, 71
+Strohmayer T. E., 2001, ApJ, 552, L49
+Takizawa M., et al., 1997, ApJ, 489, 272
+Tao L., et al., 2018, MNRAS, 480, 4443
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+15
+Verner D. A., Ferland G. J., Korista K. T., Yakovlev D. G., 1996,
+ApJ, 465, 487
+Vignarca F., Migliari S., Belloni T., Psaltis D., van der Klis M.,
+2003, A&A, 397, 729
+Vincent F. H., R´o˙za´nska A., Zdziarski A. A., Madej J., 2016, A&A,
+590, A132
+Wang J., et al., 2021, ApJ, 910, L3
+Wang J., et al., 2022, ApJ, 930, 18
+Wijnands R., Homan J., van der Klis M., 1999, ApJ, 526, L33
+Wilms J., Allen A., McCray R., 2000, ApJ, 542, 914
+Xu Y., et al., 2018, ApJ, 852, L34
+Zdziarski A. A., Johnson W. N., Magdziarz P., 1996, MNRAS, 283,
+193
+Zhang L., Wang Y., M´endez M., Chen L., Qu J., Altamirano D.,
+Belloni T., 2017, ApJ, 845, 143
+Zhang L., et al., 2020, MNRAS, 494, 1375
+Zhang Y., et al., 2022, MNRAS, 512, 2686
+˙Zycki P. T., Done C., Smith D. A., 1999, MNRAS, 309, 561
+van der Klis M., Jansen F. A., 1985, Nature, 313, 768
+APPENDIX A:
+MNRAS 000, 1–15 (0000)
+
+16
+Rawat et. al.
+Table A.1. The columns are the observation number, the chi-square of the fit to the steady-state spectrum (χ2
+SSS), rms spectrum (χ2
+rms),
+lag spectrum (χ2
+lag) with, in each case, the number of channels in each spectrum and the total reduced chi-square of the combined fit with
+degree of freedom.
+Obs no.
+χ2
+SSS (channel)
+χ2
+rms (channel)
+χ2
+lag (channel)
+χ2
+total (dof)
+1
+206.9 (238)
+15.5 (10)
+9.0 (10)
+231.4 (243)
+2
+176.5 (237)
+7.8 (10)
+7.6 (10)
+191.9 (242)
+3
+219.5 (238)
+7.7 (10)
+13.3 (10)
+240.5 (243)
+4
+205.6 (238)
+4.7 (10)
+9.4 (10)
+219.8 (243)
+5
+206.8 (238)
+13.8 (10)
+21.8 (10)
+242.3 (243)
+6
+167.9 (238)
+5.1 (10)
+4.8 (10)
+177.9 (243)
+7
+165.9 (238)
+5.0 (10)
+2.4 (10)
+173.2 (243)
+8
+227.7 (238)
+4.8 (10)
+2.3 (10)
+234.8 (243)
+9
+157.1 (238)
+5.0 (10)
+7.2 (10)
+169.3 (243)
+10
+146.1 (238)
+4.7 (10)
+4.3 (10)
+155.1 (243)
+11
+176.4 (217)
+13.0 (10)
+2.7 (10)
+192.2 (222)
+12
+129.3 (238)
+10.6 (10)
+12.5 (10)
+152.4 (243)
+13
+157.3 (238)
+7.3 (10)
+11.8 (10)
+176.3 (243)
+14
+147.0 (238)
+17.7 (10)
+3.8 (10)
+168.4 (242)
+15
+183.9 (235)
+9.3 (10)
+2.7 (10)
+195.8 (239)
+16
+146.9 (238)
+13.3 (7)
+4.8 (7)
+165.0 (236)
+17
+142.4 (238)
+23.0 (10)
+11.6 (10)
+177.0 (242)
+18
+240.5 (231)
+3.0 (7)
+0.9 (7)
+244.4 (229)
+19
+184.0 (238)
+10.5 (10)
+9.1 (10)
+203.6 (242)
+20
+185.3 (235)
+3.7 (10)
+15.5 (10)
+204.5 (240)
+21
+181.6 (216)
+11.6 (7)
+5.1 (7)
+198.2 (215)
+22
+211.3 (214)
+23.1 (10)
+23.6 (10)
+258.0 (219)
+23
+183.8 (232)
+26.2 (10)
+13.1 (11)
+223.1 (238)
+24
+184.1 (238)
+5.0 (10)
+2.3 (9)
+191.4 (241)
+25
+159.6 (238)
+5.2 (10)
+10.1 (10)
+174.9 (242)
+Note: Notice that some parameters are linked in the combined fits and therefore we cannot give the number of degrees of freedom for
+each individual fit. So, channel numbers for individual spectra are given here.
+Figure A1. The top and bottom panels show respectively the fractional rms and phase-lag spectra of the type-C QPO in MAXI J1535−571
+fitted with vkompthdk model. The 2.0–3.0 keV band is the reference band for the phase lag spectra.
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+17
+Table A.2. The columns are the observation number, QPO frequency, QPO fractional rms amplitude and time lags at the QPO frequency
+of MAXI J1535−571. Here rms1 and lag1 are in the 0.5–2.0 keV band, rms2 and lag2 are in the 2.0–4.0 keV band, and rms3 and lag3 are
+in the 4.0–10.0 keV band. The reference band for lags is 0.5–10.0 keV.
+Obs no.
+QPO frequency
+QPO fractional
+lag1
+QPO fractional
+lag2
+QPO fractional
+lag3
+(Hz)
+rms1 (%)
+(msec)
+rms2 (%)
+(msec)
+rms3 (%)
+(msec)
+1
+2.74 ± 0.01
+5.2 ± 0.1
+10.2 ± 1.0
+7.3 ± 0.2
+−1.49 ± 0.38
+9.4 ± 0.3
+−6.4 ± 0.7
+2
+2.44 ± 0.01
+5.0 ± 0.2
+12.5 ± 0.9
+6.7 ± 0.2
+−2.22 ± 0.41
+8.7 ± 0.3
+−7.1 ± 0.7
+3
+2.32 ± 0.01
+5.5 ± 0.2
+12.7 ± 1.2
+6.8 ± 0.3
+−3.20 ± 0.54
+8.8 ± 0.4
+−6.0 ± 1.1
+4
+1.83 ± 0.01
+5.8 ± 0.1
+12.5 ± 0.8
+7.4 ± 0.2
+−4.63 ± 0.38
+8.7 ± 0.3
+−2.7 ± 0.7
+5
+1.81 ± 0.00
+5.8 ± 0.1
+12.1 ± 0.5
+7.4 ± 0.1
+−4.20 ± 0.22
+9.2 ± 0.1
+−3.2 ± 0.4
+6
+2.15 ± 0.01
+5.6 ± 0.2
+14.0 ± 0.9
+7.1 ± 0.2
+−3.24 ± 0.39
+8.6 ± 0.3
+−7.1 ± 0.7
+7
+2.41 ± 0.01
+5.8 ± 0.2
+13.3 ± 1.2
+7.7 ± 0.3
+−1.59 ± 0.47
+9.8 ± 0.4
+−9.4 ± 0.9
+8
+2.77 ± 0.01
+5.5 ± 0.2
+12.6 ± 1.1
+7.6 ± 0.2
+−2.05 ± 0.42
+9.5 ± 0.4
+−6.9 ± 0.9
+9
+2.75 ± 0.02
+5.3 ± 0.2
+12.3 ± 1.3
+7.2 ± 0.2
+−1.35 ± 0.57
+10.0 ± 0.4
+−8.4 ± 1.1
+10
+3.27 ± 0.02
+4.9 ± 0.2
+9.1 ± 1.5
+7.1 ± 0.3
+−1.44 ± 0.54
+10.6 ± 0.4
+−5.5 ± 1.0
+11
+3.19 ± 0.03
+5.3 ± 0.3
+12.6 ± 1.7
+7.0 ± 0.3
+−1.42 ± 0.65
+10.5 ± 0.5
+−7.1 ± 1.1
+12
+2.72 ± 0.01
+4.7 ± 0.2
+13.7 ± 0.9
+6.9 ± 0.2
+−1.79 ± 0.33
+9.3 ± 0.3
+−8.1 ± 0.6
+13
+2.84 ± 0.01
+5.4 ± 0.2
+13.1 ± 0.9
+7.6 ± 0.2
+−2.10 ± 0.32
+10.4 ± 0.3
+−6.7 ± 0.6
+14
+4.75 ± 0.01
+3.2 ± 0.3
+9.3 ± 0.8
+5.6 ± 0.1
+0.23 ± 0.24
+9.7 ± 0.2
+−6.2 ± 0.4
+15
+9.01 ± 0.04
+−−
+4.4 ± 0.4
+1.5 ± 0.1
+0.07 ± 0.15
+3.7 ± 0.1
+−3.2 ± 0.2
+16
+7.54 ± 0.05
+1.4 ± 0.4
+6.4 ± 0.6
+2.2 ± 0.3
+0.50 ± 0.26
+6.0 ± 0.2
+−4.7 ± 0.3
+17
+7.54 ± 0.06
+1.3 ± 0.2
+5.3 ± 0.5
+2.8 ± 0.1
+0.20 ± 0.14
+5.9 ± 0.2
+−3.9 ± 0.2
+18
+7.09 ± 0.03
+1.1 ± 0.1
+4.8 ± 0.4
+2.2 ± 0.1
+0.01 ± 0.12
+5.3 ± 0.1
+−3.6 ± 0.2
+19
+5.42 ± 0.01
+2.7 ± 0.1
+7.9 ± 0.5
+4.6 ± 0.1
+−0.21 ± 0.17
+9.3 ± 0.2
+−4.6 ± 0.2
+20
+5.73 ± 0.01
+2.6 ± 0.1
+8.3 ± 0.2
+4.4 ± 0.1
+−0.40 ± 0.08
+9.1 ± 0.1
+−4.3 ± 0.1
+21
+6.77 ± 0.02
+1.9 ± 0.1
+6.4 ± 0.3
+3.3 ± 0.1
+−0.24 ± 0.10
+7.6 ± 0.1
+−3.7 ± 0.1
+22
+4.57 ± 0.01
+2.8 ± 0.1
+10.8 ± 0.4
+4.6 ± 0.1
+−0.91 ± 0.13
+8.2 ± 0.2
+−5.6 ± 0.2
+23
+4.82 ± 0.01
+2.0 ± 0.1
+9.5 ± 0.5
+4.0 ± 0.0
+−0.39 ± 0.13
+6.3 ± 0.1
+−5.3 ± 0.2
+24
+5.19 ± 0.03
+2.0 ± 0.2
+7.7 ± 1.7
+2.9 ± 0.2
+−0.23 ± 0.51
+7.2 ± 0.3
+−4.6 ± 0.8
+25
+4.50 ± 0.01
+3.1 ± 0.1
+9.1 ± 0.6
+5.2 ± 0.1
+−0.69 ± 0.22
+9.2 ± 0.2
+−5.1 ± 0.4
+MNRAS 000, 1–15 (0000)
+
+18
+Rawat et. al.
+0.1
+1
+10
+100
+1000
+104
+counts s−1 keV−1
+0.1
+0.02
+0.05
+Fractional rms
+1
+10
+2
+5
+0
+0.5
+Phase lags (rad)
+Energy (keV)
+−2
+0
+2
+(data−model)/error
+−2
+−1
+0
+1
+2
+(data−model)/error
+1
+10
+2
+5
+−2
+0
+2
+(data−model)/error
+Energy (keV)
+Figure A2. The same plot as shown in Figure 7 at ∼4.5 Hz QPO frequency in MAXI J1535−571.
+MNRAS 000, 1–15 (0000)
+
+Comptonizing medium of MAXI J1535−571
+19
+0.1
+1
+10
+100
+1000
+104
+counts s−1 keV−1
+0.01
+0.1
+0.02
+0.05
+Fractional rms
+1
+10
+2
+5
+−0.2
+0
+0.2
+0.4
+Phase lags (rad)
+Energy (keV)
+−2
+0
+2
+(data−model)/error
+−2
+−1
+0
+1
+2
+(data−model)/error
+1
+10
+2
+5
+−1
+0
+1
+(data−model)/error
+Energy (keV)
+Figure A3. The same plot as shown in Figure 7 at ∼7.0 Hz QPO frequency in MAXI J1535−571.
+2
+3
+4
+5
+6
+7
+8
+9
+QPO frequency (Hz)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+η
+Figure A4. Dependence of the η upon QPO frequency in MAXI J1535−571. The values of η are obtained from the fits to the time-averaged
+spectra, the rms and phase-lag spectra of the QPO.
+MNRAS 000, 1–15 (0000)
+

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+POLICY PRE-TRAINING FOR AUTONOMOUS DRIVING
+VIA SELF-SUPERVISED GEOMETRIC MODELING
+Penghao Wu1,2∗ Li Chen1 Hongyang Li1,3† Xiaosong Jia1,3∗ Junchi Yan1,3 Yu Qiao1
+1OpenDriveLab, Shanghai AI Laboratory
+2UC San Diego
+3Shanghai Jiao Tong University
+ABSTRACT
+Witnessing the impressive achievements of pre-training techniques on large-scale
+data in the field of computer vision and natural language processing, we won-
+der whether this idea could be adapted in a grab-and-go spirit, and mitigate the
+sample inefficiency problem for visuomotor driving. Given the highly dynamic
+and variant nature of the input, the visuomotor driving task inherently lacks view
+and translation invariance, and the visual input contains massive irrelevant in-
+formation for decision making, resulting in predominant pre-training approaches
+from general vision less suitable for the autonomous driving task. To this end,
+we propose PPGeo (Policy Pre-training via Geometric modeling), an intuitive
+and straightforward fully self-supervised framework curated for the policy pre-
+training in visuomotor driving. We aim at learning policy representations as a
+powerful abstraction by modeling 3D geometric scenes on large-scale unlabeled
+and uncalibrated YouTube driving videos. The proposed PPGeo is performed in
+two stages to support effective self-supervised training. In the first stage, the
+geometric modeling framework generates pose and depth predictions simulta-
+neously, with two consecutive frames as input. In the second stage, the visual
+encoder learns driving policy representation by predicting the future ego-motion
+and optimizing with the photometric error based on current visual observation
+only. As such, the pre-trained visual encoder is equipped with rich driving pol-
+icy related representations and thereby competent for multiple visuomotor driv-
+ing tasks. As a side product, the pre-trained geometric modeling networks could
+bring further improvement to the depth and odometry estimation tasks. Extensive
+experiments covering a wide span of challenging scenarios have demonstrated
+the superiority of our proposed approach, where improvements range from 2%
+to even over 100% with very limited data. Code and models will be available at
+https://github.com/OpenDriveLab/PPGeo.
+1
+INTRODUCTION
+Policy learning refers to the learning process of an autonomous agent acquiring the decision-making
+policy to perform a certain task in a particular environment. Visuomotor policy learning (Mnih et al.,
+2015; Levine et al., 2016; Hessel et al., 2018; Laskin et al., 2020; Toromanoff et al., 2020) takes as
+input raw sensor observations and predicts the action, simultaneously cooperating and training the
+perception and control modules in an end-to-end fashion. For visuomotor policy models, learning
+tabula rasa is difficult, where it usually requires a prohibitively large corpus of labeled data or en-
+vironment interactions to achieve satisfactory performance (Espeholt et al., 2018; Wijmans et al.,
+2019; Yarats et al., 2020).
+To mitigate the sample efficiency caveat in visuomotor policy learning, pre-training the visual per-
+ception network in advance is a promising solution. Recent studies (Shah & Kumar, 2021; Parisi
+et al., 2022; Xiao et al., 2022; Radosavovic et al., 2022; Shah et al., 2022) have demonstrated that
+applying popular visual pre-training approaches, including ImageNet (Deng et al., 2009) classifica-
+tion, contrastive learning (He et al., 2020; Chen et al., 2020c), masked image modeling (MIM) (He
+et al., 2022; Xie et al., 2022), and language-vision pre-training (Radford et al., 2021), could guar-
+antee superior representation for robotic policy learning tasks, e.g., dexterous manipulation, motor
+∗Work done during internship at Shanghai AI Laboratory.
+†Corresponding author. Email to: [email protected]
+1
+arXiv:2301.01006v1  [cs.CV]  3 Jan 2023
+
+Figure 1: Uniqueness of visuomotor driving policy learning. The planned trajectory is shown as red
+points. (a) static obstacles and background buildings (objects in yellow rectangles) are irrelevant to
+the driving decision; (b) the traffic signal in the visual input (marked with the green box) is extremely
+difficult to recognize and yet deterministic for control outputs; (c) the pre-trained visual encoder has
+to be robust to different light and weather conditions. Photo credit from (Caesar et al., 2020).
+control skills and visual navigation. However, for one crucial and challenging visuomotor task in
+particular, namely end-to-end autonomous driving1, the aforementioned predominant pre-training
+methods may not be the optimal choice (Yamada et al., 2022; Zhang et al., 2022b).
+In this paper, we aim to investigate why ever-victorious pre-training approaches for general computer
+vision tasks and robotic control tasks are prone to fail in case of end-to-end autonomous driving.
+For conventional pre-training methods in general vision tasks, e.g., classification, segmentation and
+detection, they usually adopt a wide range of data augmentations to achieve translation and view
+invariance (Zhang et al., 2016; Wu et al., 2018). For robotic control tasks, the input sequence is
+generally of small resolution; the environment setting is simple and concentrated on objects (Parisi
+et al., 2022; Radosavovic et al., 2022). We argue that the visuomotor driving investigated in this
+paper, is sensitive to geometric relationships and usually comprises complex scenarios.
+As described in Fig. 1(a), the input data often carry irrelevant information, such as background
+buildings, far-away moving vehicles, nearby static obstacles, etc., which are deemed as noises for
+the decision making task. To obtain a good driving policy, we argue that the desirable model should
+only concentrate on particular parts/patterns of the visual input. That is, taking heed of direct or
+deterministic relation to the decision making, e.g., traffic signals in Fig. 1(b). However, concurrent
+pre-training approaches fail to fulfill such a requirement. There comes a natural and necessary
+demand to formulate a pre-training scheme curated for end-to-end autonomous driving. We attempt
+to pre-train a visual encoder with a massive amount of driving data crawled freely from the web,
+such that given limited labeled data, downstream applications could generalize well and quickly
+adapt to various driving environments as depicted in Fig. 1(c).
+The pivotal question is how to introduce driving-decision awareness into the pre-training process
+to help the visual encoder concentrate on crucial visual cues for driving policy. One may resort
+to directly predicting ego-motion based on single frame sensor input, constraining the network on
+learning policy-related features. Previous literature tackles the supervision problem with pseudo
+labeling training on either an open dataset (Zhang et al., 2022b) or the target domain data (Zhang
+et al., 2022a). However, pseudo labeling approaches suffer from noisy predictions from poorly
+calibrated models - this is true especially when there exists distinct domain gap such as geographical
+locations and traffic complexities (Rizve et al., 2020).
+To address the bottleneck aforementioned, we propose PPGeo (Policy Pre-training via Geometric
+modeling), a fully self-supervised driving policy pre-training framework to learn from unlabeled
+and uncalibrated driving videos. It models the 3D geometric scene by jointly predicting ego-motion,
+depth, and camera intrinsics. Since directly learning ego-motion based on single frame input along
+with depth and intrinsics training from scratch is too difficult, it is necessary to separate the visual en-
+coder pre-training from depth and intrinsics learning in two stages. In the first stage, the ego-motion
+is predicted based on consecutive frames as does in conventional depth estimation frameworks (Go-
+dard et al., 2017; 2019). In the second stage, the future ego-motion is estimated based on the single
+frame by a visual encoder, and could be optimized with the depth and camera intrinsics network
+well-learned in the first stage. As such, the visual encoder is capable of inferring future ego-motion
+based on current input alone. The pre-trained visual encoder could be well adopted for downstream
+driving tasks since it captures driving policy related information. As a side product, the depth and
+1We use end-to-end autonomous driving and visuomotor autonomous driving interchangeably in this paper.
+2
+
+Irrelevant Object
+Deterministic Signal
+Light/Weather Variation
+(a)
+(b)
+(c)𝐼𝑡+1
+𝐼𝑡
+PoseNet
+DepthNet
+Visual Encoder 
+(Our Focus)
+Depth 𝐷𝑡
+(a) Self-supervised Visuomotor Policy Pre-training
+(b) Downstream Tasks
+Intrinsic K
+Ego Motion T
+Photometric 
+Reconstruction
+Ego Motion T
+Photometric 
+Reconstruction
+𝐼𝑡
+a.1 Stage One
+a.2 Stage Two
+- Single frame input
+- Since a car is ahead
+- We need to STOP
+- Consecutive frames input
+- Since frames barely change
+- We need to STOP
+frozen
+Visual Encoder 
+(Fine-tuned)
+Policy Learning
+Visual Input
+Figure 2: Overview of PPGeo. (a) We focus on pre-training an effective visual encoder to encode
+driving policy related information by predicting ego-motion based on single frame input (a.2 Stage
+Two). As achieving such a goal without labels is non-trivial, the visual encoder is obtained with the
+aid of a preceding procedure (a.1 Stage One) with temporal inputs and two sub-networks (pose and
+depth). In this illustrative example, the ego-vehicle needs to take action of STOP. The ego-motion
+in (a.1) is inferred by judging two consecutive frames barely change; whilst the ego-motion in (a.2)
+is predicted based on single visual input - focusing on driving policy related information. As such,
+the visual encoder could be fine-tuned and applied to a wide span of downstream tasks in (b).
+pose networks could be utilized as new initial weights for depth and odometry estimation tasks,
+bringing in an additional performance gain. To sum up, our key contributions are three-fold:
+• We propose a pre-training paradigm curated for various visuomotor driving tasks. To the best of
+our knowledge, this is the first attempt to achieve a fully self-supervised framework without any
+need of pseudo-labels2, leveraging the effect of pre-training by large-scale data to the full extent.
+• We devise a visual encoder capable of predicting ego-motion based on single visual input, being
+able to extract feature representations closely related to driving policy. Such a design of visual
+encoder is flexible to extend to various downstream applications.
+• We demonstrate the superiority of our approach on a set of end-to-end driving scenarios, covering
+different types and difficulty levels. The performance in terms of various metrics is improved from
+2% to even over 100% in challenging cases with very limited data.
+2
+METHODOLOGY
+2.1
+OVERVIEW
+The visuomotor policy learning for autonomous driving targets generating a policy π, such that it
+makes driving decisions, e.g., control actions or planned trajectory, from visual observation x. Our
+goal is to pre-train a visual encoder φ(x), which maps the raw image input to a compact repre-
+sentation containing important information for driving decision making. The representation is then
+utilized by the policy π(φ(x)) to perform driving tasks. As shown in Fig. 2, our pre-training method
+pre-trains the visual encoder on unlabeled driving videos via two stages in a self-supervised manner.
+2.2
+TWO-STAGE SELF-SUPERVISED TRAINING
+Stage One: Self-supervised Geometric Modeling. During the first stage, given a target image It
+and source images It′ in a sequence, we jointly estimate the depth of the target image, the intrinsics
+of the camera, and the 6-DoF ego-motion between these two frames. Given the estimations, we are
+able to model the 3D geometry of the scene, and reconstruct the target image by projecting pixels in
+2Pseudo-labels here mean using another model trained on additional labeled data to create “artificial” labels
+for the unlabeled dataset.
+3
+
+the source images. Formally, the pixel-wise correspondence between It and It′ is calculated as:
+pt′ = KTt→t′Dt(pt)K−1pt,
+(1)
+where pt and pt′ are the homogeneous coordinates of the pixel in It and It′ respectively, K is the
+predicted camera intrinsic matrix, and Dt(pt) represents the predicted depth value at pixel pi in
+It. With this relationship, the target image It′→t could be reconstructed with pixels in It′, and be
+optimized by the photometric reconstruction error. Following Godard et al. (2019), we choose two
+images adjacent to the current frame as the source images, i.e., t′ ∈ {t − 1, t + 1}.
+The DepthNet consists of a common encoder-decoder structure (Godard et al., 2019) and estimates
+the depth map of the input image. Two images are stacked together and fed into the encoder of
+the PoseNet, whose bottleneck feature is then utilized to predict the camera intrinsics and the ego-
+motion via two separate MLP-based heads. For camera intrinsics estimation, optical center (cx, cy)
+and focal lengths fx, fy are regressed similarly as in Gordon et al. (2019); Chanduri et al. (2021).
+Stage Two: Visuomotor Policy Pre-training. After the first stage of training, the DepthNet and
+PoseNet are well trained and fitted to the driving video data. Then, in the second stage, we replace
+the PoseNet for ego-motion estimation with the visual encoder φ(x) prepared for downstream driv-
+ing policy learning tasks. Now the visual encoder only takes a single frame image as input and
+predicts ego-motion between the current frame and subsequent frame.
+Specifically, the visual encoder estimates the ego-motion Tt→t+1 based on It alone and Tt→t−1
+based on It−1 followed by an inverse operation, respectively. The visual encoder is optimized
+by the photometric reconstruction error similar to the first stage, aside from a modification where
+the DepthNet and the intrinsics estimation are frozen and not backpropagated. This is empirically
+observed towards better performance. By doing so, the visual encoder is enforced to learn the actual
+driving policy, since the ego-motion between two consecutive frames is straightforwardly related to
+the driving decision or action taken at the current timestamp.
+One might argue that the PoseNet trained in the first stage could provide pseudo motion labels, with
+which the visual encoder could be directly supervised. However, the ego-motion predicted from
+the PoseNet is too sparse compared with the geometric projection approach. In our pipeline, every
+pixel provides supervision for the visual encoder so that inaccurate depth estimation in some pixels
+could be mitigated by the accurate ones, i.e., it constructs a “global” optimization. In contrast, direct
+supervision from the PoseNet would be greatly affected by the undesirable prediction inaccuracy
+and noise results. This is especially true for diverse uncalibrated online videos (Zhang et al., 2022a).
+Thus far, the backbone of visual encoder φ(x) has gained knowledge about the driving policy from
+the diverse driving videos. It can then be applied to downstream visuomotor autonomous driving
+tasks as the initial weights. Besides, the DepthNet and PoseNet trained on this large corpus of
+uncalibrated video data could also be utilized in depth and odometry estimation tasks.
+2.3
+LOSS FUNCTION
+Following Godard et al. (2019), the loss function is comprised of the photometric loss and the
+smoothness loss. The photometric error is comprised of an ℓ1 term and an SSIM (structural similarity
+index measure) term (Wang et al., 2004):
+ℓpe = α
+2 (1 − SSIM(It, It′→t)) + (1 − α)ℓ1(It, It′→t),
+(2)
+where we set α = 0.85 following the practice (Godard et al., 2017; 2019). The smooth loss is:
+ℓs = |∂xd∗
+t |e−|∂xIt| + |∂yd∗
+t |e−|∂yIt|,
+(3)
+where d∗
+t is the mean-normalized inverse depth map. We also adopt the minimum reprojection loss
+and auto-masking scheme (Godard et al., 2019) to improve self-supervised depth estimation.
+3
+EXPERIMENTS
+All pre-training experiments are conducted on the hours-long unlabeled YouTube driving
+videos (Zhang et al., 2022b). It covers different driving conditions e.g., geographical locations and
+weather. We sample 0.8 million frames in total at 1 Hz for training. For the first stage in PPGeo
+4
+
+pipeline, we train the model for 30 epochs by Adam (Kingma & Ba, 2015) optimizer with a learning
+rate of 10−4 which drops to 10−5 after 25 epochs. For the second stage, the encoder is trained for 20
+epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer. A cyclic learning rate scheduler
+is applied with the learning rate ranging from 10−6 to 10−4. The batch size for both stages is 128.
+We use data augmentations including ColorJitter, RamdomGrayScale, and GaussianBlur.
+3.1
+DESCRIPTION ON COMPARED BASELINES
+We use ResNet-34 (He et al., 2016) as the encoder and load different pre-trained weights for the
+initialization of downstream tasks. We compare PPGeo with pre-training methods including:
+Random. We use the default Kaiming initialization (He et al., 2015) for convolution layers and
+constant initialization for batchnorms.
+ImageNet. We use the model weight provided by Torchvision (Marcel & Rodriguez, 2010), which
+is pre-trained with the classification task on ImageNet (Deng et al., 2009).
+MIM. The model is pre-trained with the masked image modeling method on the YouTube driving
+video, which tries to reconstruct images with random masked-out patches. SimMIM (Xie et al.,
+2022) is adopted as it is suitable for convolutional networks.
+MoCo. We pre-train the model using MoCo-v2 (Chen et al., 2020c) on the YouTube driving videos.
+We exclude RandomResizedCrop and RandomHorizontalFlip augmentations as they are not suitable
+for the driving task.
+ACO. Following Zhang et al. (2022b), it is pre-trained using action-conditioned contrastive learning
+on the YouTube driving videos. ACO trains an inverse dynamic model to generate pseudo steer
+labels for driving videos, based on which steer-based discrimination is added on top of MoCo-v2.
+SelfD. SelfD (Zhang et al., 2022a) is not a pre-training method strictly since it needs to train the
+whole policy model on the driving video for each task, while other pre-training methods aforemen-
+tioned provide a general pre-training visual model for all tasks. We still include it for comparison
+due to its close relationship to our target. Specifically, we follow Zhang et al. (2022a) to train the
+model for each task with the following pipeline: training on the task data → training on the YouTube
+data with pseudo-label → fine-tuning on the task data.
+3.2
+DESCRIPTION ON DOWNSTREAM AUTONOMOUS DRIVING TASKS
+We carry out experiments under (1) three imitation learning based closed-loop driving tasks in
+CARLA (Dosovitskiy et al., 2017), (2) one reinforcement learning based driving task in CARLA,
+and (3) an open-loop planning task on real-world autonomous driving dataset nuScenes (Caesar
+et al., 2020), to fully validate the effectiveness of PPGeo. We briefly describe each task below.
+Navigation.
+It corresponds to the goal-conditioned navigation task in the CoRL2017 bench-
+mark (Dosovitskiy et al., 2017). The agent is trained in Town01 and tested in Town02 with unseen
+weather, and there are no other traffic participants. We use different sizes of training data (from
+4K to 40K) following Zhang et al. (2022b) to evaluate the generalization ability of pre-trained vi-
+sual encoders when labeled data is limited and conduct the closed-loop evaluation. The evaluation
+metric is success rate, denoting the portion of 50 pre-defined routes finished without any collision.
+And traffic lights are ignored here. CILRS (Codevilla et al., 2019), a classic image based end-to-end
+autonomous driving model, is adopted for training and evaluation.
+Navigation Dynamic. This is the navigation dynamic task in the CoRL2017 benchmark (Dosovit-
+skiy et al., 2017). The setting differentiates from Navigation that there are other dynamic objects
+such as randomly generated vehicles, which substantially increases the difficulty of driving safety.
+Leaderboard Town05-long. This challenging and realistic benchmark corresponds to the Leader-
+Board benchmark (CARLA, 2022). We collect 40K training data in Town01, 03, 04, 06 and eval-
+uate on 10 routes in the unseen Town05 (Prakash et al., 2021). Due to the challenging scenarios
+in this task, we evaluate different pre-training approaches with the state-of-the-art image-based au-
+tonomous driving model TCP (Wu et al., 2022). The major metrics of this task are Driving Score,
+Route Completion, and Infraction Score (all the higher the better). Route Completion denotes the
+portion of the route completed by the agent. Infraction Score is the number of infractions made
+5
+
+Table 1: The Successful Rate results of the closed-loop Navigation task (mean by 3 random trials).
+Pre-train Method
+Navigation - # of training samples
+10% (4K)
+20% (8K)
+40% (16K)
+100% (40K)
+Random
+0.0 ± 0.0
+9.6 ± 5.2
+15.3 ± 4.5
+73.3 ± 2.3
+ImageNet
+24.7± 2.0
+42.0 ± 2.0
+69.3 ± 6.4
+87.3 ± 4.6
+MIM
+4.7 ± 1.2
+8.0 ± 0.0
+31.3 ± 2.3
+57.3 ± 3.1
+MoCo
+7.7 ± 2.1
+39.3 ± 9.2
+48.7 ± 4.2
+69.3 ± 1.2
+ACO
+24.0 ± 2.0
+44.0 ± 1.2
+71.3 ± 1.2
+92.0 ± 3.5
+SelfD
+12.0± 0.0
+32.0 ± 0.0
+50.7 ± 2.3
+62.7 ± 1.2
+PPGeo (ours)
+42.0 ± 2.0
+73.3 ± 6.1
+91.3 ± 1.2
+96.7 ± 1.2
+along the route including pedstrain collisions, vehicle collisions, red light infractions, etc. And the
+main metric Driving Score is the product of Route Completion and Infraction Score.
+Reinforcement Learning. Proximal Policy Optimization (PPO) (Schulman et al., 2017) is used
+to train the CILRS (Codevilla et al., 2019) model initialized with different pre-trained weights in
+CARLA Town01 environment. The reward shaping details follow Roach (Zhang et al., 2021). We
+also conduct experiments to freeze the pre-trained visual encoder during training to further study the
+effectiveness of the pre-trained feature representations.
+nuScenes Planning. This task involves trajectory planning in real-world dataset nuScenes (Caesar
+et al., 2020). Given the current visual input, the model plans a 3-second trajectory (0.5 Hz), and the
+planned trajectory is compared with the ground truth log. We also calculate the collision rate, where
+a collision is defined as overlaps with future vehicles and pedestrians based on planned waypoints.
+The metric of this tasks includes (1) the L2 distance between predicted trajectory and ground truth
+trajectory, and (2) the collision rate. Metrics are measured at different time lengths from 1s to 3s.
+The planning model used here is comprised of a visual encoder and a GRU-based planner to predict
+each waypoint auto-regressively. We use the official train-val split for training and evaluation.
+3.3
+NUMERIC COMPARISON ON DOWNSTREAM TASKS
+For imitation learning based closed-loop driving tasks, the evaluation results are shown in Table 1-
+3. We present the plot between episode return and environment steps of each method in Fig. 3 for
+the reinforcement learning experiments. The open-loop nuScenes planning results are provided in
+Table 4. We could observe that PPGeo outperforms other baselines by a large margin in all tasks.
+Note that the model is tested under a different number of fine-tuning samples from 10% (4K) to full
+40K in the Navigation and Navigation Dynamic tasks. In the case of the particularly small size of
+training samples, PPGeo still demonstrates competitive performance and has a larger improvement
+gap of over 100%. This validates the generalization ability of the pre-trained visual encoder, which
+is important when adapting to a new environment with very limited labeled data. In the more chal-
+lenging and real-world style Leaderboard Town05-long task in Table 3, the model pre-trained with
+our method achieves the highest driving score and infraction score. PPGeo well handles cases where
+the agent needs to stop, leading to much fewer vehicle collisions and red light infractions.
+Since ACO considers steering angles only during pre-training, its performance degrades on more
+challenging scenarios where brake and throttles are also important. SelfD performs slightly better
+than ACO in complex cases while it significantly degenerates when the task data is limited, as
+affected by the unsatisfying pseudo labeling model. ImageNet pre-training also shows competitive
+performance, which might credit to its ability of finding salient objects in the scene when the input
+contains little irrelevant information (see examples in Sec. 3.5).
+3.4
+DEPTH AND ODOMETRY ESTIMATION
+In this part, we explore whether the large-scale training on uncalibrated data could benefit the depth
+and odometry estimation models as well and validate the effectiveness of first-stage training. Specif-
+ically, we employ the DepthNet and PoseNet trained after the first stage as initial weights for Mon-
+odepthv2 (Godard et al., 2019), and conduct experiments on KITTI (Geiger et al., 2012). Results
+in Table 5 indicate that pre-training on large-scale driving videos could bring performance improve-
+6
+
+Table 2: The Successful Rate results of the closed-loop Navigation Dynamic (mean by 3 random
+trials).
+Pre-train Method
+Navigation Dynamic - # of training samples
+10% (4K)
+20% (8K)
+40% (16K)
+100% (40K)
+Random
+0.0 ± 0.0
+2.0 ± 0.0
+10.0 ± 0.0
+32.0 ± 8.0
+ImageNet
+10.7± 1.2
+28.7 ± 5.0
+64.7 ± 2.3
+72.7 ± 1.2
+MIM
+7.3 ± 1.2
+10.3 ± 2.5
+14.7 ± 3.1
+58.7 ± 1.2
+MoCo
+4.7 ± 1.2
+12.0 ± 4.0
+28.0 ± 5.3
+66.7 ± 2.3
+ACO
+8.0 ± 1.2
+12.0 ± 0.0
+22.0 ± 2.0
+47.3 ± 5.0
+SelfD
+8.0 ± 0.0
+29.3 ± 1.2
+38.0 ± 1.6
+59.3 ± 6.4
+PPGeo (ours)
+23.3 ± 1.2
+34.0 ± 5.3
+71.3 ± 1.2
+84.0 ± 5.3
+Table 3: Closed-loop Leaderboard Town05-long task results. Besides three main metrics, infraction
+details are also reported (all the lower the better). Evaluation repeats 3 times with the mean reported.
+Pre-train
+Method
+Driving
+Score
+Infraction
+Score
+Route
+Completion
+Collisions
+pedestrian
+Collisions
+vehicle
+Collisions
+layout
+Off-road
+violations
+Agent
+blocked
+Red light
+violations
+Random
+33.50±1.67
+0.65±0.02
+60.49±2.93
+0.09±0.07
+1.16±0.40
+0.00±0.00
+0.44±0.13
+0.97±0.09
+0.53±0.12
+ImageNet
+41.29±3.20
+0.77±0.03
+57.52±4.87
+0.00±0.00
+0.71±0.20
+0.11±0.15
+0.15±0.01
+1.01±0.16
+0.29±0.10
+MIM
+36.39±0.21
+0.72±0.04
+61.75±2.26
+0.14±0.11
+0.91±0.12
+0.04±0.07
+0.18±0.17
+0.87±0.03
+0.14±0.11
+MoCo
+32.10±2.04
+0.65±0.02
+64.09±4.01
+0.13±0.11
+0.79±0.16
+0.00±0.00
+0.49±0.07
+0.81±0.15
+0.45±0.13
+ACO
+33.05±3.05
+0.67±0.06
+59.52±3.21
+0.00±0.00
+0.69±0.28
+0.05±0.07
+0.54±0.05
+0.94±0.08
+0.73±0.10
+SelfD
+38.76±3.02
+0.65±0.03
+68.72±7.36
+0.17±0.07
+0.84±0.18
+0.00±0.00
+0.32±0.03
+0.75±0.15
+0.12±0.08
+PPGeo
+47.44±5.63
+0.79±0.08
+65.05±5.11
+0.04±0.05
+0.54±0.29
+0.00±0.00
+0.16±0.11
+0.76±0.10
+0.04±0.05
+100
+200
+300
+400
+500
+600
+700
+800
+Steps (K)
+100
+0
+100
+200
+300
+400
+500
+Episode Return
+Visual Encoder Fine-tuning
+ImageNet
+MoCo
+ACO
+PPGeo
+100
+200
+300
+400
+500
+600
+700
+800
+Steps (K)
+100
+200
+300
+400
+Episode Return
+Visual Encoder Frozen
+ImageNet
+MoCo
+ACO
+PPGeo
+Figure 3: Learning curves of the RL agents using PPGeo and three other best pre-training baselines.
+Left: the pre-trained visual encoder is jointly fine-tuned during RL training; Right: the visual en-
+coder is frozen during RL training. The episode return is the mean with standard deviation in shade
+across three runs with different random seeds.
+Table 4: Open-loop nuScenes planning results. We evaluate the ℓ2 distance between model predic-
+tions and the ground truth trajectory and collision rate in horizons from 1 second to 3 seconds.
+Pre-train Method
+L2 (m) ↓
+Collision Rate (%) ↓
+1s
+2s
+3s
+1s
+2s
+3s
+Random
+1.621
+2.722
+3.851
+0.550
+1.779
+3.375
+ImagNet
+1.331
+2.202
+3.086
+0.315
+0.550
+1.366
+MIM
+1.412
+2.357
+3.331
+0.297
+0.622
+1.507
+MoCo
+1.528
+2.545
+3.585
+0.560
+1.235
+2.390
+ACO
+1.496
+2.496
+3.519
+0.446
+1.178
+2.223
+SelfD
+1.419
+2.359
+3.316
+0.353
+0.923
+2.044
+PPGeo (ours)
+1.302
+2.154
+3.018
+0.270
+0.425
+0.941
+7
+
+Table 5: Improvement from our pre-training method on depth and odometry estimation tasks.
+Pre-train
+Method
+Depth Estimation
+Odometry Estimation
+abs rel ↓
+sq rel ↓
+rmse ↓
+rmse log ↓
+a1 ↑
+a2 ↑
+a3 ↑
+Sequence 09 ↓
+Sequence 10 ↓
+ImageNet
+0.118
+0.902
+4.873
+0.196
+0.871
+0.958
+0.981
+0.017±0.010
+0.015±0.010
+PPGeo
+0.114
+0.805
+4.599
+0.186
+0.874
+0.962
+0.984
+0.016±0.009
+0.013±0.009
+Ours
+ACO
+ImageNet
+MoCo
+Origin
+Figure 4: Eigen-Cam (Muhammad & Yeasin, 2020) activation maps of the learned representation
+from different pre-training methods on the driving video data.
+Table 6: Ablative study on key designs of PPGeo on the Navigation task.
+#
+Experiment
+Navigation - # of training samples
+10% (4K)
+20% (8K)
+40% (16K)
+100% (40K)
+1
+Single stage
+24.2 ± 2.0
+53.3 ± 1.2
+79.3 ± 4.2
+92.7 ± 2.3
+2
+No frozen in 2nd stage
+32.7 ± 1.2
+58.0 ± 2.0
+86.0 ± 2.1
+92.0 ± 2.0
+3
+PoseNet direct supervision
+18.0 ± 2.0
+52.0 ± 2.0
+76.7 ± 1.2
+90.0 ± 0.0
+4
+PPGeo
+42.0 ± 2.0
+73.3 ± 6.1
+91.3 ± 1.2
+96.7 ± 1.2
+ment to both depth and odometry estimation tasks, which is an additional harvest of our pre-training
+framework. We refer readers to Godard et al. (2019) for details about the metrics of these tasks.
+3.5
+VISUALIZATION RESULTS
+Here we provide heatmaps of the feature representations learned by different pre-training methods
+using Eigen-Cam (Muhammad & Yeasin, 2020) to show the attended regions in Fig. 4. In many
+cases (Row 1&2), our model mainly concentrates on the lane in front of the ego vehicle, which is
+highly related to driving. And our model PPGeo well captures the specific cues causing the brake
+action including front vehicles (Row 3&4) and traffic lights (Row 5). We also observe that the model
+pre-trained with ImageNet classification tends to capture salient objects in the image. This is helpful
+when the salient objects are straightforwardly related to the driving decision (Row 4); but it may
+focus on wrong objects when the input contains other irrelevant information (Row 2&3).
+3.6
+ABLATIVE STUDY
+We conduct ablative study as to different designs of PPGeo on the Navigation task in Table 6. Train-
+ing the visual encoder and DepthNet in a single stage simultaneously (Row 1) leads to an inferior
+performance, indicating that it is quite challenging for the visual encoder to learn the correct ego-
+motion if depth estimation is also trained from scratch. Moreover, jointly optimizing the DepthNet
+in the second stage (Row 2, not frozen) degrades the depth estimation quality and harms the per-
+formance. In Row 3, we observe that utilizing the PoseNet obtained in the first stage to provide
+8
+
+pseudo label supervision directly leads to inferior results, since an inaccurate pseudo label impairs
+the learning process to great extent.
+4
+RELATED WORK
+Pre-training for NLP and General Vision. Pre-training or representation learning has proved to be
+an essential key to the success of artificial intelligence. In the field of Natural Language Processing
+(NLP), with the powerful capability of Transformer (Vaswani et al., 2017), pre-training on large-
+scale datasets with large models then fine-tuning on downstream tasks has become the dominant
+paradigm (Kenton & Toutanova, 2019; Brown et al., 2020). As for the field of Computer Vision,
+training specific downstream tasks with the supervised pre-trained weights of visual encoder via
+ImageNet classification task is widely adopted. Recently, unsupervised and self-supervised learn-
+ing methods such as contrastive learning (He et al., 2020; Chen et al., 2020c;b) and masked im-
+age modeling (Bao et al., 2021; He et al., 2022; Xie et al., 2022; Peng et al., 2022) have gained
+impressive improvement over ImageNet pre-training on various vision benchmarks. Very recent
+vision-language co-training approaches (Radford et al., 2021; Wang et al., 2022) demonstrate their
+extraordinary potential in the domain of multi-modal learning and applications. Yet, these generic
+representation learning methods adopt various data augmentation techniques to achieve translation
+and view invariance, while visuomotor driving sets in a highly dynamic environment. In this work,
+we show that the ever-victorious pre-training methods may not be the optimal choice, and introduce
+a curated paradigm for visuomotor driving policy learning.
+Pre-training for Visuomotor Applications. Learning a control policy directly from raw visual
+input is challenging since the model needs to reason about visual pixels and dynamic behaviors
+simultaneously. Moreover, training visuomotor models from scratch usually requires tons of labeled
+data or environment interactions. To this end, recently, Shah & Kumar (2021) shows that feature
+representations from ResNet (He et al., 2016) pre-trained on ImageNet classification is helpful for
+RL-based dexterous manipulation tasks. Parisi et al. (2022) conducts extensive experiments on
+applying “off-the-shelf” pre-trained vision models in diverse control domains and validates their
+benefits to train control policies. CLIP (Radford et al., 2021) is also adopted in some embodied
+AI and robot navigation problems (Shah et al., 2022). Besides borrowing pre-trained weights for
+visuomotor tasks, researchers in robotics now desire a paradigm learning policy representations
+from raw data directly. Xiao et al. (2022); Radosavovic et al. (2022); Seo et al. (2022); Gupta et al.
+(2022) inherit the MIM spirit to realize visual pre-training for control tasks. Yang & Nachum (2021)
+investigates unsupervised representation learning objectives from D4RL environments (Fu et al.,
+2020), and Yamada et al. (2022) further adopts task-induced approaches to learn from prior tasks.
+However, compared with visuomotor driving, the visual inputs of such control tasks are less diverse
+which usually concentrate on objects and are much more compact.
+To our best knowledge, ACO (Zhang et al., 2022b) is the only pre-training method customized
+for autonomous driving. By first training an inverse dynamic model on nuScenes (Caesar et al.,
+2020), they get pseudo steer labels of the driving videos and then construct the steer-conditioned
+discrimination for contrastive learning following MoCo. However, ACO ignores other crucial driv-
+ing factors such as throttle and brakes, and its performance is largely limited by the inverse dynamic
+model. SelfD (Zhang et al., 2022a) is not strictly designed for pre-training while it also makes
+use of vast amounts of videos to learn driving policies via semi-supervised learning. It acquires the
+pseudo labeling knowledge from the target domain. These two methods both depend on the accuracy
+of pseudo labeling. In contrast, we realize fully self-supervised learning through dense geometric
+reconstruction, evading the possible adverse effect.
+Policy Learning for Autonomous Driving. Visuomotor autonomous driving learns a driving pol-
+icy directly from sensor inputs in an end-to-end manner (Codevilla et al., 2018; 2019; Liang et al.,
+2018; Chen et al., 2020a; Prakash et al., 2021; Chen et al., 2021; Wu et al., 2022; Shao et al., 2022).
+In essence, the inherent difficulty of the urban-style autonomous driving tasks makes such meth-
+ods data-hungry. Interfuser (Shao et al., 2022), the current top-1 method on the CARLA Leader-
+board (CARLA, 2022), requires 3 million labeled data samples for imitation learning (behavior
+cloning specifically). RL-based model MaRLn (Toromanoff et al., 2020) needs 20 million environ-
+ment steps of interaction. The sample efficiency problem greatly impedes the real-world application
+of such approaches. In this work, we propose a self-supervised pre-training pipeline to learn driving
+9
+
+policy related representations on unlabeled driving videos, and pave the way for these visuomotor
+autonomous driving models to further achieve satisfying performance.
+5
+CONCLUSION AND DISCUSSION
+In this work, we have proposed a fully self-supervised visuomotor driving policy pre-training
+paradigm PPGeo by modeling the 3D geometry of large-scale unlabeled driving videos. Taking
+a direct approach to infer the ego-motion and benefiting from the two-stage pre-training pipeline,
+we enable the visual encoder to learn driving policies based on single visual input. Our method out-
+performs the peer pre-training approaches by a large margin on a series of visuomotor driving tasks.
+For its limitation, our method currently only considers the ego-motion for a single time step, and a
+future direction is to devise the framework to perform multi-step motion prediction which contains
+more information about driving decisions.
+REFERENCES
+Hangbo Bao, Li Dong, Songhao Piao, and Furu Wei. Beit: Bert pre-training of image transformers.
+In ICLR, 2021. 9
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal,
+Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are
+few-shot learners. In NeurIPS, 2020. 9
+Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush
+Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for
+autonomous driving. In CVPR, 2020. 2, 5, 6, 9
+CARLA. CARLA autonomous driving leaderboard. https://leaderboard.carla.org/,
+2022. 5, 9
+Sai Shyam Chanduri, Zeeshan Khan Suri, Igor Vozniak, and Christian M¨uller. Camlessmonodepth:
+Monocular depth estimation with unknown camera parameters. arXiv preprint arXiv:2110.14347,
+2021. 4
+Annie S Chen, Suraj Nair, and Chelsea Finn. Learning generalizable robotic reward functions from”
+in-the-wild” human videos. In RSS, 2021. 9
+Dian Chen, Brady Zhou, Vladlen Koltun, and Philipp Kr¨ahenb¨uhl. Learning by cheating. In CoRL,
+2020a. 9
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for
+contrastive learning of visual representations. In ICML, 2020b. 9
+Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum
+contrastive learning. arXiv preprint arXiv:2003.04297, 2020c. 1, 5, 9
+Felipe Codevilla, Matthias M¨uller, Antonio L´opez, Vladlen Koltun, and Alexey Dosovitskiy. End-
+to-end driving via conditional imitation learning. In ICRA, 2018. 9
+Felipe Codevilla, Eder Santana, Antonio M L´opez, and Adrien Gaidon. Exploring the limitations of
+behavior cloning for autonomous driving. In ICCV, 2019. 5, 6, 9, 14
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale
+hierarchical image database. In CVPR, 2009. 1, 5
+Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun. CARLA:
+An open urban driving simulator. In CoRL, 2017. 5
+Lasse Espeholt, Hubert Soyer, Remi Munos, Karen Simonyan, Vlad Mnih, Tom Ward, Yotam
+Doron, Vlad Firoiu, Tim Harley, Iain Dunning, et al. Impala: Scalable distributed deep-rl with
+importance weighted actor-learner architectures. In ICML, 2018. 1
+10
+
+Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep
+data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020. 9
+Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we ready for autonomous driving? the kitti
+vision benchmark suite. In CVPR, 2012. 6
+Cl´ement Godard, Oisin Mac Aodha, and Gabriel J. Brostow. Unsupervised monocular depth esti-
+mation with left-right consistency. In CVPR, 2017. 2, 4
+Cl´ement Godard, Oisin Mac Aodha, Michael Firman, and Gabriel J. Brostow. Digging into self-
+supervised monocular depth prediction. In ICCV, 2019. 2, 4, 6, 8, 14
+Ariel Gordon, Hanhan Li, Rico Jonschkowski, and Anelia Angelova. Depth from videos in the wild:
+Unsupervised monocular depth learning from unknown cameras. In ICCV, 2019. 4
+Agrim Gupta, Stephen Tian, Yunzhi Zhang, Jiajun Wu, Roberto Mart´ın-Mart´ın, and Li Fei-Fei.
+Maskvit: Masked visual pre-training for video prediction.
+arXiv preprint arXiv:2206.11894,
+2022. 9
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing
+human-level performance on imagenet classification. In ICCV, 2015. 5
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog-
+nition. In CVPR, 2016. 5, 9, 14
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick.
+Momentum contrast for
+unsupervised visual representation learning. In CVPR, 2020. 1, 9
+Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Doll´ar, and Ross Girshick.
+Masked
+autoencoders are scalable vision learners. In CVPR, 2022. 1, 9
+Matteo Hessel, Joseph Modayil, Hado Van Hasselt, Tom Schaul, Georg Ostrovski, Will Dabney, Dan
+Horgan, Bilal Piot, Mohammad Azar, and David Silver. Rainbow: Combining improvements in
+deep reinforcement learning. In AAAI, 2018. 1
+Jacob Devlin Ming-Wei Chang Kenton and Lee Kristina Toutanova. Bert: Pre-training of deep
+bidirectional transformers for language understanding. In NAACL-HLT, 2019. 9
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015. 5
+Misha Laskin, Kimin Lee, Adam Stooke, Lerrel Pinto, Pieter Abbeel, and Aravind Srinivas. Rein-
+forcement learning with augmented data. In NeurIPS, 2020. 1
+Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep visuo-
+motor policies. JMLR, 2016. 1
+Xiaodan Liang, Tairui Wang, Luona Yang, and Eric Xing. Cirl: Controllable imitative reinforcement
+learning for vision-based self-driving. In ECCV, 2018. 9
+Ilya Loshchilov and Frank Hutter.
+Decoupled weight decay regularization.
+arXiv preprint
+arXiv:1711.05101, 2017. 5
+S´ebastien Marcel and Yann Rodriguez.
+Torchvision the machine-vision package of torch.
+In
+ACMMM, 2010. 5
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Belle-
+mare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level
+control through deep reinforcement learning. Nature, 2015. 1
+Mohammed Bany Muhammad and Mohammed Yeasin. Eigen-cam: Class activation map using
+principal components. In IJCNN, 2020. 8
+Simone Parisi, Aravind Rajeswaran, Senthil Purushwalkam, and Abhinav Kumar Gupta. The unsur-
+prising effectiveness of pre-trained vision models for control. In ICML, 2022. 1, 2, 9
+11
+
+Zhiliang Peng, Li Dong, Hangbo Bao, Qixiang Ye, and Furu Wei. Beit v2: Masked image modeling
+with vector-quantized visual tokenizers. arXiv preprint arXiv:2208.06366, 2022. 9
+Aditya Prakash, Kashyap Chitta, and Andreas Geiger. Multi-modal fusion transformer for end-to-
+end autonomous driving. In CVPR, 2021. 5, 9
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal,
+Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual
+models from natural language supervision. In ICML, 2021. 1, 9
+Ilija Radosavovic, Tete Xiao, Stephen James, Pieter Abbeel, Jitendra Malik, and Trevor Darrell.
+Real world robot learning with masked visual pre-training. In CoRL, 2022. 1, 2, 9
+Mamshad Nayeem Rizve, Kevin Duarte, Yogesh S Rawat, and Mubarak Shah. In defense of pseudo-
+labeling: An uncertainty-aware pseudo-label selection framework for semi-supervised learning.
+In ICLR, 2020. 2
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy
+optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. 6
+Younggyo Seo, Danijar Hafner, Hao Liu, Fangchen Liu, Stephen James, Kimin Lee, and Pieter
+Abbeel. Masked world models for visual control. arXiv preprint arXiv:2206.14244, 2022. 9
+Dhruv Shah, Blazej Osinski, Brian Ichter, and Sergey Levine. Lm-nav: Robotic navigation with
+large pre-trained models of language, vision, and action. In CoRL, 2022. 1, 9
+Rutav Shah and Vikash Kumar. Rrl: Resnet as representation for reinforcement learning. In ICML,
+2021. 1, 9
+Hao Shao, Letian Wang, Ruobing Chen, Hongsheng Li, and Yu Liu. Safety-enhanced autonomous
+driving using interpretable sensor fusion transformer. In CoRL, 2022. 9
+Marin Toromanoff, Emilie Wirbel, and Fabien Moutarde. End-to-end model-free reinforcement
+learning for urban driving using implicit affordances. In CVPR, 2020. 1, 9
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez,
+Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. NeurIPS, 2017. 9
+Wenhui Wang, Hangbo Bao, Li Dong, Johan Bjorck, Zhiliang Peng, Qiang Liu, Kriti Aggarwal,
+Owais Khan Mohammed, Saksham Singhal, Subhojit Som, and Furu Wei. Image as a foreign lan-
+guage: Beit pretraining for all vision and vision-language tasks. arXiv preprint arXiv:2208.10442,
+2022. 9
+Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment:
+from error visibility to structural similarity. TIP, 2004. 4
+Erik Wijmans, Abhishek Kadian, Ari Morcos, Stefan Lee, Irfan Essa, Devi Parikh, Manolis Savva,
+and Dhruv Batra. Dd-ppo: Learning near-perfect pointgoal navigators from 2.5 billion frames.
+arXiv preprint arXiv:1911.00357, 2019. 1
+Penghao Wu, Xiaosong Jia, Li Chen, Junchi Yan, Hongyang Li, and Yu Qiao. Trajectory-guided
+control prediction for end-to-end autonomous driving: A simple yet strong baseline. In NeurIPS,
+2022. 5, 9, 14
+Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-
+parametric instance discrimination. In CVPR, 2018. 2
+Tete Xiao, Ilija Radosavovic, Trevor Darrell, and Jitendra Malik. Masked visual pre-training for
+motor control. arXiv preprint arXiv:2203.06173, 2022. 1, 9
+Zhenda Xie, Zheng Zhang, Yue Cao, Yutong Lin, Jianmin Bao, Zhuliang Yao, Qi Dai, and Han Hu.
+Simmim: A simple framework for masked image modeling. In CVPR, 2022. 1, 5, 9
+Jun Yamada, Karl Pertsch, Anisha Gunjal, and Joseph J Lim. Task-induced representation learning.
+In ICLR, 2022. 2, 9
+12
+
+Mengjiao Yang and Ofir Nachum. Representation matters: offline pretraining for sequential decision
+making. In ICML, 2021. 9
+Denis Yarats, Ilya Kostrikov, and Rob Fergus. Image augmentation is all you need: Regularizing
+deep reinforcement learning from pixels. In ICLR, 2020. 1
+Jimuyang Zhang, Ruizhao Zhu, and Eshed Ohn-Bar. Selfd: Self-learning large-scale driving policies
+from the web. In CVPR, 2022a. 2, 4, 5, 9
+Qihang Zhang, Zhenghao Peng, and Bolei Zhou. Learning to drive by watching youtube videos:
+Action-conditioned contrastive policy pretraining. In ECCV, 2022b. 2, 4, 5, 9
+Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In ECCV, 2016. 2
+Zhejun Zhang, Alexander Liniger, Dengxin Dai, Fisher Yu, and Luc Van Gool. End-to-end urban
+driving by imitating a reinforcement learning coach. In ICCV, 2021. 6
+13
+
+POLICY PRE-TRAINING FOR AUTONOMOUS DRIVING
+VIA SELF-SUPERVISED GEOMETRIC MODELING
+Supplementary Materials
+In this Supplementary document, we first provide detailed network structures in Sec. A. More de-
+scription and visual illustrations of the downstream tasks are discussed in Sec. B. Last, we discuss
+limitations and common failure cases in Sec. C.
+A
+NETWORK DETAILS
+For all experiments, the backbone of the visual encoder is ResNet-34 (He et al., 2016), and the
+detailed structure of it is provided in Table 7. For DepthNet and PoseNet, we follow the same model
+structure as Godard et al. (2019) with a two-layer MLP focal length head and a two-layer MLP
+optical center head added to the bottleneck of the PoseNet to predict the intrinsic matrix. Please
+refer to Godard et al. (2019) for model details.
+For the Navigation, Navigation Dynamic, and Reinforcement Learning tasks, we use CILRS (Codev-
+illa et al., 2019) and the model details are provided in Table 8. For the Leaderboard Town05-long
+task, TCP (Wu et al., 2022) is chosen as our agent, and we refer readers to Wu et al. (2022) for model
+details. For the nuScenes Planning, the trajectory planning model structure is shown in Table 9.
+Table 7: Detailed structure of the visual encoder.
+Layer Type
+Channels
+Stride
+Kernel Size
+Activation Function
+Image Encoder
+ResNet-34
+Measurement Encoder
+Conv
+256
+1
+1
+ReLU
+Conv
+256
+3
+1
+ReLU
+Conv
+256
+3
+1
+ReLU
+Conv
+6
+1
+1
+ReLU
+Average Pooling
+Table 8: Detailed structure of the CILRS model.
+Layer Type
+Dims in
+Dims out
+Activation Function
+Image Encoder
+ResNet-34
+512
+Speed Encoder
+FC
+1
+256
+ReLU
+FC
+256
+512
+-
+Speed Pred Head
+FC
+512
+256
+ReLU
+FC
+256
+256
+ReLU
+FC
+256
+256
+ReLU
+Control Pred Head
+FC
+512
+256
+ReLU
+FC
+256
+256
+ReLU
+FC
+256
+3
+Sigmoid
+14
+
+Table 9: Detailed structure of the trajectory planning model.
+Image Encoder
+ResNet-34
+Bottleneck
+Layer Type
+Dims in
+Dims out
+Activation Function
+FC
+512
+256
+ReLU
+FC
+256
+256
+-
+Decoder
+Layer Type
+Hidden dim
+Input Dim
+Output Dim
+GRU
+256
+2
+2
+B
+DOWNSTREAM TASKS DETAILS
+For Navigation and Navigation Dynamic, training data is collected in Town01, and the closed-loop
+testing is conducted in Town02. The maps of Town01 and Town02 are shown in Fig. 5. The agent
+needs to follow a series of sparse waypoints to navigate from the start point to the end point and
+avoid collisions. The difference between Navigation and Navigation Dynamic is that there are other
+dynamic vehicles and pedestrians in the town. Examples are provided in Fig. 6.
+The Leaderboard-Town05-long task is more close to real-world urban driving, with different chal-
+lenging scenarios added to the route. The map of Town05 is shown in Fig. 5.
+Town 01
+Town 02
+Town 05
+Figure 5: Maps of Town01, Town02, and Town05.
+Navigation
+Navigation Dynamic
+Figure 6: Examples of the front view image for Navigation and Navigation Dynamic tasks.
+15
+
+C
+LIMITATIONS
+In this part, we analyze some failure cases and limitations of our method. Since the visual encoder
+need to predict the future motion based on a single front-view image, there might be some factors
+that directly influence the driving decision not shown in the image (e.g., vehicles behind the ego
+vehicle, factors related to the driver, navigation information). Some of such cases are provided
+in Fig. 7. In these cases, the visual encoder does not get enough information to make the correct
+prediction. These samples during training may hamper the learning process. After training, one
+may use the difference between the prediction from PoseNet and that from visual encoder to filter
+out these samples, and re-train the visual encoder.
+𝐼𝑡
+𝐼𝑡+1
+Figure 7: Failure cases where the driving decision/future motion can not be inferred from It. For the
+cases in Row 1 and Row 2, by comparing It and It+1, we know that the ego vehicle stops. However,
+there is no clear clue in It indicating it should stop. For the case in Row 3, the ego vehicle is turning
+left, while we could hardly tell the turning direction from It alone.
+16
+

8dFAT4oBgHgl3EQfpB03/content/tmp_files/2301.08637v1.pdf.txt

@@ -0,0 +1,2871 @@
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN
+OPERATOR
+FRIEDRICH PHILIPP, MANUEL SCHALLER, KARL WORTHMANN, SEBASTIAN PEITZ, AND FELIKS N ¨USKE
+ABSTRACT. We consider the data-driven approximation of the Koopman operator for stochastic differen-
+tial equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the
+data are collected from long-term ergodic simulations. We derive both an exact expression for the variance
+of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds
+for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in
+the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS,
+we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck
+process illustrate our results.
+1. INTRODUCTION
+The Koopman operator [23] has become an essential tool in the modeling process of complex dy-
+namical systems based on simulation or measurement data. The philosophy of the Koopman approach
+is that for a (usually non-linear) dynamical system on a finite-dimensional space, the time-evolution of
+expectation values of observable functions satisfies a linear differential equation. Hence, after “lifting”
+the dynamical system into an infinite-dimensional function space of observables, linear methods become
+available for its analysis. The second step is then to notice that traditional Galerkin approximations of the
+Koopman operator can be consistently estimated from simulation or measurement data, establishing the
+fundamental connection between the Koopman approach and modern data science. Koopman methods
+have found widespread application in system identification [4], control [24, 42, 25, 17, 49], sensor place-
+ment [31], molecular dynamics [50, 44, 35, 36, 18, 56], and many other fields. We refer to [19, 33, 5] for
+comprehensive reviews of the state of the art.
+The fundamental numerical method for the Koopman approach is Extended Dynamic Mode Decom-
+position (EDMD) [54], which allows to learn a Galerkin approximation of the Koopman operator from
+finite (simulation or measurement) data on a subspace spanned by a finite set of observables, often called
+dictionary. An appropriate choice of said dictionary is a challenging problem. In light of this issue,
+representations of the Koopman operator on large approximation spaces have been considered in recent
+years, including deep neural networks [29, 32], tensor product spaces [21, 37], and reproducing kernel
+Hilbert spaces (RKHS) [55, 11, 20]. In the work [20] it was shown that by means of the integral operator
+associated to an RKHS, it is possible to construct a type of Galerkin approximation of the Koopman
+operator. The central object are (cross-)covariance operators, which can be estimated from data, using
+only evaluations of the feature map. Due to the relative simplicity of the resulting numerical algorithms
+on the one hand, and the rich approximation properties of reproducing kernels on the other hand, kernel
+methods have emerged as a promising candidate to overcome the fundamental problem of dictionary
+selection.
+A key question is the quantification of the estimation error for (compressed1) Koopman operators. For
+finite dictionaries and independent, identically distributed (i.i.d.) samples, error estimates were provided
+in [26, 38], see also [58] for the ODE case and [49] for an extension to control-affine systems. The
+1A compression of a linear operator T to a subspace M is given by PT|M, where P denotes a projection onto M.
+1
+arXiv:2301.08637v1  [math.DS]  20 Jan 2023
+
+2
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+estimation error for cross-covariance operators on kernel spaces was considered in [34], where general
+concentration inequalities were employed. The data were also allowed to be correlated, and mixing
+coefficients were used to account for the lack of independence. In this article, we take a different route
+and follow the approach of our previous paper [38], where we, in addition, also derived error estimates
+for the Koopman generator and operator for finite dictionaries and data collected from long-term, ergodic
+trajectories. This setting is relevant in many areas of science, where sampling i.i.d. from an unknown
+stationary distribution is practically infeasible, e.g., in fluid or molecular dynamics. The centerpiece of
+our results was an exact expression for the variance of the finite-data estimator, which can be bounded
+by an asymptotic variance. The asymptotic variance by itself is a highly interesting dynamical quantity,
+which can also be described in terms of Poisson equations for the generator [27, Section 3].
+We consider the Koopman semigroup (Kt)t≥0 generated by a stochastic differential equation on the
+space L2
+µ, where µ is a probability measure which is invariant w.r.t. the associated Markov process. We
+study the action of Kt on observables in an RKHS H which is densely and compactly embedded in L2
+µ. If
+this action is considered through the “lens” of the kernel integral operator E : L2
+µ → H (see Section 2.2),
+we arrive at a family of operators Ct
+H = EKtE∗ (cf. Figure 1). The action of Ct
+H : H → H is that of a
+cross-covariance operator:
+Ct
+Hψ =
+�
+(Ktψ)(x)k(x, ·) dµ(x),
+ψ ∈ H,
+where k(·, ·) is the kernel generating the RKHS H. These operators possess canonical empirical estima-
+tors based on finite simulation data, which only require evaluations of the feature map.
+L2
+µ
+L2
+µ
+H
+H
+Kt
+E
+E∗
+Ct
+H
+FIGURE 1. Diagram illustrating the different operators involved
+Our contribution, illustrated in Figure 2, is two-fold. In our first main result, Theorem 3.1, we provide
+an exact formula for the Hilbert-Schmidt variance of the canonical empirical estimator �Cm,t
+H
+of the cross-
+covariance operator Ct
+H, for m data points sampled from a long ergodic simulation. This result extends
+the findings in [38] to the kernel setting and no longer depends on the dictionary size (which would
+be infinite, at any rate). Due to the infinite-dimensional setting, additional assumptions are required,
+in particular, a spectral decomposition of the Koopman generator. Our result allows for probabilistic
+estimates for the error ∥ �Cm,t
+H
+− Ct
+H∥HS, see Proposition 3.4.
+As a second main result, we propose an empirical estimator for the restriction of the Koopman op-
+erator Kt to H, truncated to finitely many terms of its estimated Mercer series expansion, and prove a
+probabilistic bound for the resulting estimation error in Theorem 4.1, measured in the operator norm
+for bounded linear maps from H to L2
+µ. This result can be seen as a bound on the prediction error for
+the RKHS-based Koopman operator due to the use of finite data. In the situation where the RKHS is
+invariant under the Koopman operator we are able to complement the preceding error analysis with a
+bound on the full approximation error in Theorem 4.5.
+Finally, we illustrate our results for a one-dimensional Ornstein-Uhlenbeck (OU) process. For this
+simple test case, all quantities appearing in our error estimates are known analytically and can be well
+approximated numerically. Therefore, we are able to provide a detailed comparison between the error
+bound obtained from our results and the actual errors observed for finite data. Our experiments show that
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+3
+our bounds for the estimation error of the cross-covariance operator are accurate, and that the corrections
+we introduced to account for the inter-dependence of the data are indeed required. Concerning the
+prediction error, we find our theoretical bounds still far too conservative, which reflects the problem of
+accounting for the effect of inverting the mass matrix in traditional EDMD. This finding indicates that
+additional research is required on this end.
+Full Koopman 
+Approximation Error 
+Projection error 
+Theorem 4.5
+Variance representation 
+of empirical estimator 
+Theorem 3.1
+Estimation error 
+Theorem 4.1
+i.i.d. sampling
+Ergodic sampling
+Cross-covariance bound 
+Proposition 3.4
+∥Ct
+ℍ −
+̂
+C m,t
+ℍ ∥HS
+∥Kt
+N −
+̂
+K m,t
+N ∥ℍ→L2μ(X)
+∥Kt −
+̂
+K m,t
+N ∥ℍ→L2μ(X)
+∥Kt − Kt
+N∥ℍ→L2μ(X)
+FIGURE 2. Illustration of main results
+The paper is structured as follows: the setting is introduced in Section 2. The result concerning the
+variance of the empirical cross-covariance operator, Theorem 3.1, is presented and proved in Section 3,
+while our bound for the prediction error is part of Theorem 4.1 in Section 4. Numerical experiments are
+shown in Section 5, conclusions are drawn in Section 6.
+2. PRELIMINARIES
+In this section, we provide the required background on stochastic differential equations (Section 2.1),
+reproducing kernel Hilbert spaces (Section 2.2), Koopman operators (Section 2.3), and their representa-
+tions on an RKHS (Section 2.4).
+2.1. Stochastic differential equations. Let X ⊂ Rd and let a stochastic differential equation (SDE)
+with drift vector field b : X → Rd and diffusion matrix field σ : X → Rd×d be given, i.e.,
+dXt = b(Xt) dt + σ(Xt) dWt,
+(2.1)
+where Wt is d-dimensional Brownian motion. We assume that both b and σ are Lipschitz-continuous and
+that (1 + ∥ · ∥2)−1[∥b∥2 + ∥σ∥F ] is bounded on X. Then [39, Theorem 5.2.1] guarantees the existence
+of a unique solution (Xt)t≥0 to (2.1).
+The solution (Xt)t≥0 constitutes a continuous-time Markov process whose transition kernel will be
+denoted by ρt : X ×BX → R, where BX denotes the Borel σ-algebra on X. Then ρt(x, ·) is a probability
+measure for all x ∈ X, and for each A ∈ BX we have that ρt(·, A) is a representative of the conditional
+probability for A containing Xt given X0 = · , i.e.,
+ρt(x, A) = P(Xt ∈ A|X0 = x)
+for µ-a.e. x ∈ X.
+Throughout, we will assume the existence of an invariant (Borel) probability measure µ for the Markov
+process (Xt)t≥0, i.e., we have
+�
+ρt(x, A) dµ(x) = µ(A)
+(2.2)
+for all t ≥ 0.
+In addition to being invariant, we will often assume that µ is ergodic, meaning that for any t > 0
+every ρt-invariant set A (that is, ρt(x, A) = 1 for all x ∈ A) satisfies µ(A) ∈ {0, 1}. In this case, the
+Birkhoff ergodic theorem [15, Theorem 9.6] (see also (D.1)) and its generalizations apply, and allow us
+to calculate expectations w.r.t. µ using long-time averages over simulation data.
+We let ∥ · ∥p denote the Lp
+µ(X)-norm, 1 ≤ p < ∞. In the particular case p = 2, scalar product and
+norm on the Hilbert space L2
+µ(X) will be denoted by ⟨· , ·⟩µ and ∥ · ∥µ, respectively.
+
+4
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+2.2. Reproducing kernel Hilbert spaces. In what follows, let k : X × X → R be a continuous and
+symmetric positive definite kernel, that is, we have k(x, y) = k(y, x) for all x, y ∈ X and
+m
+�
+i,j=1
+k(xi, xj)cicj ≥ 0
+for all choices of x1, . . . , xm ∈ X and c1, . . . , cm ∈ R. It is well known that k generates a so-called
+reproducing kernel Hilbert space (RKHS) [1, 6, 40] (H, ⟨· , ·⟩) of continuous functions, such that for
+ψ ∈ H the reproducing property
+ψ(x) = ⟨ψ, Φ(x)⟩,
+x ∈ X,
+(2.3)
+holds, where Φ : X → H denotes the so-called feature map corresponding to the kernel k, i.e.,
+Φ(x) = k(x, ·),
+x ∈ X.
+In the sequel, we shall denote the norm on H by ∥ · ∥ and the kernel diagonal by ϕ:
+ϕ(x) = k(x, x),
+x ∈ X.
+Then for x ∈ X we have
+∥Φ(x)∥2 = ⟨Φ(x), Φ(x)⟩ = ⟨k(x, ·), k(x, ·)⟩ = k(x, x) = ϕ(x).
+We shall frequently make use of the following estimate:
+|k(x, y)| = |⟨Φ(x), Φ(y)⟩| ≤ ∥Φ(x)∥∥Φ(y)∥ =
+�
+ϕ(x)ϕ(y).
+In particular, it shows that k is bounded if and only if its diagonal ϕ is bounded.
+By Lp
+µ(X), p ∈ [1, ∞), we denote the space of all functions (not equivalence classes) on X with a
+finite p-norm ∥ · ∥p. Henceforth, we shall impose the following
+Compatibility Assumptions:
+(A1) ϕ ∈ L2
+µ(X).
+(A2) If ψ ∈ L2
+µ(X) such that
+� �
+k(x, y)ψ(x)ψ(y) dµ(x) dµ(y) = 0, then ψ = 0.
+(A3) If ψ ∈ H such that ψ(x) = 0 for µ-a.e. x ∈ X, then ψ(x) = 0 for all x ∈ X.
+Many of the statements in this subsection can also be found in [52, Chapter 4]. However, as we aim
+to present the contents in a self-contained way, we provide the proofs in Appendix A.
+The following lemma explains the meaning of the compatibility assumptions (A1) and (A2).
+Lemma 2.1. Under the assumption that ϕ ∈ L1
+µ(X) (in particular, under assumption (A1)), we have
+that H ⊂ L2
+µ(X) with
+∥ψ∥µ ≤
+�
+∥ϕ∥1 · ∥ψ∥,
+ψ ∈ H,
+(2.4)
+and assumption (A2) is equivalent to the density of H in L2
+µ(X).
+We have meticulously distinguished between functions and equivalence classes as there might be
+distinct functions φ, ψ ∈ H, which are equal µ-almost everywhere2, i.e., φ = ψ in L2
+µ(X). The com-
+patibility assumption (A3) prohibits this situation so that H can in fact be seen as a subspace of L2
+µ(X),
+which is then densely and continuously embedded.
+2For example, if µ = δa and φ(a) = ψ(a)
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+5
+Remark 2.2. (a) Condition (A1) implies k ∈ L4
+µ⊗µ(X × X), where µ ⊗ µ is the product measure on
+X × X.
+(b) The density of H in L2
+µ(X) is strongly related to the term universality in the literature, see [53].
+(c) Condition (A3) holds if supp µ = X, cf. [52, Exercise 4.6].
+It immediately follows from
+�
+|ψ(x)|∥Φ(x)∥ dµ(x) ≤ ∥ψ∥µ∥ϕ∥1/2
+1
+,
+(2.5)
+for ψ ∈ L2
+µ(X) that the linear operator E : L2
+µ(X) → H, defined by
+Eψ :=
+�
+ψ(x)Φ(x) dµ(x),
+ψ ∈ L2
+µ(X),
+is well defined (as a Bochner integral in H) and bounded with operator norm not larger than ∥ϕ∥1/2
+1
+.
+Remark 2.3. The so-called kernel mean embedding Ek, mapping probability measures ν on X to the
+RKHS H, is defined by Ekν =
+�
+Φ(x) dν(x), see, e.g., [51]. Hence, we have Eψ = Ekν with dν = ψ dµ.
+Note that the operator E is not an embedding in strict mathematical terms. The terminology embedding
+rather applies to its adjoint E∗. Indeed, the operator E enjoys the simple but important property:
+⟨Eψ, η⟩ =
+�
+ψ(x)⟨Φ(x), η⟩ dµ(x) =
+�
+ψ(x)η(x) dµ(x) = ⟨ψ, η⟩µ
+(2.6)
+for ψ ∈ L2
+µ(X) and η ∈ H. This implies that the adjoint operator E∗ : H → L2
+µ(X) is the inclusion
+operator from H into L2
+µ(X), i.e.,
+E∗η = η,
+η ∈ H.
+(2.7)
+We shall further define the covariance operator3
+CH := EE∗ ∈ L(H).
+Recall that a linear operator T ∈ L(H) on a Hilbert space H is trace class if for some (and hence for
+each) orthonormal basis (ej)j∈N of H we have that �∞
+j=1⟨(T ∗T)1/2ei, ei⟩ < ∞. A linear operator
+S ∈ L(H, K) between Hilbert spaces H and K is said to be Hilbert-Schmidt [12, Chapter III.9] if S∗S is
+trace class, i.e., ∥S∥2
+HS := �∞
+j=1 ∥Sei∥2 < ∞ for some (and hence for each) orthonormal basis (ej)j∈N.
+Lemma 2.4. Let the Compatibility Assumptions (A1)–(A3) be satisfied. Then the following hold.
+(a) The operator E is an injective Hilbert-Schmidt operator with
+∥E∥2
+HS = ∥ϕ∥1.
+(b) The space H is densely and compactly embedded in L2
+µ(X).
+(c) The operator CH is an injective non-negative selfadjoint trace class operator.
+The next theorem is due to Mercer and can be found in, e.g., [45]. It shows the existence of a particular
+orthonormal basis (ej)∞
+j=1 of L2
+µ(X) composed of eigenfunctions of E∗E, which we shall henceforth call
+the Mercer basis corresponding to the kernel k. Again for the sake of self-containedness, we give a short
+proof in Appendix A.
+3In what follows, by L(H, K) we denote the set of all bounded (i.e., continuous) linear operators between Hilbert spaces H
+and K. As usual, we also set L(H) := L(H, H).
+
+6
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Theorem 2.5 (Mercer’s Theorem). There exists an orthonormal basis (ej)∞
+j=1 of L2
+µ(X) consisting of
+eigenfunctions of E∗E with corresponding eigenvalues λj > 0 such that �∞
+j=1 λj = ∥ϕ∥1 < ∞. Fur-
+thermore, (fj)∞
+j=1 with fj =
+�
+λjej constitutes an orthonormal basis of H consisting of eigenfunctions
+of CH with corresponding eigenvalues λj. Moreover, for all x, y ∈ X,
+k(x, y) =
+�
+j
+fj(x)fj(y) =
+�
+j
+λjej(x)ej(y),
+the series converges absolutely.
+2.3. The Koopman semigroup. The Koopman semigroup (Kt)t≥0 associated with the SDE (2.1) is
+defined by
+(Ktψ)(x) = E[ψ(Xt)|X0 = x] =
+�
+ψ(y) ρt(x, dy),
+for ψ ∈ B(X), the set of all bounded Borel-measurable functions on X, and ρt(x, dy) = dρt(x, ·)(y). It
+is easy to see that the invariance of µ is equivalent to the identity
+�
+Ktψ dµ =
+�
+ψ dµ
+(2.8)
+for all t ≥ 0 and ψ ∈ B(X) (which easily extends to functions ψ ∈ L1
+µ(X), see Proposition 2.7).
+Remark 2.6. Note that in the case σ = 0 the SDE (2.1) reduces to the deterministic ODE ˙x = b(x).
+Then (2.8) implies
+�
+|ψ(φ(t, x))|2 dµ(x) =
+�
+|ψ(x)|2 dµ(x) for all t ≥ 0 and all ψ ∈ B(X), where
+φ(·, x) is the solution of the initial value problem ˙y = b(y), y(0) = x. Hence, the composition operator
+Kt : ψ �→ ψ ◦ φ(t, ·) is unitary in L2
+µ(X). However, we shall require below (see Theorem 3.1) that Kt
+has its spectrum in the interior of the unit circle. Therefore, we assume throughout that σ ̸= 0.
+The proofs of the following two propositions can be found in Appendix A.
+Proposition 2.7. For each p ∈ [1, ∞] and t ≥ 0, Kt extends uniquely to a bounded operator from
+Lp
+µ(X) to itself with operator norm ∥Kt∥Lp
+µ→Lp
+µ ≤ 1.
+By Cb(X) we denote the set of all bounded continuous functions on X. As the measure µ is finite, we
+have Cb(X) ⊂ B(X) ⊂ Lp
+µ(X) for all p ∈ [1, ∞]. In fact, Cb(X) is dense in each Lp
+µ(X), p ∈ [1, ∞),
+see [48, Theorem 3.14].
+Proposition 2.8. (Kt)t≥0 is a C0-semigroup of contractions in Lp
+µ(X) for each p ∈ [1, ∞).
+The infinitesimal generator of the C0-semigroup (Kt)t≥0 is the (in general unbounded) operator in
+L2
+µ(X), defined by
+Lψ = L2
+µ- lim
+t→0
+Ktψ − ψ
+t
+,
+(2.9)
+whose domain dom L is the set of all ψ ∈ L2
+µ(X) for which the above limit exists. By Proposition 2.8
+and the Lumer-Phillips theorem (see [28]), the operator L is densely defined, closed4, dissipative (i.e.,
+Re⟨Lψ, ψ⟩µ ≤ 0 for all ψ ∈ dom L), and its spectrum is contained in the closed left half-plane.
+Lemma 2.9. The constant function 1 is contained in dom L and L1 = 0. Moreover, if M := span{1} ⊂
+L2
+µ(X), then both M and M⊥ are invariant under L and all Kt, t ≥ 0.
+4Recall that a linear operator T, defined on a subspace dom T of a Hilbert space H, which maps to a Hilbert space K, is
+closed if its graph is closed in H × K.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+7
+Proof. It is easy to see that Kt1 = 1 for each t ≥ 0 and hence 1 ∈ dom L with L1 = 0. Hence KtM ⊂
+M for all t ≥ 0 and LM ⊂ M. Now, if ψ ⊥ 1, then ⟨Ktψ, 1⟩µ =
+�
+Ktψ dµ =
+�
+ψ dµ = ⟨ψ, 1⟩µ = 0,
+which shows that also KtM⊥ ⊂ M⊥. The relation LM⊥ ⊂ M⊥ follows from (2.9).
+□
+2.4. Representation of Koopman Operators on the RKHS. Using the integral operator E, it is possi-
+ble to represent the Koopman operator with the aid of a linear operator on H, which is based on kernel
+evaluations. This construction mimics the well-known kernel trick used frequently in machine learning.
+To begin with, for any x, y ∈ X define the rank-one operator Cxy : H → H by
+Cxyψ := ⟨ψ, Φ(y)⟩Φ(x) = ψ(y)Φ(x).
+For t ≥ 0 and ψ ∈ H we further define the cross-covariance operator Ct
+H : H → H by
+Ct
+Hψ :=
+� �
+Cxyψ ρt(x, dy) dµ(x) =
+�
+(Ktψ)(x)Φ(x) dµ(x) = EKtψ = EKtE∗ψ.
+Thus, we have
+Ct
+H = EKtE∗.
+(2.10)
+In other words, the cross-covariance operator Ct
+H represents the action of the Koopman semigroup
+through the lens of the RKHS integral operator E (see [20] for details). Being the product of the two
+Hilbert-Schmidt operators EKt and E∗, the operator Ct
+H is trace class for all t ≥ 0 (cf. [16, p. 521]).
+Note that due to ρ0(x, · ) = δx, for t = 0 this reduces to the already introduced covariance operator
+� �
+Cxy ρ0(x, dy) dµ(x) =
+�
+Cxx dµ(x) = EE∗ = CH.
+The identity (2.10) shows that for all η, ψ ∈ H we have
+⟨η, Ct
+Hψ⟩ = ⟨η, Ktψ⟩µ,
+(2.11)
+which shows that the role of Ct
+H is analogous to that of the stiffness matrix in a traditional finite-
+dimensional approximation of the Koopman operator. In this analogy, the covariance operator CH plays
+the role of the mass matrix.
+2.5. Empirical estimators. Next, we introduce empirical estimators for Ct
+H based on finite data (xk, yk),
+k = 1, . . . , m. We consider two sampling scenarios for fixed t > 0:
+(1) The xk are drawn i.i.d. from µ, and each yk ∼ µ is obtained from the conditional distribution
+ρt(xk, ·), i.e., yk|(xk = x) ∼ ρt(x, ·) for µ-a.e. x ∈ X. For example, yk can be obtained by
+simulating the SDE (2.1) starting from xk until time t.
+(2) µ is ergodic and both xk and yk are obtained from a single (usually long-term) simulation of the
+dynamics Xt at discrete integration time step ∆t > 0, using a sliding-window estimator, i.e.,
+x0 = X0 ∼ µ,
+xk = Xk∆t,
+and
+yk = Xk∆t+t.
+Moreover, we assume that there exists a Riesz basis (ψj)∞
+j=0 of L2
+µ(X) consisting of eigenfunc-
+tions of the generator L with corresponding eigenvalues µj satisfying �∞
+j=0 e2(Re µj)∆t < ∞.
+Remark 2.10. It easily follows from the discussion in Appendix B that the last assumption on the
+generator L and on the decay of its eigenvalues µj is equivalent to the similarity of L to an (unbounded)
+normal operator N such that eN∆t ∈ L(L2
+µ(X)) is Hilbert-Schmidt. If the assumption holds with
+ψj = Sej, where (ej) is an orthonormal basis of L2
+µ(X), the operator N is given by N = �
+j µj⟨ · , ej⟩ej
+with dom N = {ψ : (µj⟨ψ, ej⟩) ∈ ℓ2} and L = SNS−1. The condition �∞
+j=0 e2(Re µj)∆t < ∞ then
+obviously means that the eigenvalues of eN∆t form an ℓ2 sequence.
+
+8
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Recall that the joint distribution of two random variables X and Y is given by
+dPX,Y (x, y) = dPY |X=x(y) · dPX(x).
+Set X = xk and Y = yk. Then, in both cases (1) and (2), we have PX = µ and
+PY |X=x(B) = P(yk ∈ B|xk = x) = P(Xt ∈ B|X0 = x) = ρt(x, B).
+In other words, for the joint distribution µ0,t of xk and yk we have
+dµ0,t(x, y) = dρt(x, ·)(y) · dµ(x) = ρt(x, dy) · dµ(x).
+More explicitly,
+µ0,t(A × B) =
+�
+A
+ρt(x, B) dµ(x).
+Now, since
+Ct
+H =
+� �
+Cxy ρt(x, dy) dµ(x) =
+�
+Cxy dµ0,t(x, y) = E
+�
+Cxk,yk
+�
+,
+for the empirical estimator for Ct
+H we choose the expression
+�Cm,t
+H
+= 1
+m
+m−1
+�
+k=0
+Cxk,yk.
+(2.12)
+3. VARIANCE OF THE EMPIRICAL ESTIMATOR
+In case (1), the law of large numbers [3, Theorem 2.4] and, in case (2), ergodicity [2] ensures the
+expected behavior
+lim
+m→∞ ∥ �Cm,t
+H
+− Ct
+H∥HS = 0
+a.s.
+However, this is a purely qualitative result, and nothing is known a priori on the rate of this convergence.
+The main result of this section, Theorem 3.1, contains an exact expression for the Hilbert-Schmidt vari-
+ance of the empirical estimator �Cm,t
+H
+based on m data points, which then yields probabilistic estimates
+for the expression ∥ �Cm,t
+H
+− Ct
+H∥HS, see Proposition 3.4. Here, our focus is on the estimation from a
+single ergodic trajectory, i.e., case (2) above. While the broader line of reasoning partially resembles that
+of our previous paper [38], we require additional steps due to the infinite-dimensional setting introduced
+by the RKHS.
+In Theorem 3.1 and its proof, we will be concerned with evolving kernels kt : X × X → R, defined
+by
+kt(x, x′) :=
+� �
+k(y, y′) ρt(x, dy) ρt(x′, dy′).
+We have
+kt(x, x′) =
+� �
+⟨Φ(y), Φ(y′)⟩ ρt(x, dy) ρt(x′, dy′) =
+��
+Φ(y) ρt(x, dy),
+�
+Φ(y′) ρt(x′, dy′)
+�
+.
+The integrals in the last expression are well defined as limits in H for µ-a.e. x, x′ ∈ X as
+� �
+∥Φ(y)∥ ρt(x, dy) dµ(x) =
+� � �
+ϕ(y) ρt(x, dy) dµ(x) =
+� �
+ϕ(x) dµ(x) ≤ ∥ϕ∥1/2
+1
+,
+see (2.8). This shows that kt is well defined ((µ ⊗ µ)-a.e.) and that it is a positive definite kernel on its
+domain. Moreover, k0 = k and
+|kt(x, x′)| ≤
+� �
+ϕ(y′)
+� �
+ϕ(y) ρt(x, dy) ρt(x′, dy′) = (Kt√ϕ)(x) · (Kt√ϕ)(x′).
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+9
+In particular, kt ∈ L2
+µ⊗µ(X 2) with ∥kt∥L2
+µ⊗µ ≤ ∥ϕ∥1. By Φt we denote the corresponding feature map,
+i.e.,
+Φt(x) = kt(x, ·).
+Note that not necessarily Φt(x) ∈ H. Finally, we define
+Φt,x := Φ(x)Φt(x).
+We are now in the position to formulate our first main result.
+Theorem 3.1. Setting zk = (xk, yk), k = 1, . . . , m, the Hilbert-Schmidt variance of the empirical
+estimator can be written as
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= 1
+m
+�
+E0(t) + 2
+m−1
+�
+k=1
+m−k
+m
+· E
+�
+⟨Czk − Ct
+H, Cz0 − Ct
+H⟩HS
+�
+�
+,
+(3.1)
+where
+E0(t) := E
+�
+∥Cz0 − Ct
+H∥2
+HS
+�
+= ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2
+µ⊗µ.
+In case (1), E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= 1
+mE0(t), whereas in case (2) we have
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= 1
+m
+�
+�E0(t) + 2
+∞
+�
+j=1
+dj,tqj
+1 − qj
+�
+1 − 1
+m ·
+1 − qm
+j
+1 − qj
+��
+� ,
+(3.2)
+with
+qj = eµj∆t,
+dj,t = ⟨cj,t, ψj⟩µ,
+and
+cj,t(x) = ⟨Φt,x, �ψj⟩µ.
+Before proving Theorem 3.1 in Subsection 3.1 below, let us comment on its statements and draw some
+conclusions.
+Remark 3.2. (a) Note that, by ergodicity of the invariant measure µ, the generator L has no eigenvalues
+on the imaginary axis, except the simple zero eigenvalue (see Proposition D.1 in the Appendix). In
+contrast, if we drop the ergodicity assumption, we have
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= 1
+m
+�
+�E0(t) + 2
+∞
+�
+j=ν0
+dj,tqj
+1 − qj
+�
+1 − 1
+m ·
+1 − qm
+j
+1 − qj
+��
+� + m − 1
+m
+ν0−1
+�
+j=1
+dj,t,
+where ν0 = #{j : µj ∈ 2πi
+∆t Z} is the number of eigenvalues of L of the form 2kπi
+∆t , k ∈ Z, counting
+multiplicities. Obviously, the last term does not decay to zero as m → ∞ if �ν0−1
+j=1 dj,t ̸= 0.
+(b) The definition of cj,t requires Φt,x to be in L2
+µ(X) for µ-a.e. x ∈ X. This will in fact be proved in
+Lemma 3.6 below.
+In the following, we let
+σ2
+m := E0(t) + 2
+∞
+�
+j=1
+dj,tqj
+1 − qj
+�
+1 − 1
+m ·
+1 − qm
+j
+1 − qj
+�
+and
+σ2
+∞ := E0(t) + 2
+∞
+�
+j=1
+dj,tqj
+1 − qj
+.
+Then
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= σ2
+m
+m
+and σ2
+m → σ2
+∞ as m → ∞. Both infinite series converge absolutely as (qj) ∈ ℓ2 by assumption, and
+(dj,t) ∈ ℓ2 as shown in the proof of Theorem 3.1. We can therefore interpret σ2
+∞ as asymptotic variance
+of the estimator ˆCm,t
+H , similar to our previous results in [38, Lemma 6].
+An upper bound on the variance can be obtained as follows:
+
+10
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Corollary 3.3. In case (2), for all m ∈ N we have
+σ2
+m ≤ ⟨Ktϕ, ϕ⟩µ
+�
+1 + 4B
+Aδq
+∥q∥ℓ2
+�
+,
+(3.3)
+where A and B denote the lower and upper Riesz bounds of (ψj), respectively,
+q = (qj)∞
+j=1 ,
+and
+δq = inf
+j≥1 |1 − qj| > 0.
+Proof. First of all, by Lemma 3.6,
+E0(t) = ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2
+µ⊗µ ≤ ⟨Ktϕ, ϕ⟩µ.
+We have |1 − qj| ≥ δq and |qj| ≤ 1 for all j ≥ 1 and hence
+1
+|1 − qj| ·
+����1 − 1
+m ·
+1 − qm
+j
+1 − qj
+���� ≤ 1
+δq
+�
+1 + 1
+m
+m−1
+�
+k=0
+|qj|k
+�
+≤ 2
+δq
+.
+This and (3.7) imply (3.3).
+□
+Proposition 3.4. We have the following probabilistic bound on the estimation error:
+P
+�
+∥Ct
+H − �Cm,t
+H ∥HS > ε
+�
+≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+σ2
+m
+mε2 ,
+in case (2),
+(3.4)
+E0(t)
+mε2 ,
+in case (1),
+(3.5)
+2 e
+−
+mε2
+8∥k∥2∞ ,
+in case (1) with bounded kernel.
+(3.6)
+In particular, the above also holds upon replacing the left-hand side by P
+�
+∥EKtψ − �Cm,t
+H ψ∥ > ε
+�
+for
+ψ ∈ H, ∥ψ∥ = 1.
+Proof. The inequalities (3.4) and (3.5) are an immediate consequence of Markov’s inequality, applied to
+the random variable ∥Ct
+H − �Cm,t
+H ∥2
+HS. The inequality (3.6) follows from Ct
+H − �Cm,t
+H
+= 1
+m
+�m−1
+k=0 (Ct
+H −
+Czk), Hoeffding’s inequality for Hilbert space-valued random variables [43, Theorem 3.5] (see also [30,
+Theorem A.5.2]), and (cf. Lemma 3.6 below)
+∥Ct
+H − Cxy∥HS ≤ ∥Ct
+H∥HS + ∥Cxy∥HS =
+�
+⟨k, kt⟩L2
+µ⊗µ +
+�
+ϕ(x)ϕ(y) ≤ 2∥k∥∞,
+since also ∥kt∥∞ ≤ ∥k∥∞. The estimate
+∥EKtψ − �Cm,t
+H ψ∥ = ∥EKtE∗ψ − �Cm,t
+H ψ∥ = ∥(Ct
+H − �Cm,t
+H )ψ∥ ≤ ∥Ct
+H − �Cm,t
+H ∥HS
+finally yields the last claim.
+□
+Remark 3.5. Under additional assumptions (boundedness of the kernel, mixing, etc.), other concen-
+tration inequalities than Markov’s, such as, e.g., [3, Theorem 2.12] (α-mixing) or [46, Th´eor`eme 3.1]
+(β-mixing), might lead to better estimates than (3.4).
+3.1. Proof of Theorem 3.1.
+Lemma 3.6. Let t ≥ 0. Then Φt,x ∈ L2
+µ(X) for µ-a.e. x ∈ X with
+∥Φt,x∥2
+µ ≤ ϕ(x)(Ktϕ)(x) · ⟨Ktϕ, ϕ⟩µ.
+Moreover, for every t ≥ 0 we have
+∥Cxy∥2
+HS = ϕ(x)ϕ(y)
+and
+∥Ct
+H∥2
+HS = ⟨k, kt⟩L2
+µ⊗µ =
+� �
+Φt,x(y) dµ(y) dµ(x).
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+11
+Proof. We estimate
+|Φt,x(x′)|2 = |k(x, x′)kt(x, x′)|2 ≤ ϕ(x)ϕ(x′)(Kt√ϕ)2(x) · (Kt√ϕ)2(x′)
+≤ ϕ(x)(Ktϕ)(x) · ϕ(x′)(Ktϕ)(x′),
+where we have applied Jensen’s inequality to (Kt√ϕ)(x). This proves the first inequality. Next, if
+(fj) ⊂ H denotes the Mercer basis corresponding to k, then
+⟨Cxy, Cx′y′⟩HS =
+�
+i
+⟨Cxyfi, Cx′y′fi⟩ =
+�
+i
+fi(y)fi(y′)k(x, x′) = k(x, x′)k(y, y′)
+This proves ∥Cxy∥2
+HS = ϕ(x)ϕ(y). Moreover, it yields
+∥Ct
+H∥2
+HS =
+����
+�
+Cxy dµ0,t(x, y)
+����
+2
+HS
+=
+� �
+k(x, x′)k(y, y′) dµ0,t(x, y) dµ0,t(x′, y′)
+=
+� �
+k(x, x′)
+�� �
+k(y, y′) ρt(x, dy) ρt(x′, dy′)
+�
+dµ(x′) dµ(x) = ⟨k, kt⟩L2
+µ⊗µ,
+as claimed.
+□
+Proof of Theorem 3.1. First of all, we have
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= E
+���� 1
+m
+m−1
+�
+k=0
+(Czk − Ct
+H)
+���
+2
+HS
+�
+= E
+� 1
+m2
+m−1
+�
+k,ℓ=0
+�
+Czk − Ct
+H, Czℓ − Ct
+H
+�
+HS
+�
+= E
+�
+1
+m2
+m−1
+�
+k=0
+∥Czk − Ct
+H∥2
+HS + 2
+m2
+m−1
+�
+k=0
+m−1
+�
+ℓ=k+1
+�
+Czk − Ct
+H, Czℓ − Ct
+H
+�
+HS
+�
+= 1
+mE
+�
+∥Cz0 − Ct
+H∥2
+HS
+�
++ 2
+m2
+m−1
+�
+k=1
+(m − k)E
+�
+⟨Czk − Ct
+H, Cz0 − Ct
+H⟩HS
+�
+.
+where we exploited that E[⟨Czk − Ct
+H, Czℓ − Ct
+H⟩HS] only depends on the difference ℓ − k.
+Let us compute the first term. Since E[Cz0] = Ct
+H and thus E[⟨Cz0, Ct
+H⟩HS] = ∥Ct
+H∥2
+HS,
+E
+�
+∥Cz0 − Ct
+H∥2
+HS
+�
+= E
+�
+∥Cz0∥2
+HS
+�
+− ∥Ct
+H∥2
+HS.
+For ψ ∈ H we have
+∥Cz0ψ∥2 = ∥ψ(y0)Φ(x0)∥2 = ψ(y0)2ϕ(x0).
+Using the Mercer basis (fi) ⊂ H corresponding to k in H (cf. Theorem 2.5), we obtain
+E
+�
+∥Cz0∥2
+HS
+�
+= E
+� �
+i
+∥Cz0fi∥2�
+= E
+� �
+i
+fi(y0)2ϕ(x0)
+�
+= E[ϕ(x0)ϕ(y0)].
+Note that the latter equals (ϕ(x) = k(x, x) by definition)
+E[ϕ(x0)ϕ(y0)] =
+�
+ϕ(x)
+�
+ϕ(y) ρt(x, dy) dµ(x) =
+�
+ϕ(x)(Ktϕ)(x) dµ(x) = ⟨Ktϕ, ϕ⟩µ.
+We obtain
+E
+�
+∥Cz0 − Ct
+H∥2
+HS
+�
+= E[ϕ(x0)ϕ(y0)] − ⟨k, kt⟩L2
+µ⊗µ = ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2
+µ⊗µ = E0(t)
+and thus (3.1).
+Case (1). In this case, zk and zℓ are independent for k ̸= ℓ, so that
+E
+�
+⟨Czk − Ct
+H, Czℓ − Ct
+H⟩HS
+�
+= 0.
+
+12
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Hence, the statement of the theorem for case (1) follows.
+Case (2). Here, the cross terms do not vanish. In fact,
+E
+�
+⟨Czk − Ct
+H, Cz0 − Ct
+H⟩HS
+�
+= E[⟨Czk, Cz0⟩HS] − ∥Ct
+H∥2
+HS = E
+� �
+i
+⟨Czkfi, Cz0fi⟩
+�
+− ∥Ct
+H∥2
+HS
+= E
+�� �
+i
+fi(yk)fi(y0)
+�
+k(xk, x0)
+�
+− ∥Ct
+H∥2
+HS
+= E
+�
+k(yk, y0)k(xk, x0)
+�
+− ∥Ct
+H∥2
+HS.
+Now,
+E
+�
+k(yk, y0)k(xk, x0)
+�
+=
+� � � �
+k(y′, y)k(x′, x) ρt(x′, dy′) ρk∆t(x, dx′) ρt(x, dy) dµ(x)
+=
+� �
+k(x, x′)
+�� �
+k(y, y′) ρt(x, dy) ρt(x′, dy′)
+�
+ρk∆t(x, dx′) dµ(x)
+=
+� �
+k(x, x′)kt(x, x′) ρk∆t(x, dx′) dµ(x)
+=
+� ��
+[Φ(x)Φt(x)](x′) ρk∆t(x, dx′)
+�
+dµ(x)
+=
+�
+[Kk∆tΦt,x](x) dµ(x).
+Hence,
+E
+�
+⟨Czk − Ct
+H, Cz0 − Ct
+H⟩HS
+�
+=
+�
+(Kk∆tΦt,x)(x) dµ(x) − ⟨k, kt⟩L2
+µ⊗µ.
+Let us now exploit the assumptions on the spectral properties of the generator L in case (2). For µ-a.e.
+x ∈ X, we have
+Φt,x =
+∞
+�
+j=0
+cj,t(x)ψj,
+the series converging in L2
+µ(X). Therefore,
+KsΦt,x =
+∞
+�
+j=0
+cj,t(x)Ksψj =
+∞
+�
+j=0
+cj,t(x)eµjsψj,
+and thus (for k ≥ 1)
+� �
+Kk∆tΦt,x
+�
+(x) dµ(x) =
+�
+∞
+�
+j=0
+cj,t(x)eµjk∆tψj(x) dµ(x) =
+∞
+�
+j=0
+dj,t · eµjk∆t =
+∞
+�
+j=0
+dj,t · qk
+j .
+This series converges absolutely for each t ≥ 0 due to our assumption that �
+j |qj|2 < ∞ and since for
+each j ∈ N0 we have by Lemma 3.6 that
+∞
+�
+j=0
+|dj,t|2 ≤ B2
+∞
+�
+j=0
+∥cj,t∥2
+µ = B2
+�
+∞
+�
+j=0
+|⟨Φt,x, �ψj⟩µ|2 dµ(x)
+≤ B2
+A2
+�
+∥Φt,x∥2
+µ dµ(x) ≤ B2
+A2 ⟨Ktϕ, ϕ⟩2
+µ,
+(3.7)
+where A and B are the Riesz bounds of (ψj).
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+13
+Without loss of generality, we may assume that µ0 = 0 with ψ0 = 1 and µ1, . . . , µν0−1 ∈ 2πi
+∆t Z,
+and ψk ∈ 1⊥ for k ≥ 1, see Lemma 2.9. The duality relations then imply �ψ0 = 1. Now, c0,t(x) =
+⟨Φt,x, 1⟩µ =
+�
+k(x, y)kt(x, y) dµ(y) and hence
+d0,t = ⟨c0,t, 1⟩µ =
+�
+c0,t(x) dµ(x) =
+� �
+k(x, y)kt(x, y) dµ(y) dµ(x) = ⟨k, kt⟩L2
+µ⊗µ.
+(3.8)
+This implies
+E
+�
+⟨Czk − Ct
+H, Cz0 − Ct
+H⟩HS
+�
+=
+∞
+�
+j=0
+dj,t · qk
+j − ⟨k, kt⟩L2
+µ⊗µ =
+∞
+�
+j=1
+dj,t · qk
+j
+and therefore
+E
+�
+∥ �Cm,t
+H
+− Ct
+H∥2
+HS
+�
+= 1
+mE0(t) + 2
+m
+∞
+�
+j=1
+dj,t
+m−1
+�
+k=1
+(1 − k
+m)qk
+j
+= 1
+m
+�
+�E0(t) + 2
+ν0−1
+�
+j=1
+dj,t
+m−1
+�
+k=1
+(1 − k
+m)qk
+j + 2
+∞
+�
+j=ν0
+dj,t
+m−1
+�
+k=1
+(1 − k
+m)qk
+j
+�
+� .
+The identity
+m−1
+�
+k=1
+�
+1 − k
+m
+�
+qk =
+�
+q
+1−q
+�
+1 − 1
+m · 1−qm
+1−q
+�
+if q ̸= 1
+m−1
+2
+if q = 1
+finally yields (3.2).
+□
+4. BOUND ON THE KOOPMAN PREDICTION ERROR
+The kernel cross-covariance operator Ct
+H can also be used to approximate the predictive capabilities
+of the Koopman operator, for observables in H. Approximating the full Koopman operator involves
+the inverse of the co-variance operator, which becomes an unbounded operator on a dense domain of
+definition in the infinite-dimensional RKHS case. Moreover, its empirical estimator �Cm
+H is finite-rank and
+thus not even injective. While Fukumizu et al. tackle this problem in [10] by means of a regularization
+procedure, we choose to use pseudo-inverses instead (cf. Remark 4.2). We truncate the action of the
+Koopman operator using N terms of the Mercer series expansion and derive a bound for the prediction
+error for fixed truncation parameter N. While we use similar ideas as presented in [11], we heavily
+rely on our new results on the cross-covariance operator, cf. Section 3. Afterwards, we deal with the
+case of Koopman-invariance of the RKHS [22]. Here, we establish an estimate for the truncation error,
+which then yields a bound on the deviation from the full Koopman operator. We emphasize that this
+error bound is extremely useful in comparison to its prior counterparts based on the assumption that the
+space spanned by a finite number of so-called observables (dictionary) is invariant under the Koopman
+operator. The latter essentially requires to employ only Koopman eigenfunctions as observables, see,
+e.g., [25, 14].
+Let (ej) be the Mercer orthonormal basis of L2
+µ(X) corresponding to the kernel k and let λj = ∥Eej∥µ
+as well as fj :=
+�
+λjej (cf. Theorem 2.5). We arrange the Mercer eigenvalues in a non-increasing way,
+i.e.,
+λ1 ≥ λ2 ≥ . . . .
+Let ψ ∈ H. Then
+Ktψ =
+∞
+�
+j=1
+⟨Ktψ, ej⟩µej =
+∞
+�
+j=1
+⟨Ct
+Hψ, ej⟩ej =
+N
+�
+j=1
+⟨Ct
+Hψ, ej⟩ej +
+∞
+�
+j=N+1
+⟨Ct
+Hψ, ej⟩ej.
+(4.1)
+
+14
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+4.1. Estimation error. In the next theorem, we estimate the probabilistic error between the first sum-
+mand
+Kt
+Nψ =
+N
+�
+j=1
+⟨Ct
+Hψ, ej⟩ej,
+ψ ∈ H,
+and its empirical estimator, which is of the form �N
+j=1⟨ �Cm,t
+H ψ, �ej⟩�ej with approximations �ej of the ej.
+Theorem 4.1. Assume that the eigenvalues λj of CH are simple, i.e., λj+1 > λj for all j. Fix an
+arbitrary N ∈ N and let
+δN =
+min
+j=1,...,N
+λj − λj+1
+2
+.
+(4.2)
+Further, let ε ∈ (0, δN) and δ ∈ (0, 1) be arbitrary and fix some5 m ≥ max{N, 2σ2
+m
+ε2δ }. Let now
+�λ1 ≥ . . . ≥ �λm denote the largest m eigenvalues of �Cm
+H in descending order and let �e1, . . . , �em be
+corresponding eigenfunctions, respectively, such that ∥�ej∥ = �λ−1/2
+j
+for j = 1, . . . , m. If we define
+�Km,t
+N ψ =
+N
+�
+j=1
+⟨ �Cm,t
+H ψ, �ej⟩�ej,
+ψ ∈ H,
+(4.3)
+then, with probability at least 1 − δ, we have that
+∥Kt
+N − �Km,t
+N ∥H→L2µ(X) ≤
+�
+1
+√λN
++ N + 1
+δNλN
+(1 + ∥ϕ∥1)∥ϕ∥1/2
+1
+�
+ε.
+(4.4)
+All of the above statements equally apply to case (1) upon replacing σm by E0(t).
+Remark 4.2. (a) If we set �fj = �λ1/2
+j
+· �ej, then
+�Cm
+H =
+m
+�
+j=1
+�λj⟨ · , �fj⟩ �fj,
+and thus
+N
+�
+j=1
+⟨ · , �ej⟩�ej =
+N
+�
+j=1
+1
+�λj
+⟨ · , �fj⟩ �fj = ( �Cm
+H )† �QN,
+where �QN = �N
+j=1⟨ · , �fj⟩ �fj is the orthogonal projector onto the span of the first N eigenfunctions of
+�Cm
+H in H. Therefore,
+�Km,t
+N ψ =
+m
+�
+j=1
+⟨ �Cm,t
+H ψ, �ej⟩�ej = ( �Cm
+H )† �QN �Cm,t
+H ψ.
+In particular, for N = m we have �Km,t
+N
+= ( �Cm
+H )† �Cm,t
+H , which surely is one of the first canonical choices
+for an empirical estimator of Kt.
+(b) The functions �ej have unit length in the empirical L2
+µ-norm:
+1
+m
+m
+�
+k=1
+�ej(xk)�ej(xk) =
+�
+�Cm
+H �ej, �ej
+�
+= 1.
+Therefore, projecting onto the first N empirical Mercer features is the whitening transformation com-
+monly used in traditional EDMD [19].
+5By Corollary 3.3, an amount of at least m ≥ max
+�
+N ,
+2∥ϕ∥2
+µ
+ε2δ
+�
+1 +
+4B
+Aδq ∥q∥ℓ2
+��
+data points suffices.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+15
+Proof of Theorem 4.1. By Proposition 3.4, both events ∥Ct
+H − �Cm,t
+H ∥HS ≤ ε and ∥CH − �Cm
+H ∥HS ≤ ε
+occur with probability at least 1 − δ/2, respectively. Hence, they occur simultaneously with probability
+at least 1 − δ.
+In the remainder of this proof we assume that both events occur. Then all the statements deduced in
+the following hold with probability at least 1 − δ.
+Let us define the intermediate approximation
+�Km,t
+N ψ =
+N
+�
+j=1
+⟨ �Cm,t
+H ψ, ej⟩ej,
+ψ ∈ H.
+Let ψ ∈ H be arbitrary. Setting C := Ct
+H − �Cm,t
+H , we have
+∥Kt
+N�� − �Km,t
+N ψ∥2
+µ =
+�����
+N
+�
+j=1
+�
+Cψ, ej
+�
+ej
+�����
+2
+µ
+=
+N
+�
+j=1
+���
+Cψ, ej
+���2 =
+N
+�
+j=1
+���
+ψ, C∗ej
+���2
+≤ ∥ψ∥2
+N
+�
+j=1
+∥C∗ej∥2 ≤ ∥ψ∥2
+N
+�
+j=1
+1
+λj
+∥C∗fj∥2 ≤ ∥ψ∥2
+λN
+N
+�
+j=1
+∥C∗fj∥2
+≤ ∥ψ∥2
+λN
+∞
+�
+j=1
+∥C∗fj∥2 = ∥ψ∥2
+λN
+· ∥Ct
+H − �Cm,t
+H ∥2
+HS,
+and thus,
+∥Kt
+Nψ − �Km,t
+N ψ∥µ ≤ ∥ψ∥
+√λN
+· ε.
+Next, we aim at estimating the remaining error
+�Km,t
+N ψ − �Km,t
+N ψ =
+N
+�
+j=1
+⟨ �Cm,t
+H ψ, ej⟩ej −
+N
+�
+j=1
+⟨ �Cm,t
+H ψ, �ej⟩�ej
+=
+N
+�
+j=1
+λ−1
+j ⟨ �Cm,t
+H ψ, fj⟩fj −
+N
+�
+j=1
+�λ−1
+j ⟨ �Cm,t
+H ψ, �fj⟩ �fj
+=
+N
+�
+j=1
+λ−1
+j ⟨f, fj⟩fj −
+N
+�
+j=1
+�λ−1
+j ⟨f, �fj⟩ �fj
+=
+N
+�
+j=1
+�
+λ−1
+j Pjf − �λ−1
+j
+�Pjf
+�
+=
+N
+�
+j=1
+λ−1
+j (Pj − �Pj)f +
+N
+�
+j=1
+(λ−1
+j
+− �λ−1
+j ) �Pjf,
+where f = �Cm,t
+H ψ,
+Pjf = ⟨f, fj⟩fj
+and
+�Pjf = ⟨f, �fj⟩ �fj.
+By (2.4), it suffices to estimate the above error in the ∥ · ∥-norm. By Theorem C.3, the first summand can
+be estimated as
+���
+N
+�
+j=1
+λ−1
+j (Pj − �Pj)f
+��� ≤
+N
+�
+j=1
+1
+λj
+∥Pj − �Pj∥∥f∥ ≤ N · ∥CH − �Cm
+H ∥
+λNδN
+∥f∥ ≤
+N
+λNδN
+∥f∥ε.
+
+16
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+For the second summand we have
+���
+N
+�
+j=1
+(λ−1
+j
+− �λ−1
+j ) �Pjf
+���
+2
+=
+N
+�
+j=1
+|λ−1
+j
+− �λ−1
+j |2∥ �Pjf∥2 =
+N
+�
+j=1
+|λj − �λj|2
+λ2
+j�λ2
+j
+∥ �Pjf∥2.
+Now, note that ϵ < δN by assumption and therefore ∥CH − �Cm
+H ∥HS ≤ δN ≤ λN−λN+1
+2
+≤ λN
+2 . For
+j = 1, . . . , N, according to Theorem C.1 this implies
+�λj ≥ λj − |λj − �λj| ≥ λj − ∥CH − �Cm
+H ∥HS ≥ λj − λN
+2
+≥ λj
+2 .
+Hence,
+���
+N
+�
+j=1
+(λ−1
+j
+− �λ−1
+j ) �Pjf
+���
+2
+≤ 4
+N
+�
+j=1
+|λj − �λj|2
+λ4
+j
+∥ �Pjf∥2 ≤ 4∥CH − �Cm
+H ∥2
+HS
+λ4
+N
+∥ �QNf∥2,
+and thus,
+���
+N
+�
+j=1
+(λ−1
+j
+− �λ−1
+j ) �Pjf
+��� ≤
+2
+λ2
+N
+∥f∥ε ≤
+1
+λNδN
+∥f∥ε.
+From
+∥ �Cm,t
+H ∥ ≤ ∥ �Cm,t
+H
+− Ct
+H∥ + ∥Ct
+H∥ ≤ ∥ �Cm,t
+H
+− Ct
+H∥HS + ∥EKtE∗∥ ≤ ε + ∥ϕ∥1
+we conclude
+�� �Km,t
+N ψ − �Km,t
+N ψ
+�� ≤ N + 1
+λNδN
+∥ �Cm,t
+H ψ∥ε ≤ N + 1
+λNδN
+(ε + ∥ϕ∥1)∥ψ∥ε.
+All together, we obtain (recall (2.4))
+∥Kt
+Nψ − �Km,t
+N ψ∥µ ≤ ∥Kt
+Nψ − �Km,t
+N ψ∥µ + ∥ϕ∥1/2
+1
+∥ �Km,t
+N ψ − �Km,t
+N ψ∥
+≤ ∥ψ∥
+√λN
+· ε + N + 1
+λNδN
+(ε + ∥ϕ∥1)∥ϕ∥1/2
+1
+∥ψ∥ε
+=
+�
+1
+√λN
++ N + 1
+δNλN
+(1 + ∥ϕ∥1)∥ϕ∥1/2
+1
+�
+ε · ∥ψ∥,
+which implies (4.4).
+□
+4.2. Projection error in case of Koopman-invariance of the RKHS. In the preceeding section, we
+have seen that the empirical operator �Km,t
+N
+can be written as ( �Cm
+H )† �Cm,t
+H
+if m = N. In the limit m → ∞,
+we would arrive at the operator C−1
+H Ct
+H, which is not even well-defined for all ψ ∈ H, in general.
+However, if the RKHS is invariant under Kt, the above operator limit is well-defined as a bounded
+operator on H. In this situation we are able to extend Theorem 4.1 to an estimate on the full error made
+by our empirical estimator.
+We start by defining the operator
+Kt
+H := C−1
+H Ct
+H
+on its natural domain
+dom Kt
+H := {ψ ∈ H : Ct
+Hψ ∈ ran CH}.
+(4.5)
+We consider Kt
+H as an operator from H into itself (with domain of definition in H).
+Lemma 4.3. We have
+dom Kt
+H = {ψ ∈ H : Ktψ ∈ H},
+(4.6)
+and Kt
+H is closed.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+17
+Proof. Note that Ct
+Hψ ∈ ran CH if and only if EKtψ = CHφ for some φ ∈ H. Since CHφ = Eφ and
+ker E = {0}, the latter is equivalent to Ktψ = φ ∈ H, which proves the representation of the domain.
+As to the closedness of Kt
+H, let (ψn) ⊂ dom Kt
+H and φ ∈ H such that ψn → ψ in H and Kt
+Hψn → φ in
+H as n → ∞. The latter implies Ct
+Hψn → CHφ, while the first implies Ct
+Hψn → Ct
+Hψ in H as n → ∞,
+from which we conclude that Ct
+Hψ = CHφ, i.e., ψ ∈ dom Kt
+H and Kt
+Hψ = φ.
+□
+If the Koopman operator leaves the RKHS H invariant (i.e., KtH ⊂ H), Kt
+H is defined on all of H.
+Moreover, since the canonical inclusion map E∗ : H → L2(µ) is injective, it possesses an unbounded
+inverse on its range H, and therefore:
+C−1
+H Ct
+Hφ = C−1
+H EKtE∗φ = (EE∗)−1EE∗(E∗)−1KtE∗φ = (E∗)−1KtE∗φ.
+(4.7)
+Remarkably, invariance of H under the Koopman operator implies that the left-hand side not only repro-
+duces the Koopman operator on H, but actually defines a bounded operation.
+Parts of the next proposition can be found in [22, Theorem 5.3] and [8, Theorem 1].
+Proposition 4.4. For t > 0, the following statements are equivalent:
+(i) KtH ⊂ H.
+(ii) Kt
+H ∈ L(H).
+(iii) ran Ct
+H ⊂ ran CH.
+Proof. With regard to the two representations (4.5) and (4.6) of the domain, it is immediate that both (i)
+and (iii) are equivalent to dom Kt
+H = H. The equivalence of the latter to (ii) follows from the closed
+graph theorem.
+□
+Note that if one of (i)–(iii) holds, then Kt
+H = Kt|H.
+Theorem 4.5. In addition to the assumptions in Theorem 4.1, assume that H is invariant under the
+Koopman operator Kt. For fixed N ∈ N, let δN be as in (4.2), choose ε, δ, and m as in Theorem 4.1
+and define the empirical estimator �Km,t
+N
+as in (4.3). Then, with probability at least 1 − δ we have that
+∥Kt − �Km,t
+N ∥H→L2µ(X) ≤
+�
+λN+1 ∥Kt
+H∥ +
+�
+1
+√λN
++ N + 1
+δNλN
+(1 + ∥ϕ∥1)∥ϕ∥1/2
+1
+�
+ε.
+(4.8)
+Proof. First of all, Theorem 4.1 implies that
+∥Kt − �Km,t
+N ∥H→L2µ(X) ≤ ∥Kt − Kt
+N∥H→L2µ(X) + ∥Kt
+N − �Km,t
+N ∥H→L2µ(X)
+≤ ∥Kt − Kt
+N∥H→L2µ(X) +
+�
+1
+√λN
++ N + 1
+δNλN
+(1 + ∥ϕ∥1)∥ϕ∥1/2
+1
+�
+ε.
+Now, for ψ ∈ H,
+∥Ktψ − Kt
+Nψ∥2
+µ =
+�����
+∞
+�
+j=N+1
+⟨Ct
+Hψ, ej⟩ej
+�����
+2
+µ
+=
+∞
+�
+j=N+1
+|⟨Ct
+Hψ, ej⟩|2 =
+∞
+�
+j=N+1
+1
+λj
+|⟨Ct
+Hψ, fj⟩|2
+=
+∞
+�
+j=N+1
+1
+λj
+|⟨Kt
+Hψ, CHfj⟩|2 =
+∞
+�
+j=N+1
+λj|⟨Kt
+Hψ, fj⟩|2 ≤ λN+1∥Kt
+Hψ∥2,
+which proves the theorem.
+□
+We have just proved that the projection error ∥Ktψ − Kt
+Nψ∥µ decays at least as fast as the square
+roots of the eigenvalues of CH. Recall that (λj)j∈N ∈ ℓ1(N), since CH is trace class with �∞
+j=1 λj =
+Tr(CH) = ∥E∗∥2
+HS = ∥ϕ∥1, see Lemma 2.4(c).
+
+18
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+5. ILLUSTRATION WITH THE ORNSTEIN-UHLENBECK PROCESS
+For the numerical illustration of our results, we consider the Ornstein-Uhlenbeck (OU) process on
+X = R, which is given by the SDE
+dXt = −αXt dt + dWt,
+where α > 0 is a positive parameter.
+5.1. Analytical Results. Since all relevant properties of the OU process are available in analytical form,
+we can exactly calculate all of the terms appearing in our theoretical error bounds. Moreover, we can
+also compute the exact estimation and prediction errors for finite data in closed form. Let us begin by
+recapping the analytical results required for our analysis, which can be found in [41].
+The invariant measure µ, and the density of the stochastic transition kernel ρt, are given by
+dµ(x) =
+�α
+π e−αx2 dx
+and
+dρt(x, y) =
+� α
+πv2
+t
+exp
+�
+− α
+v2
+t
+(y − e−αtx)2�
+dx dy,
+with v2
+t = (1 − e−2αt)/2α. The Koopman operators Kt are self-adjoint in L2
+µ(R), their eigenvalues and
+corresponding eigenfunctions are given by
+qj = e−αjt
+and
+ψj(x) =
+1
+�
+2jαjj!
+Hj(
+√
+2αx),
+j ∈ N0,
+where Hj are the physicist’s Hermite polynomials.
+We consider the Gaussian radial basis function (RBF) kernel with bandwidth σ > 0, i.e.,
+k(x, y) = exp
+�
+−(x − y)2
+σ2
+�
+.
+Let us quickly verify that this choice of the kernel satisfies the compatibility assumptions (A1)–(A3).
+Indeed, (A1) is trivial as k(x, x) = 1 and (A3) follows easily from the continuity of the functions in H.
+To see that H is dense in L2
+µ(R) (i.e., (A2)), let ψ ∈ L2
+µ(R) be such that ⟨ψ, Φ(y)⟩µ = 0 for all y ∈ R.
+The latter means that φ∗ϕσ = 0, where φ(x) = ψ(x)e−αx2 and ϕσ(x) = e−x2/σ2. We apply the Fourier
+transform and obtain �φ · �
+ϕσ = 0. Noting that the Fourier transform of a Gaussian is again a Gaussian,
+we get �φ = 0 and thus ψ = 0.
+The Mercer eigenvalues and features with respect to the invariant measure µ of the OU process, i.e.,
+the eigenvalues and eigenfunctions of the integral operator E∗E in L2
+µ(R), are also available in analytical
+form [9]. They are given by
+λi =
+� α
+C1
+�
+1
+σ2C1
+�i
+and
+ϕi(x) = γie−ζ2x2Hi
+�√αηx
+�
+,
+i ∈ N0,
+using the following constants:
+η =
+�
+1 +
+4
+ασ2
+�1/4
+,
+γi =
+�
+η
+2iΓ(i + 1)
+�1/2
+,
+ζ2 = α
+2 (η2 − 1),
+C1 = α + ζ2 + σ−2.
+With these results, we can compute the variance of the empirical estimator for Ct
+H as described in The-
+orem 3.1. The eigenvalues qj were already given above. The coefficients dj,t can be calculated using
+Mercer’s theorem as
+dj,t =
+� �
+k(x, x′)k(y, y′)ψj(x)ψj(x′) dµ0,t(x, y) dµ0,t(x′, y′)
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+19
+=
+�
+k,ℓ
+λkλℓ
+��
+ϕk(x)ϕℓ(y)ψj(x) dµ0,t(x, y)
+�2
+.
+The series needs to be truncated at a finite number of terms and the integrals can be calculated by numer-
+ical integration. As d0,t = ⟨k, kt⟩L2
+µ⊗µ = ∥Ct
+H∥2
+HS (cf. (3.8)), and hence
+∥Ct
+H∥2
+HS =
+�
+k,ℓ
+λkλℓ
+��
+ϕk(x)ϕℓ(y) dµ0,t(x, y)
+�2
+,
+(5.1)
+the Hilbert-Schmidt norm of the cross-covariance operator Ct
+H can be computed similarly. Since, for the
+Gaussian RBF kernel, we have ϕ(x) = k(x, x) = 1 for all x, we therefore find
+E0(t) =
+�
+Ktϕ, ϕ
+�
+µ − ∥Ct
+H∥2
+HS = 1 − ∥Ct
+H∥2
+HS,
+completing the list of terms required by Theorem 3.1. In addition, we notice that upon replacing ei-
+ther one or two of the integrals in (5.1) by finite-data averages, we can also calculate ∥ ˆCm,t
+H ∥2
+HS and
+⟨Ct
+H, ˆCm,t
+H ⟩HS. Therefore, the estimation error for finite data {(xk, yk)}m
+k=1 can be obtained by simply
+expanding the inner product
+∥Ct
+H − ˆCm,t
+H ∥2
+HS = ∥Ct
+H∥2
+HS + ∥ ˆCm,t
+H ∥2
+HS − 2⟨ ˆCm,t
+H , Ct
+H⟩HS,
+allowing us to precisely compare the estimation error to the error bounds obtained in Theorem 3.1.
+Besides the estimation error for Ct
+H, we are also interested in the prediction error, which is bounded
+according to Theorem 4.1. We will compare these bounds to the actual error ∥(Kt
+N − ˆKm,t
+N )φ∥L2µ(X), for
+a specific observable φ ∈ H and a fixed number of N Mercer features. For the OU process, it is again
+beneficial to consider Gaussian observables φ:
+φ(x) =
+1
+�
+2πσ2
+0
+exp
+�
+−(x − m0)2
+2σ2
+0
+�
+.
+Application of the Koopman operator leads to yet another, unnormalized Gaussian observable, which is
+given by
+Ktφ(x) =
+1
+�
+2πσ2
+t
+exp
+�
+−(m0 − e−αtx)2
+2σ2
+t
+�
+,
+σ2
+t = σ2
+0 + v2
+t .
+The inner products of Ktφ with the Mercer eigenfunctions ϕi can be evaluated by numerical integration,
+providing full access to the truncated observable Kt
+Nφ. On the other hand, the empirical approximation
+ˆKm,t
+N φ can be computed directly based on the data. We note that
+ˆKm,t
+N φ =
+N
+�
+j=1
+�
+ˆCm,t
+H φ, ˆej
+�
+ˆej = 1
+m
+m
+�
+k=1
+φ(yk)
+N
+�
+j=1
+⟨Φ(xk), ˆej⟩ ˆej = 1
+m
+m
+�
+k=1
+φ(yk)
+N
+�
+j=1
+ˆej(xk)ˆej.
+The functions ˆej can be obtained from the eigenvalue decomposition of the standard kernel Gramian
+matrix
+1
+mKX := 1
+m [k(xk, xl)]m
+k,l=1 ,
+as the latter is the matrix representation of the empirical covariance operator ˆCm
+H on the subspace
+span{Φ(xk)}m
+k=1. If 1
+mKX = V ΛV ⊤ is the spectral decomposition of the Gramian, then
+ˆej =
+1
+m1/2ˆλj
+m
+�
+l=1
+VljΦ(xl)
+
+20
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+are the correctly normalized eigenfunctions according to Theorem 4.1. Plugging this into the above, we
+find
+ˆKm,t
+N φ(x) = 1
+m
+m
+�
+k=1
+φ(yk)
+N
+�
+j=1
+1
+m1/2ˆλj
+m
+�
+l=1
+Vljk(xl, xk)
+1
+m1/2ˆλj
+m
+�
+r=1
+Vrjk(xr, x)
+= 1
+mφ(Y )⊤ 1
+mKX
+�
+VNΛ−2
+N V ⊤
+N
+�
+KX,x
+= 1
+mφ(Y )⊤VNΛ−1
+N V ⊤
+N KX,x,
+where φ(Y ) = [φ(yk)]m
+k=1, KX,x = [k(xk, x)]m
+k=1, VN = V [IN 0m−N]⊤, ΛN = diag(�λj)N
+j=1.
+5.2. Numerical Results. For the actual numerical experiments, we set α = 1, choose the elementary
+integration time step as ∆t = 10−2, and set the lag time to t = 0.05. We compute the exact variance
+E[∥Ct
+H− ˆCm,t
+H ∥2
+HS] by the expression given in Theorem 3.1, and also the coarser estimate for the variance
+given in Corollary 3.3. We test three different kernel bandwidths, σ ∈ {0.05, 0.1, 0.5}. All Mercer series
+are truncated after the first 10 terms for σ ∈ {0.1, 0.5}, and 20 terms for σ = 0.05, while Koopman
+eigenfunction expansions are truncated after 15 terms.
+In the first set of experiments, we use Chebyshev’s inequality to compute the maximal estimation
+error ∥Ct
+H − ˆCm,t
+H ∥HS that can be guaranteed with confidence 1 − δ = 0.9, for a range of data sizes
+m between m = 20 and m = 50.000. As a comparison, we generate 200 independent simulations of
+length m +
+t
+∆t , corresponding to the sliding-window estimator with m data points, for each data size.
+We then compute the resulting estimation error using the expressions given in the previous section. We
+extract the 1 − δ-percentile of the estimation error for all trajectories, i.e., the maximal error that is not
+exceeded by 100 ∗ (1 − δ) percent of the trajectories. In addition, we also use Chebyshev’s inequality
+with the i.i.d. variance 1
+mE0(t) to predict the estimation error. The comparison of these results for all
+data sizes m and the different kernel bandwidths is shown in Figure 3. We observe that the bound from
+Theorem 3.1 is quite accurate, over-estimating the actual error by about a factor three, and captures the
+detailed qualitative dependence of the estimation error on m. The coarser bound from Corollary 3.3,
+however, appears to discard too much information, it over-estimates the error by one to two orders of
+magnitude, and also does not capture the initial slope for small m. Finally, we note that for the larger
+kernel bandwidths, the i.i.d. variance is indeed too small, leading to an under-estimation of the error.
+This observation confirms that it is indeed necessary to take the effect of the correlation between data
+points into account.
+In a second set of experiments, we test the performance of our theoretical bounds concerning the
+prediction of expectations for individual observables, obtained in Theorem 4.1. For the same three
+Gaussian RBF kernels as in the first set of experiments, we consider the observable φ = ϕ0, i.e., the first
+Mercer feature, and choose N = 10 in the Mercer series expansion Kt
+Nφ and its empirical approximation
+ˆKm,t
+N φ. Note that φ is a different observable depending on the bandwidth. Again, we set 1 − δ = 0.9,
+and use the bound from Theorem 4.1 to bound the L2
+µ-error between Kt
+Nφ and ˆKm,t
+N φ. As a comparison,
+we compute the actual L2
+µ-error by numerical integration, using the fact that we can evaluate Kt
+Nφ and
+ˆKm,t
+N φ based on the discussion above. We repeat this procedure 15 times and provide average errors
+and standard deviations. The results for all three kernels are shown in Figure 4, and we find that our
+theoretical bounds are much too pessimistic in all cases. This finding highlights our previous observation
+that bounding the prediction error outside the RKHS still requires more in-depth research.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+21
+102
+103
+104
+m
+10
+2
+10
+1
+100
+101
+(m)
+Error for Ct , t = 0.05, 
+= 0.050
+T 3.1
+C 3.3
+i.i.d.
+Data
+102
+103
+104
+m
+10
+2
+10
+1
+100
+101
+(m)
+Error for Ct , t = 0.05, 
+= 0.100
+T 3.1
+C 3.3
+i.i.d.
+Data
+102
+103
+104
+m
+10
+2
+10
+1
+100
+101
+(m)
+Error for Ct , t = 0.05, 
+= 0.500
+T 3.1
+C 3.3
+i.i.d.
+Data
+FIGURE 3. Probabilistic error estimates for Ct
+H associated to the OU process, at lag time
+t = 0.05, and the Gaussian RBF kernel with different bandwidths σ ∈ {0.05, 0.1, 0.05}
+(corresponding to left, center and right panels). The blue and green curves show the es-
+timated error using the fine and coarse bounds from Theorem 4.1 and Corollary 3.3, re-
+spectively, while the purple curves represent the bound obtained from the i.i.d.-variance
+1
+mE0(t). The red curve shows the 0.9-percentile of the estimation error based on 200
+independent simulations.
+102
+103
+m
+10
+1
+101
+103
+105
+107
+Prediction Error t = 0.05, 
+= 0.050
+Prediction N = 10
+Data Bound N = 10
+102
+103
+m
+10
+1
+101
+103
+105
+107
+Prediction Error t = 0.05, 
+= 0.100
+Prediction N = 10
+Data Bound N = 10
+102
+103
+m
+10
+1
+101
+103
+105
+107
+Prediction Error t = 0.05, 
+= 0.500
+Prediction N = 10
+Data Bound N = 10
+FIGURE 4. Comparison of the theoretical bound on the prediction error ∥Kt
+Nφ −
+ˆKm,t
+N φ∥µ, if φ is chosen as the first Mercer feature ϕ0, using N = 10 in the Mercer
+series representation. The predicted error is shown in blue, error bars for the actual error
+obtained from 15 independent data sets are shown in red. Different panels correspond
+to the same kernel bandwidths as in Figure 3 above.
+6. CONCLUSIONS
+We have analyzed the finite-data estimation error for data-driven approximations of the Koopman
+operator on reproducing kernel Hilbert spaces. More specifically, we have provided an exact expression
+for the variance of empirical estimators for the cross-covariance operator, if a sliding-window estimator
+is applied to a long ergodic trajectory of the dynamical system. This setting is relevant for many complex
+systems, such as molecular dynamics simulations. Our results present a significant improvement over
+the state of the art, since they concern a setting where the notorious problem of dictionary selection
+can be circumvented, and therefore no longer depend on the dictionary size. We have also extended
+the concept of asymptotic variance to an infinite-dimensional approximation space for the Koopman
+operator. Our numerical study on the Ornstein Uhlenbeck process has shown that, even using a simple
+mass concentration inequality, accurate bounds on the estimation error can be obtained.
+In our second result, we have extended our estimates to a uniform bound on the prediction error for
+observables in the RKHS. Thereby, we have circumvented dealing with an unbounded inverse of the
+covariance operator by applying a finite-dimensional truncation of the associated Mercer series. In case
+of Koopman-invariance of the RKHS, however, we were able to find a bound on the truncation error
+which then yields estimates for the full approximation error.
+
+22
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Still, the resulting error bounds have proven very conservative in the numerical examples. Therefore,
+obtaining sharper bounds on the prediction error constitutes a primary goal for future research.
+REFERENCES
+[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
+[2] A. Beck, and J. T. Schwartz, A vector-valued random ergodic theorem, Proc. Amer. Math. Soc. 8(6) (1957), 1049–1059.
+[3] D. Bosq, Linear Processes in Function Spaces – Theory and Applications, Springer-Verlag New York, Inc., 2000.
+[4] S. L. Brunton, J.L. Proctor, and J.N. Kutz, Discovering governing equations from data by sparse identification of nonlin-
+ear dynamical systems, Proc. Natl. Acad. Sci. 113 (2016), 3932–3937.
+[5] S. L. Brunton, M. Budisic, E. Kaiser, J. N. Kutz, Modern Koopman theory for dynamical systems, SIAM Rev. 64(2)
+(2022), 229–340.
+[6] A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic
+Publishers, 2004.
+[7] O. Christensen, An Introduction to Frames and Riesz Bases, 2nd ed., Springer International Publishing Switzerland,
+2016.
+[8] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc.
+17 (1966), 413–415.
+[9] G. E. Fasshauer and M. J. McCourt, Stable Evaluation of Gaussian Radial Basis Function Interpolants SIAM J. Sci.
+Comput. 2012 34:2, A737–A762.
+[10] K. Fukumizu, L. Song, and A. Gretton, Kernel Bayes’ Rule: Bayesian inference with positive definite kernels, J. Mach.
+Learn. Res. 14 (2013), 3753–3783.
+[11] D. Giannakis, Data-driven spectral decomposition and forecasting of ergodic dynamical systems, Appl. Comput. Har-
+mon. Anal, 47(2) (2019), 338–396.
+[12] I.C. Gohberg and M.G. Krein, Introduction To The Theory of Linear Nonselfadjoint Operators, volume 18 of Translations
+of Mathematical Monographs. American Mathematical Society, 1969.
+[13] M. Hairer, Ergodic properties of Markov processes, Lecture notes, https://www.hairer.org/notes/Markov.pdf
+[14] M. Haseli and J. Cort´es, Learning Koopman eigenfunctions and invariant subspaces from data: Symmetric subspace
+decomposition, IEEE Trans. Automat. Control 67(7) (2022), 3442-3457.
+[15] O. Kallenberg, Foundations of modern probability, Springer-Verlag, New York, 1997.
+[16] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.
+[17] E. Kaiser, J. N. Kutz, S. L. Brunton, Data-driven discovery of Koopman eigenfunctions for control, Mach. Learn. Sci.
+Technol. 2 (2021), 035023.
+[18] S. Klus, P. Koltai, and C. Sch¨utte, On the numerical approximation of the Perron–Frobenius and Koopman operator, J.
+Comput. Dyn. 1(3) (2016), 51–79.
+[19] S. Klus, F. N¨uske, P. Koltai, H. Wu, I. Kevrekidis, C. Sch¨utte, and F. No´e, Data-driven model reduction and transfer
+operator approximation, J. Nonlinear Sci. 28(3) (2018), 985–1010.
+[20] S. Klus, I. Schuster, K. Muandet, Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces, J.
+Nonlinear Sci. 30(1) (2020), 283–315.
+[21] S. Klus, and C. Sch¨utte, Towards tensor-based methods for the numerical approximation of the Perron–Frobenius and
+Koopman operator, J. Comput. Dyn. 3(2) (2016), 139–161.
+[22] II. Klebanov, I. Schuster, and T.J. Sullivan, A rigorous theory of conditional mean embeddings, SIAM J. Math. Data Sci.
+2 (2020), 583–606.
+[23] B. O. Koopman, Hamiltonian Systems and Transformations in Hilbert Space, Proc. Natl. Acad. Sci. 5 (17) (1931),
+315–318.
+[24] M. Korda and I. Mezi´c, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive
+control, Automatica 93 (2018), 149–160.
+[25] M. Korda and I. Mezi´c, Optimal construction of Koopman eigenfunctions for prediction and control, IEEE Trans. Au-
+tomat. Control 65 (12) (2020), 5114–5129.
+[26] A. J. Kurdila and P. Bobade, Koopman theory and linear approximation spaces, Preprint, arXiv:1811.10809 (2018).
+[27] T. Leli`evre, and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numer. 25
+(2016), 681–880.
+[28] G. Lumer and R.S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698.
+[29] B. Lusch, J. N. Kutz, and S. L. Brunton, Deep learning for universal linear embeddings of nonlinear dynamics, Nat.
+Commun. 9(1) (2018), 1–10.
+[30] M. Mollenhauer, On the Statistical Approximation of Conditional Expectation Operators, Dissertation, Freie Universit¨at
+Berlin, 2021.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+23
+[31] K. Manohar, B. W. Brunton, J. N. Kutz, S. L. Brunton, Data-driven sparse sensor placement for reconstruction: Demon-
+strating the benefits of exploiting known patterns. IEEE Control Systems Magazine, 38(3) (2018), 63–86.
+[32] A. Mardt, L. Pasquali, H. Wu, and F. No´e, VAMPnets for deep learning of molecular kinetics, Nat. Commun. 9, 5 (2018).
+[33] A. Mauroy, Y. Susuki, and I. Mezi´c, Koopman operator in systems and control, Springer International Publishing, Berlin,
+2020.
+[34] M. Mollenhauer, S. Klus, C. Sch¨utte, and P. Koltai, Kernel autocovariance operators of stationary processes: Estimation
+and convergence, J. Mach. Learn. Res. 23(327) (2022), 1–34.
+[35] F. No´e and F. N¨uske, A variational approach to modeling slow processes in stochastic dynamical systems, Multiscale
+Model. Simul. 11 (2013), 635–655.
+[36] F. N¨uske, B. G. Keller, G. P´erez-Hern´andez, A. S. J. S. Mey, and F. No´e, Variational approach to molecular kinetics, J.
+Chem. Theory Comput. 10 (2014), 1739–1752.
+[37] F. N¨uske, P. Gelß, S. Klus, and C. Clementi, Tensor-based computation of metastable and coherent sets, Physica D 427
+(2021), 133018.
+[38] F. N¨uske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, Finite-data error bounds for Koopman-based prediction
+and control, J. Nonlinear Sci. 33 (2023), 1–34.
+[39] B. Øksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Corrected Printing,
+Springer-Verlag Heidelberg New York, 2000.
+[40] V. Paulsen, An introduction to the theory of reproducing kernel Hilbert spaces, https://www.math.uh.edu/ vern/rkhs.pdf
+[41] G. A. Pavliotis, Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations,
+Texts in Applied Mathematics, vol. 60, Springer, 2014.
+[42] S. Peitz, and S. Klus, Koopman operator-based model reduction for switched-system control of PDEs, Automatica 106
+(2019), 184–191.
+[43] I. Pinelis, Optimum bounds for the distributions of martingales in Banach spaces, The Annals of Probability 22 (1994),
+1679–1706.
+[44] J. H. Prinz, H. Wu, M. Sarich, B. G. Keller, M. Senne, M. Held, J. D. Chodera, C. Sch¨utte, and F. No´e, Markov models
+of molecular kinetics: Generation and validation, J. Chem. Phys. 17(134) (2011), 174105.
+[45] F. Riesz and B. Nagy, Functional Analysis, Blackie & Son Ltd., Glasgow, Bombay, Toronto, 1955.
+[46] N. Rhomari, Approximation et in´egalit´es exponentielles pour les sommes des vecteurs al´eatoires d´ependants, Comptes
+rendus de l’Acad´emie des science, S´erie 1, 334 (2002), 149–154.
+[47] W. Rudin, Functional Analysis, Second edition, McGraw-Hill, Inc., 1991.
+[48] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, Inc., 1987.
+[49] M. Schaller, K. Worthmann, F. Philipp, S. Peitz, and F. N¨uske, Towards reliable data-based optimal and predictive control
+using extended DMD, IFAC PapersOnLine, to appear, arXiv preprint arXiv:2202.09084.
+[50] C. Sch¨utte, A. Fischer, W. Huisinga, and P. Deuflhard, A direct approach to conformational dynamics based on hybrid
+Monte Carlo, J. Comput. Phys. 1(151) (1999), 146–168.
+[51] A. Smola, A. Gretton, L. Song, and B. Sch¨olkopf, A Hilbert Space Embedding for Distributions, in: M. Hutter, R.A.
+Servedio, and E. Takimoto (Eds.): ALT 2007, Lecture Notes in Computer Science, vol. 4754, pp. 13–31, 2007. Springer,
+Berlin, Heidelberg.
+[52] I. Steinwart and A. Christmann, Support Vector Machines, Springer Science+Business Media, LLC, 2008.
+[53] B.K. Sriperumbudur, K. Fukumizu, and G.R.G. Lanckriet, Universality, characteristic kernels and RKHS embedding of
+measures, J. Mach. Learn. Res. 12 (2011), 2389–2410.
+[54] M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending
+dynamic mode decomposition, J. Nonlinear Sci. 25(6) (2015), 1307–1346
+[55] M. O. Williams, C. W. Rowley, and I Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J.
+Comput. Dyn. 2(2) (2015), 247–265.
+[56] H. Wu, and F. No´e, Variational approach for learning Markov processes from time series data, J. Nonlinear Sci. 30(1)
+(2020), 23–66.
+[57] Y. Yu, T. Wang, and R. J. Samworth, A useful variant of the Davis-Kahan theorem for statisticians, Biometrika 102
+(2015), 315–323.
+[58] C. Zhang and E. Zuazua, A quantitative analysis of Koopman operator methods for system identification and predictions,
+Comptes Rendus. M´ecanique, Online first (2023), 1–31.
+
+24
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+APPENDIX A. PROOFS
+Proof of Lemma 2.1. Let ψ ∈ H. Then (2.4) follows from
+�
+|ψ(x)|2 dµ(x) =
+�
+|⟨ψ, Φ(x)⟩|2 dµ(x) ≤ ∥ψ∥2
+�
+ϕ(x) dµ(x) = ∥ψ∥2∥ϕ∥1.
+Assume that (A2) holds and that ψ ∈ L2
+µ(X) is such that ⟨ψ, Φ(x)⟩µ = 0 for all x ∈ X. Then
+0 =
+�
+⟨ψ, Φ(x)⟩µψ(x) dµ(x) =
+� �
+k(x, y)ψ(x)ψ(y) dµ(x) dµ(y).
+Hence, ψ = 0 by (A2). Conversely, assume that H is dense in L2
+µ(X). Let ψ ∈ L2
+µ(X) such that
+� �
+k(x, y)ψ(x)ψ(y) dµ(x) dµ(y) = 0.
+Since the integrand equals ⟨ψ(x)Φ(x), ψ(y)Φ(y)⟩ and the
+integral
+�
+ψ(x)Φ(x) dµ(x) exists by (2.5), we obtain
+�
+ψ(x)Φ(x) dµ(x) = 0H.
+This implies that
+⟨ψ, Φ(y)⟩µ =
+�
+ψ(x)k(x, y) dµ(x) = 0 for each y ∈ X. Hence, ⟨ψ, φ⟩µ = 0 for each φ ∈ H :=
+span{Φ(x) : x ∈ X}. Now, let φ ∈ H. Then there exists a sequence (φn) ⊂ H such that ∥φn − φ∥ → 0
+as n → ∞. Therefore,
+|⟨ψ, φ⟩µ| = |⟨ψ, φ − φn⟩µ| ≤ ∥ψ∥µ∥φ − φn∥µ ≤ ∥ψ∥µ
+�
+∥ϕ∥1∥φ − φn∥.
+Hence, ⟨ψ, φ⟩µ = 0, and the density of H in L2
+µ(X) implies ψ = 0.
+□
+Proof of Lemma 2.4. (a) For ψ ∈ L2
+µ(X) we have
+∥Eψ∥2 =
+� �
+ψ(x)ψ(y)⟨Φ(x), Φ(y)⟩ dµ(x) dµ(y) =
+� �
+k(x, y)ψ(x)ψ(y) dµ(x) dµ(y).
+Hence, the injectivity of E follows from (A2). If (ei) is an orthonormal basis of H, then
+�
+i
+∥E∗ei∥2
+µ =
+�
+i
+∥ei∥2
+µ =
+�
+i
+�
+|ei(x)|2 dµ(x) =
+�
+i
+�
+|⟨Φ(x), ei⟩|2 dµ(x) =
+�
+∥Φ(x)∥2 dµ(x).
+The claim is now a consequence of ∥Φ(x)∥2 = ϕ(x).
+(b) By Lemma 2.1, H is dense in L2
+µ(X). Moreover, E∗ is compact by (a) and Schauder’s theorem
+[47, Theorem 4.19].
+(c) This follows from (a) and ker CH = ker EE∗ = ker E∗ = {0} by (A3).
+□
+Proof of Theorem 2.5. By Lemma 2.4, the operator E∗E ∈ B(L2
+µ(X)) is a positive self-adjoint trace-
+class operator. Hence, by the well known spectral theory of compact operators (see, e.g., [12]) there
+exists an orthonormal basis (ej)∞
+j=1 of L2
+µ(X) consisting of eigenfunctions of E∗E corresponding to a
+summable sequence (λj)∞
+j=1 of strictly positive eigenvalues. Since E∗ψ = ψ for ψ ∈ H, we have
+Eej = λjej and thus ej ∈ H for all j and CHej = EE∗ej = Eej = λjej. Moreover, ⟨fi, fj⟩ =
+�
+λj/λi⟨Eei, ej⟩ =
+�
+λj/λi⟨ei, ej⟩µ = δij by (2.6) so that the fj indeed form an orthonormal system in
+H. The completeness of (fj) in H follows from the injectivity of E. Finally, �∞
+j=1 λj = Tr CH = ∥ϕ∥1
+and
+k(x, y) = ⟨Φ(x), Φ(y)⟩ =
+�
+j
+⟨Φ(x), fj⟩⟨fj, Φ(y)⟩ =
+�
+j
+fj(x)fj(y),
+which completes the proof.
+□
+Proof of Proposition 2.7. Let ψ ∈ B(X). For p = ∞ we have |(Ktψ)(x)| = |Ex[ψ(Xt)]| ≤ Ex[|ψ(Xt)|] ≤
+∥ψ∥∞. If p < ∞, by Jensen’s inequality, for every convex φ : R → R we have φ ◦ Ktψ ≤ Kt(φ ◦ ψ)
+and thus |Ktψ|p ≤ Kt|ψ|p, which, by invariance of µ, leads to
+∥Ktψ∥p
+p =
+�
+|Ktψ|p dµ ≤
+�
+Kt|ψ|p dµ =
+�
+|ψ|p dµ = ∥ψ∥p
+p.
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+25
+The claim now follows by density of B(X) in Lp
+µ(X).
+□
+Proof of Proposition 2.8. Let ψ ∈ Cb(X) and fix x ∈ X. Denote the stochastic solution process of the
+SDE (2.1) with initial value x by Xx
+t . Since Xx
+t (ω) is continuous in t for P-a.e. ω ∈ Ω (see [39, Theorem
+5.2.1]), ψ(Xx
+t (ω)) → ψ(Xx
+0 (ω)) = ψ(x) as t → 0 for P-a.e. ω ∈ Ω. Hence, by dominated convergence,
+Ktψ(x) = E[ψ(Xx
+t )] =
+�
+ψ(Xx
+t (ω)) dP(ω) → ψ(x)
+as t → 0. It now follows from Proposition 2.7 and, again, dominated convergence that ∥Ktψ −ψ∥p → 0
+as t → 0. If ψ ∈ Lp
+µ(X) and ε > 0, there exists η ∈ Cb(X) such that ∥ψ − η∥p < ε/3. Choose δ > 0
+such that ∥Ktη − η∥p < ε/3 for t < δ. Then
+∥Ktψ − ψ∥p ≤ ∥Kt(ψ − η)∥p + ∥Ktη − η∥p + ∥η − ψ∥p < ε
+for t < δ, which proves the claim.
+□
+APPENDIX B. RIESZ BASES
+Recall that a Riesz basis [7] of a Hilbert space H is a sequence (ψj) ⊂ H satisfying span{ψj} = H
+and for which there exist A, B > 0 such that for all c ∈ ℓ2,
+A∥c∥2 ≤
+���
+�
+j
+cjψj
+���
+H ≤ B∥c∥2.
+The constant A (B, resp.) is called a lower (upper, resp.) Riesz bound of the basis. Also recall that to
+every Riesz basis (ψj) there exists a dual Riesz basis ( �ψj) such that ⟨ψj, �ψk⟩H = δjk. If (ψj) has the
+bounds A and B, then ( �ψj) has bounds 1/B and 1/A. Every element f of H admits a representation
+f = �
+j⟨f, �ψj⟩Hψj = �
+j⟨f, ψj⟩H �ψj and
+A2∥f∥2
+H ≤
+�
+j
+|⟨f, ψj⟩|2 ≤ B2∥f∥2
+H
+and
+B−2∥f∥2
+H ≤
+�
+j
+|⟨f, �ψj⟩|2 ≤ A−2∥f∥2
+H.
+It can furthermore be easily seen that a sequence (ψj) ⊂ H is a Riesz basis of H if and only if there
+exists a boundedly invertible linear operator S ∈ L(H) and an orthonormal basis (ej) of H such that
+ψj = Sej for all j. Then �ψj = (S−1)∗ej for all j, B = ∥S∥, and A = ∥S−1∥−1.
+APPENDIX C. SOME FACTS FROM SPECTRAL THEORY
+In this section, let H be a Hilbert space. If P is an orthogonal projection in H, we set P ⊥ = I − P.
+For v ∈ H, ∥v∥ = 1, denote by Pv the rank-one orthogonal projection onto span{v}.
+We say that a linear operator on H is non-negative if it is self-adjoint and its spectrum is contained
+in [0, ∞). For a non-negative compact operator T on H we denote by λ1(T) ≥ λ2(T) ≥ . . . the
+eigenvalues of T in descending order (counting multiplicities). We set λj(T) = 0 if j > rank(T).
+Moreover, if T has only simple eigenvalues6, we let Pj(T) denote the orthogonal projection onto the
+eigenspace ker(T − λj(T)) and Qn(T) = �n
+j=1 Pj(T) the spectral projection corresponding to the n
+largest eigenvalues of T.
+Theorem C.1 ([12, Cor. II.2.3]). If T and �T are two non-negative compact operators on H, then for all
+j ∈ N,
+|λj(T) − λj( �T)| ≤ ∥T − �T∥.
+6i.e., dim ker(T − λ) = 1 for each eigenvalue λ of T
+
+26
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+Lemma C.2. For v, w ∈ H with ∥v∥ = ∥w∥ = 1 we have
+∥Pv − Pw∥ = ∥P ⊥
+w Pv∥ =
+�
+1 − |⟨v, w⟩|2.
+(C.1)
+Proof. First of all, the second equation in (C.1) is clear, since
+∥P ⊥
+w Pvf∥2 = ∥⟨f, v⟩P ⊥
+w v∥2 = |⟨f, v⟩|2(1 − ∥Pwv∥2) = |⟨f, v⟩|2(1 − |⟨v, w⟩|2).
+Second, if Pv,w denotes the orthogonal projection onto Hv,w := span{v, w}, we have
+∥Pv − Pw∥ = ∥(Pv − Pw)Pv,w∥ = ∥(Pv − Pw)|Hv,w∥ =
+sup
+x∈Hv,w, ∥x∥=1
+∥(Pv − Pw)x∥,
+which is a two-dimensional problem in Hv,w. Now, if x ∈ Hv,w, ∥x∥ = 1, we write x = av + bw and
+obtain a2 + 2abγ + b2 = 1, where γ = ⟨v, w⟩. Moreover, ⟨x, v⟩ = a + bγ, ⟨x, w⟩ = aγ + b and so
+∥(Pv − Pw)x∥2 = ∥⟨x, v⟩v − ⟨x, w⟩w∥2 = ∥(a + bγ)v − (aγ + b)w∥2
+= (a + bγ)2 − 2(a + bγ)(aγ + b)γ + (aγ + b)2
+= a2 + 2abγ + b2γ2 − 2γ(a2γ + abγ2 + ab + b2γ) + a2γ2 + 2abγ + b2
+= (1 − γ2)a2 + 2abγ − 2abγ3 + b2(1 − γ2)
+= (1 − γ2)(a2 + b2 + 2abγ)
+= 1 − |⟨v, w⟩|2.
+Hence, the objective function is constant on {x ∈ Hv,w : ∥x∥ = 1} and (C.1) is proved.
+□
+The next theorem is a variant of the Davis-Kahan sin(Θ) theorem (cf. [57]).
+Theorem C.3. Let T and �T be non-negative Hilbert-Schmidt operators on H, let n ∈ N, assume that
+the largest n + 1 eigenvalues of T are simple, and set
+δ =
+min
+j=1,...,n
+λj(T) − λj+1(T)
+2
+.
+If ∥T − �T∥HS < δ, then for j = 1, . . . , n we have
+∥Pj(T) − Pj( �T)∥ ≤ ∥T − �T∥
+δ
+.
+Proof. For j ∈ N put λj = λj(T), Pj = Pj(T), �λj = λj( �T), and �Pj = Pj( �T). By Theorem C.1, we
+have |λj − �λj| ≤ ∥T − �T∥HS < δ for all j, hence �λj is contained in the interval Ij = (λj − δ, λj + δ)
+for j = 1, . . . , n + 1. By assumption, sup Ij+1 ≤ inf Ij for j = 1, . . . , n. In particular, the intervals
+I1, . . . , In+1 are pairwise disjoint.
+Now, let j ∈ {1, . . . , n}. Then for k ∈ N \ {j} we have |�λk − λj| > δ. Therefore, we have
+dist(λj, σ( �T)\{�λj}) ≥ δ and thus, for f ∈ �P ⊥
+j H,
+∥( �T − λj)f∥ ≥ dist
+�
+λj, σ( �T| �P ⊥
+j H)
+�
+∥f∥ = dist(λj, σ( �T)\{�λj})∥f∥ ≥ δ∥f∥.
+As TPj = λjPj and �P ⊥
+j �T = �T �P ⊥
+j , we obtain
+∥T − �T∥ ≥ ∥ �P ⊥
+j ( �T − T)Pj∥ = ∥ �P ⊥
+j �TPj − �P ⊥
+j TPj∥ = ∥( �T − λj) �P ⊥
+j Pj∥ ≥ δ∥ �P ⊥
+j Pj∥.
+The claim now follows from Lemma C.2.
+□
+
+ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR
+27
+APPENDIX D. ERGODICITY AND THE GENERATOR
+In this section, we prove the following proposition on the spectral properties of the generator L under
+the ergodicity assumption.
+Proposition D.1. Assume that the invariant measure µ is ergodic. Then ker L = span{1} and ker(L −
+iωI) = {0} for ω ∈ R\{0}.
+Proof. First of all, it is worth mentioning that Lψ = 0 implies Ktψ = ψ for all t ≥ 0 and that Lψ = iωψ,
+ω ∈ R \ {0}, implies K2π/ωψ = ψ. Therefore, it suffices to show that Ktψ = ψ for some t > 0 and
+ψ ∈ L2
+µ(X) is only possible for constant ψ. For this, we consider the Markov process (Xnt)∞
+n=0. For
+convenience, we assume w.l.o.g. that t = 1 holds. By invariance of µ, the process (Xn)∞
+n=0 is stationary,
+i.e., (Xn)∞
+n=0 and (Xn+1)∞
+n=0 are equally distributed as X N0-valued random variables. According to
+[15, Lemma 9.2] there exist X-valued random variables X−k, k ∈ N, such that X := (Xn)n∈Z is also
+stationary. By Pµ denote the law of the X Z-valued random variable X.
+On S := X Z define the left shift T : S → S by T(xn)n∈Z := (xn+1)n∈Z. Stationarity of X means
+that also TX ∼ Pµ.
+A set A ∈ BZ
+X := �
+k∈Z BX is called shift-invariant if T −1A = A. It is easy to see that the set of
+shift-invariant sets forms a sub-σ-algebra I of BZ
+X . Now, by [13, Corollary 5.11] and the ergodicity of
+µ we have Pµ(A) ∈ {0, 1} for any A ∈ I. Now, Birkhoff’s Ergodic Theorem [15, Theorem 9.6] states
+that
+lim
+n→∞
+1
+n
+n−1
+�
+k=0
+f(T kX) = E
+�
+f(X)|X−1I
+�
+(D.1)
+almost surely and in L1(Ω) for any f ∈ L1(S). Given ψ ∈ L1
+µ(X), let us apply this theorem to the
+function f = ψ ◦ π0, where the projection π0 : S → X is defined by π0(xn)n∈Z = x0. First of all,
+�
+|f| dPµ =
+�
+|ψ(x0)| dPµ((xn)n∈Z) =
+�
+|ψ(x)| dµ(x) < ∞
+as Pµ ◦ π−1
+0
+= µ. Hence, we have f ∈ L1(S). Furthermore, we compute f(T kX) = ψ(π0(T kX)) =
+ψ(Xk). For A ∈ I we have P(X−1A) = Pµ(A) ∈ {0, 1}. Thus, we obtain
+lim
+n→∞
+1
+n
+n−1
+�
+k=0
+ψ(Xk) = E[f(X)] =
+�
+f dPµ =
+�
+ψ ◦ π0 dPµ =
+�
+ψ dµ
+almost surely and in L1(Ω).
+Therefore, if ψ ∈ L2
+µ(X) such that Ktψ = ψ, then Kktψ = ψ for all k ∈ N0, hence for µ-a.e. x ∈ X
+we have
+ψ(x) = 1
+n
+n−1
+�
+k=0
+ψ(x) = 1
+n
+n−1
+�
+k=0
+Kktψ(x) = 1
+n
+n−1
+�
+k=0
+E[ψ(Xkt)|X0 = x]
+= E
+�
+1
+n
+n−1
+�
+k=0
+ψ(Xkt)
+����� X0 = x
+�
+n→∞
+−→
+�
+ψ dµ.
+Thus, ψ must indeed be (µ-essentially) constant.
+□
+
+28
+F. PHILIPP, M. SCHALLER, K. WORTHMANN, S. PEITZ, AND F. N ¨USKE
+AUTHOR AFFILIATIONS
+F. Philipp TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25, D-98693
+ILMENAU, GERMANY
+Email address: [email protected]
+M. Schaller TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25, D-
+98693 ILMENAU, GERMANY
+Email address: [email protected]
+K. Worthmann TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25,
+D-98693 ILMENAU, GERMANY
+Email address: [email protected]
+S. Peitz PADERBORN UNIVERSITY, DEPARTMENT OF COMPUTER SCIENCE, DATA SCIENCE FOR ENGINEERING, GER-
+MANY
+Email address: [email protected]
+F. N¨uske MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS, MAGDEBURG, GERMANY
+Email address: [email protected]
+

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+A Mapping of Assurance Techniques for Learning Enabled
+Autonomous Systems to the Systems Engineering Lifecycle
+Christian Ellis1, Maggie Wigness2, and Lance Fiondella1
+Abstract—Learning enabled autonomous systems provide in-
+creased capabilities compared to traditional systems. However,
+the complexity of and probabilistic nature in the underlying
+methods enabling such capabilities present challenges for current
+systems engineering processes for assurance, and test, evalua-
+tion, verification, and validation (TEVV). This paper provides
+a preliminary attempt to map recently developed technical
+approaches in the assurance and TEVV of learning enabled
+autonomous systems (LEAS) literature to a traditional systems
+engineering v-model. This mapping categorizes such techniques
+into three main approaches: development, acquisition, and
+sustainment. We review the latest techniques to develop safe,
+reliable, and resilient learning enabled autonomous systems,
+without recommending radical and impractical changes to exist-
+ing systems engineering processes. By performing this mapping,
+we seek to assist acquisition professionals by (i) informing
+comprehensive test and evaluation planning, and (ii) objectively
+communicating risk to leaders.
+I. INTRODUCTION
+It is widely recognized [1] that the complexity and resulting
+capabilities of autonomous systems created using machine
+learning methods, which we refer to as learning enabled
+autonomous systems (LEAS), pose new challenges to sys-
+tems engineering compared to their traditional counterparts.
+Moreover, the inability to translate qualitative assessments
+to quantitative metrics which measure system performance
+hinder adoption. Such limitations make it difficult to produce
+reliable systems, and even harder to assure [2]. Without under-
+standing the capabilities and limitations of existing assurance
+techniques, defining safety and performance requirements that
+are both clear and testable remains out of reach.
+Mature test, evaluation, verification, and validation (TEVV)
+methods have been in use for decades to ensure the safety
+analysis and acquisition of hardware systems [3], but fewer
+TEVV methods for software are available, and even fewer
+for software that improves itself through learning [4]. Initial
+approaches to autonomous systems use control theory to
+physically model the world and its underlying dynamics [5],
+while LEAS infer and generalize statistical patterns, which
+lead to the achievement of goals from a sample of pre-
+collected training data points. However, due to the nature of
+the environments where LEAS are fielded and the massive
+size of their underlying state spaces, systems will likely
+encounter states during operation they have never experienced
+before, yet still being required to take action.
+1 Christian Ellis is a PhD Student and Lance Fiondella is an Asso-
+ciate Professor in the Department of Electrical and Computer Engineer-
+ing at the University of Massachusetts Dartmouth, USA. cellis3,
+[email protected]
+2 Maggie Wigness is a researcher at the United States Army Research
+Laboratory (ARL).
+Fig. 1: Recent work in assurance for LEAS is mapped to
+relevant stages of the MITRE systems engineering lifecy-
+cle [6], into 3 distinct categories—development, acquisition,
+and sustainment.
+Early work from the autonomy community identified issues
+that arise from incorporating learning into autonomous sys-
+tems [4] including state space explosion, operation in unpre-
+dictable environments, emergent behavior, and effective hu-
+man machine interaction. Assurance methods such as formal
+methods [7], or reliability analysis [8] seek to provide either
+certain or probabilistic guarantees on system performance.
+Formal methods support verification by exhaustively search
+and identifying dangerous regions of the state space and
+provide techniques to avoid such states. Reliability analysis
+supports test and evaluation by quantifying the probability
+a system will be operational at a point in time from oper-
+ational data collected throughout the systems lifecycle. The
+aforementioned state space explosion makes formal methods
+challenging to scale, and reliability analysis difficult to ac-
+curately predict estimates. A different field of research seeks
+to develop methods which explicitly consider safety during
+the learning and operational stages [9]. Lastly, investments
+such as the DARPA Assured Autonomy program1 seeks to
+continually assure learning enabled cyber-physical systems
+by constructing formal methods that assure correctness at
+design time and perform runtime monitoring at operation
+time. While this program has advanced the state of the
+1https://www.darpa.mil/program/assured-autonomy
+arXiv:2301.00057v1  [cs.SE]  30 Dec 2022
+
+Transition
+Concept
+Operation&
+Development
+Maintenance
+Requirements
+Test&
+Engineering
+Evaluation
+System
+System
+Architecture
+Integration
+SystemDesign
+&Development
+Development
+Acquisition
+Sustainmentart in formal methods [10] and runtime monitoring [11], a
+systematic approach to identify outstanding gaps will remain
+unclear unless the community makes explicit and coordinated
+efforts to understand how such methods may be incorporated
+into the broader systems engineering process.
+Our work seeks to communicate recent technical devel-
+opments in LEAS assurance with a focus on autonomous
+vehicles, accompanying recent literature reviews [12] [13], by
+mapping such developments to distinct steps of a well known
+systems engineering model chosen due to its prevalence,
+namely the v-model. Fig. 1 shows the mapping and identifies
+three top level lifecycle phases: development, acquisition, and
+sustainment. For each top level lifecycle phase, a section of
+the paper has been dedicated to outlining recent technical de-
+velopments and how they contribute to the goals of the phase.
+This representation helps identify where the latest methods for
+TEVV fit in the broader systems engineering process while
+also enabling systematic consideration of potential sources of
+defects, faults, and attacks. Note that we use the v-model only
+to assist the classification of where TEVV methods fit. This is
+not a recommendation to use a certain software development
+lifecycle over another.
+The remainder of the paper is organized as follows. Sec-
+tion II outlines the specific scientific fields supporting LEAS.
+Section III provides an overview of the mapping between tra-
+ditional systems engineering and the state of the art in assur-
+ance for LEAS. Section IV maps assurance techniques, which
+assist development to design and development, and system
+integration in the systems engineering lifecycle. Section V
+maps assurance techniques, which assist acquisition to test
+and evaluation. Section VI maps assurance techniques, which
+assist sustainment to transition, operation, & maintenance.
+Lastly, section VII concludes with areas this mapping can
+impact.
+II. METHODS SUPPORTING THE DEVELOPMENT OF
+LEARNING ENABLED AUTONOMOUS SYSTEMS
+This section seeks to define the fields of engineering
+with significant impact on development of LEAS with a
+focus on vehicles and their corresponding challenges for
+assurance. In later sections (Sec. IV—VI), solutions to such
+challenges are identified and categorized according to where
+they reside within the systems engineering lifecycle such as
+development, acquisition, or sustainment. Rather than seek to
+obtain an exhaustive list of engineering fields, of which there
+are many, we first provide an overview of learning enabled
+autonomous vehicles and then review two key contributing
+fields, including machine learning and reinforcement learning.
+While there are other non-learning methods such as optimal
+control theory [14], which have made large and long lasting
+impacts on the development on LEAS, they are not considered
+in this paper.
+LEAS normally follow one of two design approaches, end-
+to-end (E2E) or modular. In the E2E approach [15], a system’s
+sensors act as the input to a learning algorithm. For example, a
+deep neural network outputs the corresponding actions such as
+steering wheel angle (lateral control) [16], torque (longitudi-
+nal control) [17], or both [18]. In the modular approach [19], a
+system’s sensors act as input to a perception sub-system which
+is responsible for building a map and model of the world.
+Such subsystems commonly include perception components
+that use ML techniques such as semantic segmentation [20] or
+object detection [21]. This model is then used by a planning
+subsystem, which outputs a kinematically feasible trajectory
+to which controls are applied [22], [23].
+While it may be possible to break down layers of E2E
+neural networks into sub-components using interpretability
+techniques [24], this paper specifically focuses on the modular
+approach for two reasons: i) it is clear to a human what the
+responsibility of each component is (increased interpretabil-
+ity), and ii) the modular approach is currently more common
+in autonomous vehicle designs in industry and government
+implementations. While some of the underlying problems for
+assurance are the same for both approaches, including those
+previously mentioned [4], we explicitly consider software
+assurance methods which are applicable to either perception
+or planning components, or the joint-combination thereof.
+A. Machine Learning
+In machine learning (ML), tasks are completed by training
+a model from data to perform function approximation using
+a combination of mathematical optimization and statistical
+techniques [25]. This results in computer programs which are
+able to complete a task without constructing a set of exact
+solution instructions ahead of time. There are three main
+forms of learning, including supervised, unsupervised, and
+reinforcement learning. In supervised learning, each training
+sample from the dataset is associated with a set of features and
+a corresponding label to train a model. For example, a neural
+network can be trained on a dataset of images containing
+handwritten digits, where each sample’s corresponding label
+is 0 through 9. In unsupervised learning, each training sample
+is only represented by a set of extracted features, which are
+subsequently used to identify the underlying feature patterns
+throughout the dataset. For example, clustering techniques
+divide a dataset into k distinct groups, where all data points
+in a group are similar with respect to some distance measure.
+Finally, in reinforcement learning, an autonomous agent learns
+the optimal way to act over time via interaction with the
+environment, such as an autonomous robot learning how to
+move its actuators and joints to navigate in an environment
+without hitting obstacles.
+Although the ability to perform complex tasks solely from
+data has made ML highly successful, it is for this same
+reason that ML models are difficult to assure. Among other
+factors, a model’s performance depends on the data experi-
+enced during training and the environment in which it was
+trained [26]. Naive metrics such as the model’s accuracy
+on a test set may be perceived as overconfident because
+they assume most future data will be like the experienced
+data. This is especially true in complex systems such as
+government systems tasked with operating in contested op-
+
+erational environments, demonstrating the need for metrics
+to assess model performance in new environments. Another
+assurance challenge includes determining relevant test cases
+given the state space explosion and curse of dimensionality
+problems, of which the Range Adversarial Planning Tool has
+been proposed [27]. Furthermore, such models are brittle to
+perturbations in input, which may come from sources such as,
+sensor noise or adversarial attacks [28]. Lastly, it is inevitable
+that such models will fail from time to time, and explanations
+of why they fail (interpretability techniques) and how to fail
+gracefully (resilience techniques) are also valuable. Although
+there are a variety of new assurance techniques [12] [13] that
+seek to alleviate such issues, a framework does not exist to
+assess their thoroughness and relative effectiveness.
+B. Reinforcement Learning
+Reinforcement learning (RL) is given a dedicated subsec-
+tion because it is an enabler of intelligent-like capabilities
+required for complex autonomous systems. Reinforcement
+learning provides a framework for autonomous agents to make
+decisions under uncertainty and learn from environmental
+interaction [29]. Specified by a reward function, an agent
+seeks to obtain an optimal policy which maximizes its reward
+by taking actions over a time horizon in an environment. A
+policy is a function that maps the current state to the single
+action that maximizes the expected future reward. Formally,
+this structure is part of a Markov Decision Process (MDP)
+consisting of a state space S, an action space A, a state
+transition distribution over next states T(st+1|st, a), and a
+reward function R(s, a, s′) whose solution is the optimal
+policy which maximizes the expected future reward π∗. Exact
+RL seeks to converge to the optimal policy using tabular
+techniques, requiring an agent to visit each state many times.
+Conversely, approximate techniques such as deep RL [30]
+allow an agent to operate in large (possibly infinite) state
+and action spaces without explicitly visiting each state by
+obtaining a parameterized policy.
+Although RL has demonstrated its ability to mimic intel-
+ligent capabilities such as beating players at Go [31], and
+autonomous driving [32], there are limitations. Designing
+reward functions explicitly by hand is a challenging task
+that can lead to a misalignment between the reward function
+specified and the true reward function the algorithm designer
+intended [33]. Such value misalignment leads to unintended
+consequences such as reward hacking [34], where the robot
+maximizes reward in a way that the algorithm designer did
+not intend while often failing to meet its goals. Furthermore,
+many solutions sample inefficiently and are often brittle [35],
+limiting their real world applicability. Lastly, RL can cause
+a disconnect between how a programmer may interpret what
+an agent has learned and the true learned concept [36]. For
+example, a programmer may believe an agent has learned to
+traverse to a goal grid cell, but because of the environment
+setup, the agent may have actually simply learned to traverse
+to a green grid cell. For systems incorporating RL, such limi-
+tations and corresponding tests for each should be considered
+explicitly in the assurance process.
+III. OVERVIEW OF MAPPING
+This paper provides a preliminary attempt to map recently
+developed technical approaches in the assurance and TEVV
+of learning enabled autonomous systems (LEAS) literature
+to a traditional systems engineering v-model. The mapping
+identifies three top level lifecycle phases: development, ac-
+quisition, and sustainment. Proceeding according to the colors
+in Fig. 1, the stages surrounded in the black box, including
+system design & development, and system integration, assist
+development and therefore are mapped to methods which
+explicitly provide safety assurance during the learning process
+(Sec. IV). The stage in the green box, test and evaluation,
+assists the acquisition of systems and therefore is mapped to
+TEVV analysis techniques which quantify the performance
+of an already built system, or component (Sec. V). The stage
+in the orange box, transition operation & maintenance, is
+mapped to safety assurance techniques which aid sustainment
+by monitoring or adapt performance of a fielded system
+(Sec. VI). For each stage, applicable classes of techniques
+are organized by respective subsections.
+In addition to the stage of the system engineering lifecycle,
+this mapping also seeks to categorize technical developments
+according to their granularity. When evaluating different ap-
+proaches to the same problem, the choice of performance
+metrics depend on the scope of the unit under test—whole
+system, learning enabled component, or a traditional com-
+ponent. Interfaces at various lifecycle levels of granularity
+promote systems thinking [37] about architecture. Namely, the
+way a system’s components and subsystems relate, interact,
+and work over time. By understanding the input paths that
+contribute to a unit’s decisions, the outputs that may lead to
+failures within the larger system become clearer.
+Generally speaking, there are two main approaches to as-
+sure LEAS—white-box techniques and black-box techniques.
+White-box techniques require either a model of the system
+under test, or direct access to the source code. In contrast,
+black-box techniques only look at the inputs and outputs of
+the system under test, and are unaware of the underlying
+methods of how the system generates the outputs. White-
+box techniques are better for component level assurance,
+while black-box techniques are often better for system-wide
+assurance.
+The implementation of assurance techniques and their
+accompanying metrics to quantify system performance and
+safety (Sec. IV—Sec. VI) can all be used as supporting
+evidence for a safety assurance case [38] to determine system
+readiness level and maturity. Tools which automate trace-
+ability and reproducibility throughout the system lifecycle
+such as [39] can reduce the burden of collecting evidence.
+The appropriate choice of assurance methods and associated
+metrics is dependent on the system maturity. Initial project
+milestones may focus on demonstrating anti-fragility, while
+later milestones may focus on demonstrating the ability to
+
+accomplish a mission and accompanying capabilities. Fur-
+thermore, quantitative metrics may only be applicable at
+certain levels of system granularity. For example, an entire
+system may be best evaluated by the outcomes of a pre-
+determined mission and supporting data, while a learning
+enabled component may be better evaluated by measures
+specific to machine learning such as uncertainty quantifica-
+tion, robustness to environmental shift, and the ability to fail
+gracefully and recover from faults. Metrics which are able
+to capture the performance of all approaches under test may
+be preferred over metrics that measure the performance of a
+certain class of algorithms. Lastly, if performance data can
+be collected during the development process, one could also
+perform a quantitative analysis of a system at any given time
+using traditional reliability [8] and defect removal [40].
+IV. ASSURANCE ACTIVITIES TO SUPPORT SYSTEM
+DEVELOPMENT
+This section maps assurance methods which assist de-
+velopment to system design and development, and system
+integration in the systems engineering lifecycle.
+A. Artificial Intelligence Safety
+AI safety is a sub-field of AI which seeks to ensure that a
+deployed AI systems (i) operates as the designer intended
+and (ii) completes its task without harming humans. The
+importance of AI safety is backed by impactful institutions
+such as the Future of Life Institute2 and Machine Intelligence
+Research Institute3. In the academic literature, AI safety
+has been popularized by the agenda of Amodedi et al. [9],
+who discuss five failure modes for AI; negative side effects,
+reward hacking, scalable supervision, safe exploration, and
+distributional shift. Moreover, in the context of RL, the
+value alignment problem arises due to a gap in the specified
+reward function and what the human actually intended [41].
+Specifically, Taylor et al. [42] discuss eight different ap-
+proaches focusing on two areas of value alignment—reward
+specification and techniques to avoid side effects. Burden et
+al. [43] argue that the scope of AI safety problems residing
+in a specific system can be characterized by three quantitative
+factors; generality, capability, and control. For a literature
+review of AI safety, the reader is directed to [44].
+While the works above seek to obtain safer agents by
+altering the underlying methodologies, the focus is on agents
+in artificial environments rather than physical robots, thereby
+creating a gap between theoretical and applied research.
+Moreover, most approaches assume that the system is fol-
+lowing a RL paradigm, demonstrating the importance to
+understand the underlying learning paradigm employed by a
+project. Lastly, although AI safety approaches alone will not
+be sufficient for LEAS assurance, if the methods are applied
+during the learning process, such approaches are likely to
+perform and test better than their non-safe counterparts,
+leading to higher assurance measures.
+2https://futureoflife.org/
+3https://intelligence.org/
+B. Learning from Human Feedback
+Incorporating human interaction can positively impact the
+performance of a LEAS because it is often easier to provide
+feedback on desired behavior rather than explicitly defining it.
+This is one solution to the value alignment problem mentioned
+in Sec. IV-A. Such human interaction may include learning
+from demonstration, intervention, or evaluation [45]. In learn-
+ing from demonstration [46], the human provides a dataset
+of examples mimicking how the system should operate. In
+learning from intervention [47], the system operates fully
+autonomously and the human takes over as required to correct
+system behavior. In learning from human evaluation [48],
+[49], the system completes various tasks fully autonomously,
+and then a human ranks the tasks. This ranking may be
+from best to worst, or answering the yes/no question, “Was
+this the behavior you wanted to see the system perform?”
+All of the methods mentioned fall into a sub-field known
+as imitation learning [50]. Lastly, recent developments in
+imitation learning attempt to incorporate safety as part of the
+learning process using uncertainty quantification, creating a
+new sub field known as safe imitation learning [51], [52].
+C. Uncertainty Estimation
+System requirements often demand that a learning en-
+abled autonomous system make a prediction, classification,
+or decision at every time-step during operation. Since many
+implementations contain perception systems that will likely
+never be 100% accurate, the certainty or lack thereof, of a
+prediction may assist in the final decision made—especially if
+the outcome of such a prediction may lead to risky behavior.
+The ability for a system to measure what it does and does
+not know can be captured by quantifying uncertainty with
+Bayesian analysis techniques [53]. There are two main types
+of uncertainty, aleatoric and epistemic. Aleatoric uncertainty
+measures the variance between samples in a population.
+This type of uncertainty cannot be reduced with more data.
+An example is the outcome of a fair coin flip. Epistemic
+uncertainty measures the lack of knowledge of a population,
+which is often captured in a system’s parameters. This type
+of uncertainty can be alleviated by collecting more data.
+An understanding of the different types of uncertainty helps
+system designers understand if performance can be increased
+by simply collecting more data. Additional details can be
+found in the reviews on uncertainty quantification applied to
+machine learning [54], neural networks [55], and computer
+vision [56]. Such techniques aid at the learning enabled com-
+ponent level, and can be used to quantify system confidence in
+the current operational environment and thereby communicate
+uncertainty (risk) to the system end users.
+D. Cost-sensitive Learning
+At the system level, the impact of a learning enabled
+component on the whole system is measured in terms of
+its ability to assist in the completion of a task. Additional
+failure modes introduced by such components must be ex-
+plicitly considered. Cost-sensitive learning [57] is applicable
+
+in classification problems where the cost associated with the
+misclassification is not equal among classes. For example,
+in the context of commercial autonomous vehicles, a false
+positive resulting in the vehicle stopping when it did not need
+to likely has lower cost than a false negative resulting in a
+vehicle colliding with a pedestrian.
+E. Formal Methods
+Static analysis techniques such as formal methods are able
+to provide guarantees on system performance without ever
+operating the system [58]. Rather than attempting to discover
+faults while the system is placed under operation, claims about
+a system are proved or disproved algorithmically using rig-
+orous mathematical methods. Such methods develop a model
+of the system being tested, such as a finite-state automaton,
+and then test that model against a set of specifications defined
+in a formal language. There are two main approaches, formal
+verification [7], which checks if a given system satisfies a set
+of specifications, while program synthesis seeks to construct a
+system from a set of specifications [59]. For a literature review
+of formal methods in the context of autonomous robotics, the
+reader is directed to [60].
+In the context of LEAS, the system is often complex
+and safety is critical, thereby making formal methods an
+attractive solution. Specifically, synthesis methods provide a
+“correct-by-construction” approach [61], where capabilities
+and required operating conditions such as safety constraints
+are described as specifications and act as input to a synthesis
+algorithm which outputs the appropriate system model and
+optimal control policy. The vehicle’s actions are thereby guar-
+anteed to stay within the operating conditions determined by
+the obtained policy. However, many approaches are currently
+limited to static environments, meaning a robot which is
+guaranteed to satisfy the specifications in one environment
+does not necessarily carry over to other environments. More-
+over, many formal methods have issues scaling to large state
+spaces [62] due to their exhaustive nature. However, solutions
+have been proposed using clever optimization techniques such
+as
+[63] [64]. Nevertheless, synthesis methods can be used
+to assure safety during a systems development phase, while
+formal verification techniques such as model checking [7]
+may be more applicable at the acquirement level.
+V. ASSURANCE ACTIVITIES TO SUPPORT SYSTEM
+ACQUISITION
+This section maps assurance activities to support system
+acquisition to test and evaluation in the systems engineering
+lifecycle.
+A. Autonomy Standards
+Standards seek to provide safety assurances, verify capa-
+bilities, and promote understanding. Several standards have
+been developed to assist the design and development of
+commercial autonomous vehicles such as ISO 26262 [65] and
+IEC 61508 [66]. Specifically, UL 4600 [67] and ISO/PAS
+21448 [68] explicitly consider autonomous vehicle capabil-
+ities incorporating learning. UL 4600 employs the idea of
+safety assurance cases, where system performance is argued
+like a court case given evidence. Minimizing risk is the
+goal while also accepting that it cannot be eliminated all to-
+gether. Military focused standards include ALFUS [69] paired
+with the updated ARP6128 [70], and MIL-STD-882E [71].
+Most recently, IEEE 2817 [72] (in development) seeks to
+standardize verification methods specifically for autonomous
+systems. Although this discussion is part of the development
+subsection, the standards listed here may also be applicable
+to the other two lifecycle categories identified in Figure 1.
+B. Software Testing
+Capabilities of autonomous systems are enabled by soft-
+ware. There is no debate on the importance of software
+testing, when acknowledging the severity of historic software
+failures such as the patriot missile or Boeing 737 MAX.
+Traditional methods such as those outlined in [73] seek
+to partition the input space using graph or logic coverage
+to exhaustively test a program. While traditional methods
+may work for testing traditional software systems, exhaustive
+methods are rendered infeasible due to the state space ex-
+plosion problem. Moreover, for statistical learning algorithms
+commonly applied in machine learning methods, the set of
+all possible samples is often much larger than the number
+of samples collected. For example, the set of all possible
+images a camera may sense using the RGB spectrum with
+an image size of 256 × 256 is 16, 777, 216(256∗256). This
+demonstrates the importance of analyzing dataset features
+and their associated effectiveness [74] to obtain a generalized
+model.
+In regards to software engineering—a machine learning
+model is similar to traditional components, they both have
+inputs and outputs. The difference is the size of the input
+space and that the outputs may change on the same input at
+different points in time if the model is continually learning.
+However, if the model is not learning from new data, it can
+be considered as a deterministic component. An inaccurate
+prediction from a model can be thought of as equivalent to
+a software fault [75]. However, due to the large state space,
+the issue remains in the detection of such faults. The next
+subsection that follows seek to identify such faults.
+C. Automated Test Generation
+Automated test case generation seeks to increase the ef-
+fectiveness of test and evaluation by minimizing testing time,
+and identifying the most impactful test cases which are likely
+to contain faults. A survey of automatic test-case generation
+[76] identifies five main categories—structural testing, model-
+based testing, combinatorial testing, random testing, and
+search-based testing. Aforementioned for LEAS, the number
+of configurations is often intractable and therefore exhaustive
+or tree methods are infeasible. Search-based methods seek to
+alleviate this issue by using clever optimization techniques,
+which identify test cases in areas (boundaries) of a systems
+
+configuration space that are likely to lead to system failure.
+Therefore, this subsection focuses on search-based methods.
+Most relevant to LEAS, Mullins et al. [27] provide a tool
+which automatically identifies test cases for a system under
+test with a search based optimization approach dependent on
+a set of mission scenario configurations and a performance
+score for each configuration. A case study using the afore-
+mentioned tool in an autonomous surface vessel domain can
+be found in [77]. Bridging the gap between formal verification
+and automated test case generation, Akellea et al. [78] provide
+a black-box method to identify test cases which do not satisfy
+a provided temporal logic specification based on a dataset of
+observed demonstrations. Most recently, Badithela et al. [79]
+identify test cases for mission objectives by constructing a
+set of constraints given a user-defined sequence of waypoints
+and a reachability objective. In conclusion, recent research in
+search-based automated test generation is able to handle the
+state space explosion problem, while also finding the most
+impactful test cases. The results from such test cases help
+provide impactful evidence towards, or against the construc-
+tion of safety case (UL 4600).
+D. Metrics for Machine Learning
+Metrics provide a quantitative analysis of performance,
+clearly identifying the best solution out of a set of possible
+solutions. Performance is best measured by the system’s abil-
+ity to accurately make predictions in the current operational
+environment which positively contribute to the larger mission
+goals. Initial metrics in supervised machine learning focused
+on confusion matrices and receiver operating characteristics
+(ROC) with metrics such as accuracy, sensitivity, specificity,
+precision, and F1 score. For regression models, statistical
+measures such as mean absolute error or mean squared error
+were sufficient. In the context of neural network regression,
+the statistical significance of input features and an accompany-
+ing statistical test may be identified [80]. In the reinforcement
+learning framework, Chan et al. [81] provide a set of test and
+evaluation metrics to statistically measure the variability and
+risk of RL algorithms both during and after training.
+Agnostic to the task (classification, regression, clustering,
+etc.), neuron coverage was introduced [82], as a testing metric
+analogous to code coverage [83] for traditional systems. Code
+coverage measures the percentage of a code base that has been
+covered by tests. High code coverage implies that few soft-
+ware bugs remain, and vice versa. Similarly, neural coverage
+measures the percentage neuron activations occur from the
+testing dataset, seeking to obtain the same implications of
+code coverage. However, recent research [84] [85] has shown
+that neuron coverage is an insufficient metric for testing.
+Wang et al. [86] seek to address this limitation by quantifying
+the value of a test set. Addition metrics research is needed to
+quantitatively measure the assurance problems outlined in [4].
+VI. ASSURANCE ACTIVITIES TO SUPPORT SYSTEM
+SUSTAINMENT
+This section maps assurance activities which support sys-
+tem sustainment to transition, operating, and maintenance in
+the systems engineering lifecycle.
+A. Runtime Monitoring
+Runtime monitoring observes the current state of a system
+and determines if the system is satisfying or violating a set
+of pre-determined specifications. This is similar to the formal
+methods approach outlined in Sec. IV-E. However, runtime
+monitoring occurs online (while the system is operating),
+whereas most techniques from formal methods occur offline.
+Kane et al. introduced EgMon [87], which detects the vio-
+lation of specifications using propositional metric temporal
+logic. Similarly, Zapridou et al. [88] develop an adaptive
+cruise control system in the CARLA simulator and perform
+runtime monitoring using signal temporal logic. Yel et al. [89]
+provide a runtime monitoring technique using neural networks
+for safe motion planning. In the U.S. government sector,
+a Boeing team as part of the DARPA assured autonomy
+program, implemented runtime monitoring in a flight simula-
+tor [11]. For an overview of runtime monitoring techniques,
+the reader is directed to [90].
+B. Resilience Engineering
+Resilience engineering techniques seek to build systems
+which remain operational subject to faults and distur-
+bances [91]. Such techniques quantify the impact of degraded
+performance and robustness to faults while providing predic-
+tions such as the expected time until recovery [92]. In the con-
+text of LEAS, resilience techniques can accommodate sensor
+inaccuracies which may come from measurement limitations
+in the hardware, dust or debris, and adversarial attacks [28].
+Resilience monitoring enables a system to recognize that
+performance is degraded, and then adapt appropriately, such
+as moving from perception based navigation to odometry
+based navigation. At the time of writing there is little technical
+research on the incorporation of resilience techniques to
+LEAS [93], However, [94] offers an initial taxonomy on
+resilience for multi-robot systems.
+VII. CONCLUSION
+This paper provides preliminary attempt to map recently
+developed technical approaches for the assurance of LEAS
+to a traditional systems engineering v-model. By doing so,
+we seek to improve the acquisition process by: (i) informing
+comprehensive assurance planning, (ii) promoting detailed
+analysis of alternatives, and (iii) objectively communicating
+risk to leaders. As indicated by the number of references in
+each section, most research has been done in the development
+of methods which explicitly consider safety assurance, while
+further research is needed in methods which aid the acquire-
+ment, and sustainment of such systems. Future work seeks
+to perform a case study assuring a LEAS using some of the
+methodologies referenced in this paper.
+
+REFERENCES
+[1] L. Freeman, “Test and evaluation for artificial intelligence,” INSIGHT,
+vol. 23, no. 1, pp. 27–30, 2020.
+[2] E. Denney, G. Pai, and M. Johnson, “Towards a rigorous basis for
+specific operations risk assessment of uas,” in 2018 IEEE/AIAA 37th
+Digital Avionics Systems Conference (DASC).
+IEEE, 2018, pp. 1–10.
+[3] Test and Evaluation Management Guide, 6th ed.
+Defense Acquisition
+University, 2016.
+[4] M. Clark, J. Alley, P. Deal, J. DePriest, E. Hansen, C. Heitmeyer,
+R. Nameth, M. Steinberg, C. Turner, S. Young et al., “Autonomy
+community of interest (coi) test and evaluation, verification and val-
+idation (tevv) working group: Technology investment strategy 2015-
+2018,” Office of the Assistant Secretary of Defense for Research and
+Engineering, Tech. Rep., 2015.
+[5] N. S. Nise, Control systems engineering, 8th ed.
+John Wiley & Sons,
+2020.
+[6] L. S. Metzger, G. Rebovich Jr, R. A. Cormier, S. J. T. Norman,
+D. L. Schuh, P. A. Smyton, R. S. Swarz, and F. C. Wendt, “Systems
+engineering guide: Collected wisdom from mitres systems engineering
+experts,” MITRE CORP BEDFORD MA United States, Tech. Rep.,
+2014.
+[7] C. Baier and J.-P. Katoen, Principles of model checking.
+MIT press,
+2008.
+[8] K. S. Trivedi and A. Bobbio, Reliability and availability engineering:
+modeling, analysis, and applications.
+Cambridge University Press,
+2017.
+[9] D. Amodei, C. Olah, J. Steinhardt, P. Christiano, J. Schulman,
+and D. Man´e, “Concrete problems in ai safety,” arXiv preprint
+arXiv:1606.06565, 2016.
+[10] E. Botoeva, P. Kouvaros, J. Kronqvist, A. Lomuscio, and R. Misener,
+“Efficient verification of relu-based neural networks via dependency
+analysis,” in Proceedings of the AAAI Conference on Artificial Intelli-
+gence, vol. 34, no. 04, 2020, pp. 3291–3299.
+[11] S. Beland, I. Chang, A. Chen, M. Moser, J. Paunicka, D. Stuart, J. Vian,
+C. Westover, and H. Yu, “Towards assurance evaluation of autonomous
+systems,” in Proceedings of the 39th International Conference on
+Computer-Aided Design, 2020, pp. 1–6.
+[12] F. A. Batarseh, L. Freeman, and C.-H. Huang, “A survey on artificial
+intelligence assurance,” Journal of Big Data, vol. 8, no. 1, pp. 1–30,
+2021.
+[13] D. J. Porter and J. W. Dennis, “Test & evaluation of ai-enabled and
+autonomous systems: A literature review,” 2020.
+[14] D. E. Kirk, Optimal control theory: an introduction.
+Courier Corpo-
+ration, 2004.
+[15] A. Tampuu, T. Matiisen, M. Semikin, D. Fishman, and N. Muhammad,
+“A survey of end-to-end driving: Architectures and training methods,”
+IEEE Transactions on Neural Networks and Learning Systems, 2020.
+[16] M. Bojarski, D. Del Testa, D. Dworakowski, B. Firner, B. Flepp,
+P. Goyal, L. D. Jackel, M. Monfort, U. Muller, J. Zhang et al., “End
+to end learning for self-driving cars,” arXiv preprint arXiv:1604.07316,
+2016.
+[17] L. George, T. Buhet, E. Wirbel, G. Le-Gall, and X. Perrotton, “Imitation
+learning for end to end vehicle longitudinal control with forward
+camera,” arXiv preprint arXiv:1812.05841, 2018.
+[18] H. Yu, S. Yang, W. Gu, and S. Zhang, “Baidu driving dataset and
+end-to-end reactive control model,” in 2017 IEEE Intelligent Vehicles
+Symposium (IV).
+IEEE, 2017, pp. 341–346.
+[19] E. Yurtsever, J. Lambert, A. Carballo, and K. Takeda, “A survey of
+autonomous driving: Common practices and emerging technologies,”
+IEEE access, vol. 8, pp. 58 443–58 469, 2020.
+[20] S. Minaee, Y. Y. Boykov, F. Porikli, A. J. Plaza, N. Kehtarnavaz, and
+D. Terzopoulos, “Image segmentation using deep learning: A survey,”
+IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.
+[21] Z. Zou, Z. Shi, Y. Guo, and J. Ye, “Object detection in 20 years: A
+survey,” arXiv preprint arXiv:1905.05055, 2019.
+[22] C. Urmson, J. Anhalt, D. Bagnell, C. Baker, R. Bittner, M. Clark,
+J. Dolan, D. Duggins, T. Galatali, C. Geyer et al., “Autonomous driving
+in urban environments: Boss and the urban challenge,” Journal of Field
+Robotics, vol. 25, no. 8, pp. 425–466, 2008.
+[23] E. Guizzo, “How google’s self-driving car works,” IEEE Spectrum,
+vol. 18, no. 7, pp. 1132–1141, 2011.
+[24] F.-L. Fan, J. Xiong, M. Li, and G. Wang, “On interpretability of
+artificial neural networks: A survey,” IEEE Transactions on Radiation
+and Plasma Medical Sciences, 2021.
+[25] C. M. Bishop, Pattern recognition and Machine Learning.
+Springer,
+2006.
+[26] “Enabling ai data readiness in the department of defense.”
+[27] G. E. Mullins, P. G. Stankiewicz, R. C. Hawthorne, J. D. Appler, M. H.
+Biggins, K. Chiou, M. A. Huntley, J. D. Stewart, and A. S. Watkins,
+“Delivering test and evaluation tools for autonomous unmanned vehi-
+cles to the fleet,” Johns Hopkins APL technical digest, vol. 33, no. 4,
+pp. 279–288, 2017.
+[28] T. B. Brown, D. Man´e, A. Roy, M. Abadi, and J. Gilmer, “Adversarial
+patch,” arXiv preprint arXiv:1712.09665, 2017.
+[29] R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction.
+MIT press, 2018.
+[30] K. Arulkumaran, M. P. Deisenroth, M. Brundage, and A. A. Bharath,
+“Deep reinforcement learning: A brief survey,” IEEE Signal Processing
+Magazine, vol. 34, no. 6, pp. 26–38, 2017.
+[31] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van
+Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam,
+M. Lanctot et al., “Mastering the game of go with deep neural networks
+and tree search,” nature, vol. 529, no. 7587, pp. 484–489, 2016.
+[32] B. R. Kiran, I. Sobh, V. Talpaert, P. Mannion, A. A. Al Sallab, S. Yo-
+gamani, and P. P´erez, “Deep reinforcement learning for autonomous
+driving: A survey,” IEEE Transactions on Intelligent Transportation
+Systems, 2021.
+[33] S. Russell, Human compatible: Artificial intelligence and the problem
+of control.
+Penguin, 2019.
+[34] J.
+Clark
+and
+D.
+Amodei,
+“Faulty
+reward
+functions
+in
+the
+wild,”
+Dec
+2016.
+[Online].
+Available:
+https://openai.com/blog/
+faulty-reward-functions/
+[35] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, “Soft actor-critic: Off-
+policy maximum entropy deep reinforcement learning with a stochastic
+actor,” in International conference on machine learning. PMLR, 2018,
+pp. 1861–1870.
+[36] J. Koch, L. Langosco, J. Pfau, J. Le, and L. Sharkey, “Objective robust-
+ness in deep reinforcement learning,” arXiv preprint arXiv:2105.14111,
+2021.
+[37] E. Rechtin and M. W. Maier, The art of systems architecting.
+CRC
+press, 2010.
+[38] P. Koopman, U. Ferrell, F. Fratrik, and M. Wagner, “A safety standard
+approach for fully autonomous vehicles,” in International Conference
+on Computer Safety, Reliability, and Security.
+Springer, 2019, pp.
+326–332.
+[39] C. Hartsell, N. Mahadevan, S. Ramakrishna, A. Dubey, T. Bapty,
+T. Johnson, X. Koutsoukos, J. Sztipanovits, and G. Karsai, “Cps design
+with learning-enabled components: A case study,” in Proceedings of the
+30th International Workshop on Rapid System Prototyping (RSP’19),
+2019, pp. 57–63.
+[40] M. Nafreen, M. Luperon, L. Fiondella, V. Nagaraju, Y. Shi, and
+T. Wandji, “Connecting software reliability growth models to software
+defect tracking,” in 2020 IEEE 31st International Symposium on
+Software Reliability Engineering (ISSRE).
+IEEE, 2020, pp. 138–147.
+[41] D. Hadfield-Menell, “The principal-agent alignment problem in artifi-
+cial intelligence,” Ph.D. dissertation, University of California, Berkeley,
+2021.
+[42] J. Taylor, E. Yudkowsky, P. LaVictoire, and A. Critch, “Alignment for
+advanced machine learning systems,” Ethics of Artificial Intelligence,
+pp. 342–382, 2016.
+[43] J. Burden and J. Hern´andez-Orallo, “Exploring ai safety in degrees:
+Generality, capability and control,” in SafeAI@ AAAI, 2020.
+[44] T. Everitt, G. Lea, and M. Hutter, “Agi safety literature review,” in
+IJCAI, 2018.
+[45] N. R. Waytowich, V. G. Goecks, and V. J. Lawhern, “Cycle-of-
+learning for autonomous systems from human interaction,” in AI-HRI
+Symposium, AAAI Fall Symposium Series, 2018.
+[46] H. Ravichandar, A. S. Polydoros, S. Chernova, and A. Billard, “Recent
+advances in robot learning from demonstration,” Annual Review of
+Control, Robotics, and Autonomous Systems, vol. 3, pp. 297–330, 2020.
+[47] W. Saunders, G. Sastry, A. Stuhlm¨uller, and O. Evans, “Trial without
+error: Towards safe reinforcement learning via human intervention,”
+in Proceedings of the 17th International Conference on Autonomous
+Agents and MultiAgent Systems, 2018, pp. 2067–2069.
+
+[48] P. F. Christiano, J. Leike, T. B. Brown, M. Martic, S. Legg, and
+D. Amodei, “Deep reinforcement learning from human preferences,”
+in NIPS, 2017.
+[49] G. Warnell, N. Waytowich, V. Lawhern, and P. Stone, “Deep tamer:
+Interactive agent shaping in high-dimensional state spaces,” in Thirty-
+Second AAAI Conference on Artificial Intelligence, 2018.
+[50] T. Osa, J. Pajarinen, G. Neumann, J. Bagnell, P. Abbeel, and J. Peters,
+“An algorithmic perspective on imitation learning,” Foundations and
+Trends in Robotics, vol. 7, no. 1-2, pp. 1–179, 2018.
+[51] D. Brown, R. Coleman, R. Srinivasan, and S. Niekum, “Safe imitation
+learning via fast bayesian reward inference from preferences,” in
+International Conference on Machine Learning.
+PMLR, 2020, pp.
+1165–1177.
+[52] C. Ellis, M. Wigness, J. G. Rogers III, C. Lennon, and L. Fiondella,
+“Risk averse bayesian reward learning for autonomous navigation
+from human demonstration,” in International Conference on Intelligent
+Robots and Systems, Prague Czech Republic, 2021.
+[53] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian data
+analysis, 3rd ed.
+Chapman and Hall/CRC, 2013.
+[54] E. H¨ullermeier and W. Waegeman, “Aleatoric and epistemic uncertainty
+in machine learning: An introduction to concepts and methods,” Ma-
+chine Learning, vol. 110, no. 3, pp. 457–506, 2021.
+[55] M. Abdar, F. Pourpanah, S. Hussain, D. Rezazadegan, L. Liu,
+M. Ghavamzadeh, P. Fieguth, X. Cao, A. Khosravi, U. R. Acharya et al.,
+“A review of uncertainty quantification in deep learning: Techniques,
+applications and challenges,” Information Fusion, 2021.
+[56] A. Kendall and Y. Gal, “What uncertainties do we need in bayesian
+deep learning for computer vision?” in NIPS, 2017.
+[57] G. Haixiang, L. Yijing, J. Shang, G. Mingyun, H. Yuanyue, and
+G. Bing, “Learning from class-imbalanced data: Review of methods
+and applications,” Expert Systems with Applications, vol. 73, pp. 220–
+239, 2017.
+[58] X. Rival and K. Yi, Introduction to static analysis: an abstract
+interpretation perspective.
+Mit Press, 2020.
+[59] S. Gulwani, O. Polozov, R. Singh et al., “Program synthesis,” Foun-
+dations and Trends® in Programming Languages, vol. 4, no. 1-2, pp.
+1–119, 2017.
+[60] M. Luckcuck, M. Farrell, L. A. Dennis, C. Dixon, and M. Fisher,
+“Formal specification and verification of autonomous robotic systems:
+A survey,” ACM Computing Surveys (CSUR), vol. 52, no. 5, pp. 1–41,
+2019.
+[61] U. Topcu, “Formal specification and correct-by-construction synthesis
+of control protocols for adaptable, human-embedded autonomous sys-
+tems,” Trustees of The University of Pennsylvania Philadelphia United
+States, Tech. Rep., 2019.
+[62] M. Fisher, L. Dennis, and M. Webster, “Verifying autonomous systems,”
+Communications of the ACM, vol. 56, no. 9, pp. 84–93, 2013.
+[63] C. Dehnert, S. Junges, J.-P. Katoen, and M. Volk, “A storm is coming:
+A modern probabilistic model checker,” in International Conference on
+Computer Aided Verification.
+Springer, 2017, pp. 592–600.
+[64] M. Suilen, N. Jansen, M. Cubuktepe, and U. Topcu, “Robust policy
+synthesis for uncertain pomdps via convex optimization.” International
+Joint Conferences on Artificial Intelligence, 2020.
+[65] I. ISO, “26262: Road vehicles-functional safety,” International Orga-
+nization for Standardization/FDIS, vol. 26262, 2011.
+[66] I. IEC, “61508: Functional safety of electrical/electronic/programmable
+electronic safety-related system,” International Electrotechnical Com-
+mission, 2010.
+[67] U. UL, “4600: Standard for safety for the evaluation of autonomous
+vehicles and other products,” Underwriters Laboratories, vol. 4600,
+2019.
+[68] I. ISO, “21448: road vehicles-safety of the intended functionality,”
+International Organization for Standardization, 2019.
+[69] H.-M. Huang, K. Pavek, B. Novak, J. Albus, and E. Messin, “A frame-
+work for autonomy levels for unmanned systems (alfus),” Proceedings
+of the AUVSI’s unmanned systems North America, pp. 849–863, 2005.
+[70] SAE, “Unmanned systems terminology based on the alfus framework,”
+vol. ARP6128, 2019.
+[71] U. D. of Defense, “System saftey,” vol. MIL-STD-882E, 2012.
+[72] I. S. Association, “Guide for verification of autonomous systems,” vol.
+P2817, 2019.
+[73] A. P. Mathur, Foundations of software testing, 2/e.
+Pearson Education
+India, 2013.
+[74] P. Barbiero, G. Squillero, and A. Tonda, “Modeling generalization in
+machine learning: A methodological and computational study,” arXiv
+preprint arXiv:2006.15680, 2020.
+[75] A. Gula, C. Ellis, S. Bhattacharya, and L. Fiondella, “Software and
+system reliability engineering for autonomous systems incorporating
+machine learning,” in 2020 Annual Reliability and Maintainability
+Symposium (RAMS).
+IEEE, 2020, pp. 1–6.
+[76] S. Anand, E. K. Burke, T. Y. Chen, J. Clark, M. B. Cohen,
+W. Grieskamp, M. Harman, M. J. Harrold, P. McMinn, A. Bertolino
+et al., “An orchestrated survey of methodologies for automated software
+test case generation,” Journal of Systems and Software, vol. 86, no. 8,
+pp. 1978–2001, 2013.
+[77] P. G. Stankiewicz and G. E. Mullins, “Improving evaluation method-
+ology for autonomous surface vessel colregs compliance,” in OCEANS
+2019-Marseille.
+IEEE, 2019, pp. 1–7.
+[78] P. Akella, M. Ahmadi, R. M. Murray, and A. D. Ames, “Formal
+test synthesis for safety-critical autonomous systems based on control
+barrier functions,” in 2020 59th IEEE Conference on Decision and
+Control (CDC).
+IEEE, 2020, pp. 790–795.
+[79] A. Badithela and R. M. Murray, “Synthesis of static test environments
+for observing sequence-like behaviors in autonomous systems,” arXiv
+preprint arXiv:2108.05911, 2021.
+[80] E. Horel and K. Giesecke, “Significance tests for neural networks,”
+Journal of Machine Learning Research, vol. 21, no. 227, pp. 1–29,
+2020.
+[81] S. C. Chan, S. Fishman, J. Canny, A. Korattikara, and S. Guadarrama,
+“Measuring the reliability of reinforcement learning algorithms,” in
+International Conference on Learning Representations, Addis Ababa,
+Ethiopia, 2020.
+[82] K. Pei, Y. Cao, J. Yang, and S. Jana, “Deepxplore: Automated whitebox
+testing of deep learning systems,” in proceedings of the 26th Symposium
+on Operating Systems Principles, 2017, pp. 1–18.
+[83] J. C. Miller and C. J. Maloney, “Systematic mistake analysis of digital
+computer programs,” Communications of the ACM, vol. 6, no. 2, pp.
+58–63, 1963.
+[84] S. Abrecht, M. Akila, S. S. Gannamaneni, K. Groh, C. Heinzemann,
+S. Houben, and M. Woehrle, “Revisiting neuron coverage and its ap-
+plication to test generation,” in International Conference on Computer
+Safety, Reliability, and Security.
+Springer, 2020, pp. 289–301.
+[85] F. Harel-Canada, L. Wang, M. A. Gulzar, Q. Gu, and M. Kim,
+“Is neuron coverage a meaningful measure for testing deep neural
+networks?” in Proceedings of the 28th ACM Joint Meeting on European
+Software Engineering Conference and Symposium on the Foundations
+of Software Engineering, 2020, pp. 851–862.
+[86] J. Wang, J. Chen, Y. Sun, X. Ma, D. Wang, J. Sun, and P. Cheng,
+“Robot: Robustness-oriented testing for deep learning systems,” in 2021
+IEEE/ACM 43rd International Conference on Software Engineering
+(ICSE).
+IEEE, 2021, pp. 300–311.
+[87] A. Kane, O. Chowdhury, A. Datta, and P. Koopman, “A case study on
+runtime monitoring of an autonomous research vehicle (arv) system,”
+in Runtime Verification.
+Springer, 2015, pp. 102–117.
+[88] E. Zapridou, E. Bartocci, and P. Katsaros, “Runtime verification of
+autonomous driving systems in carla,” in International Conference on
+Runtime Verification.
+Springer, 2020, pp. 172–183.
+[89] E. Yel, T. J. Carpenter, C. Di Franco, R. Ivanov, Y. Kantaros, I. Lee,
+J. Weimer, and N. Bezzo, “Assured runtime monitoring and planning:
+toward verification of neural networks for safe autonomous operations,”
+IEEE Robotics & Automation Magazine, vol. 27, no. 2, pp. 102–116,
+2020.
+[90] I. Cassar, A. Francalanza, L. Aceto, and A. Ing´olfsd´ottir, “A survey
+of runtime monitoring instrumentation techniques,” arXiv preprint
+arXiv:1708.07229, 2017.
+[91] A. M. Madni and S. Jackson, “Towards a conceptual framework for
+resilience engineering,” IEEE Systems Journal, vol. 3, no. 2, pp. 181–
+191, 2009.
+[92] S. Hosseini, K. Barker, and J. E. Ramirez-Marquez, “A review of
+definitions and measures of system resilience,” Reliability Engineering
+& System Safety, vol. 145, pp. 47–61, 2016.
+[93] S. Johnsen and S. Kilskar, “A review of resilience in autonomous
+transport to improve safety and security.”
+ESREL, 2020.
+[94] A. Prorok, M. Malencia, L. Carlone, G. S. Sukhatme, B. M. Sadler,
+and V. Kumar, “Beyond robustness: A taxonomy of approaches towards
+resilient multi-robot systems,” arXiv preprint arXiv:2109.12343, 2021.
+

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@@ -0,0 +1,1308 @@
+Eigenvalues of QCD Dirac matrix with improved staggered quarks in the continuum
+limit
+Olaf Kaczmarek,1 Ravi Shanker,2, ∗ and Sayantan Sharma2
+1Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany
+2The Institute of Mathematical Sciences, a CI of Homi Bhabha National Institute, Chennai, 600113, India
+We calculate the eigenmodes of the Highly Improved Staggered Quark (HISQ) matrix near the
+chiral crossover transition in QCD with 2 + 1 flavors with the aim to gain more insights into its
+temperature dependence.
+On performing the continuum extrapolation, we do not observe any
+gap opening up in the infrared part of the eigenvalue density of QCD Dirac operator, instead we
+observe a peak.
+The existence of the peak and oscillations of the infrared eigenmodes can be
+understood in terms of an interacting ensemble of instantons. From the properties of the continuum
+extrapolated eigen spectrum we further show that the anomalous UA(1) part of the chiral symmetry
+is not effectively restored simultaneously along with its non-singlet counterpart.
+We provide an
+explanation for this observation, further showing interesting connections between the anomalous
+UA(1) restoration and the change in the infrared part of the eigenvalue distribution.
+PACS numbers:
+12.38.Gc, 11.15.Ha, 11.30.Rd, 11.15.Kc
+Introduction The eigenvalue spectrum of the quark
+Dirac operator contains valuable information about the
+fundamental properties of Quantum Chromodynamics
+(QCD). The chiral condensate which acts as a (pseudo)
+order parameter for the chiral (crossover) transition in
+QCD is related to the density of near-zero eigenvalues [1].
+In fact it was shown from very general considerations that
+the formation of the chiral condensate is related to the
+occurrence of small eigenvalues that scale proportional
+to the volume [2]. The breaking of the non-singlet part
+of chiral symmetry i.e. SUA(2) × SUV (2) → SUV (2) of
+QCD with physical quark masses at the crossover tem-
+perature Tc = 156.5 ± 1.5 MeV [3] can also be explained
+in terms of modifications in the deep infrared part of the
+eigenvalue density. The flavor-singlet UA(1) part of the
+chiral symmetry on the other hand, is anomalous yet is
+believed to play an important role in determining the
+nature of the chiral phase transition [4–6]. The temper-
+ature dependence of the amount of UA(1) breaking near
+the chiral crossover transition in QCD can be only deter-
+mined using non-perturbative lattice techniques and is a
+topic of contemporary interest in lattice QCD see for e.g.
+Ref. [7, 8] for recent reviews. Whereas there are some
+very compelling evidence that show UA(1) remains effec-
+tively broken in 2 + 1 flavor QCD with physical quark
+mass m
+[9–15], even when m → 0 [16], there are lat-
+tice studies which also favor an effective restoration at
+Tc [17–22].
+The eigenvalue spectrum of the QCD Dirac matrix also
+encodes within it some remarkable universal properties.
+It was shown that the route towards achieving thermo-
+dynamic limit for the infrared modes of the Dirac op-
+erator is universal [23], for any number of light quark
+flavors.
+The existence of a non-zero chiral condensate
+leads to a sum rule involving sum of inverse squares of
+∗Electronic address: [email protected]
+these small eigenvalues [2]. These sum rules are univer-
+sal irrespective of the details of the nature and type of
+gauge interactions [23, 24] and could be derived from chi-
+ral random matrix theory [25]. A good agreement was
+demonstrated for the distribution of the small eigenvalues
+and the spectral density of lattice QCD Dirac operator
+and chiral random matrix theory at zero temperature on
+small lattice volumes [26]. In fact universal correlations
+between higher order spectral functions in a random ma-
+trix theory has been derived [27] and its connection to
+QCD was discussed. At finite temperature the universal
+features of infrared eigenvalues can be also accounted for
+within a random matrix theory [28–30]. Additionally the
+infrared eigenvalue spectrum of QCD has more subtle
+features. A near-zero peak of localized eigenvalues has
+been observed for finite lattices, mixing with but very
+different from the delocalized bulk modes whose spectral
+density follows random matrix statistics [7, 31]. Whether
+or not such a feature survives in the continuum limit is
+yet to be ascertained. Previous studies of quark Dirac
+spectrum in an instanton liquid ensemble [29, 32] at zero
+temperature have observed similar peak-like feature.
+With increasing temperature the localized modes
+starts separating out from the random bulk modes lead-
+ing to the opening up of a mobility edge [31]. The corre-
+sponding temperature where a finite mobility edge sepa-
+rates the bulk modes from the localized one was initially
+estimated from lattice studies to be identical to Tc in dy-
+namical [33–38] as well as in quenched QCD [39], remi-
+niscent of an Anderson-like transition that is observed in
+disordered semi-metals [40]. However independent lat-
+tice studies do discuss another possible scenario where
+the opening of a finite mobility edge may occur at tem-
+peratures higher that Tc [41], with an intermediate phase
+consisting of scale-invariant infinitely extended infrared
+modes [42, 43] strongly interacting with the bulk modes
+leading to a singularity at the mobility edge.
+Most of the previous lattice QCD studies were ei-
+ther performed in the quenched limit or with dynam-
+arXiv:2301.11610v1  [hep-lat]  27 Jan 2023
+
+2
+ical quarks but away from the physical point and for
+finite lattice spacings. On a finite lattice, the most of-
+ten used lattice discretization i.e. the staggered fermions
+only has a remnant of the continuum chiral symmetry
+group due to mixing of spin and flavor degrees of free-
+dom.
+Furthermore the anomalous part of the chiral
+symmetry in the continuum is not realized exactly by
+the staggered/Wilson quarks and is expected to be re-
+covered only in the continuum limit. We, for the first
+time study the properties of the eigenvalue spectrum of
+(highly) improved dynamical staggered Dirac operator
+in large volume lattices by carefully performing a con-
+tinuum extrapolation. We show that the deep infrared
+spectrum of QCD Dirac operator has indeed a peak of
+near-zero modes which survives in continuum. These are
+distinct from other infrared modes which has a linearly
+rising density and a quadratic level repulsion similar to a
+certain class of random matrix theories. These so-called
+bulk modes are delocalized in volume as compared to the
+near-zero modes and they tend to distinctly disentangle
+from each other at a temperature ∼ 1.15 Tc, which is also
+where UA(1) is effectively restored.
+In the subsequent
+sections we discuss our results and also provide a unified
+physical explanation of these phenomena we observe.
+Numerical Details In this work we use the gauge
+configurations for 2 + 1 flavor QCD with physical quark
+masses generated by the HotQCD collaboration using
+Highly Improved Staggered quark (HISQ) discretization
+for the fermions and tree-level Symanzik improved gauge
+action. These ensembles have been previously used to
+measure the equation of state of QCD both at zero and
+finite baryon density [3, 44]. The Goldstone pion mass is
+set to 140 MeV and the kaon mass is 435 MeV for these
+configurations. We focus on five different temperatures,
+one below Tc and others above Tc.
+For most of these
+temperatures we consider three different lattice spacings
+corresponding to Nτ = 8, 12, 16, details of which are men-
+tioned in Table I in Appendix A. The number of spatial
+lattice sites was chosen to be Ns = 4Nτ such that the
+spatial volume in each case was about 4 fm, which en-
+sures that the system is close to the thermodynamic limit.
+We next measure the eigenvalues of the massless HISQ
+Dirac matrix on these gauge ensembles using conjugate
+gradient method based algorithms.
+General features of the eigenvalue spectrum of
+QCD using HISQ Dirac operator in continuum
+limit In this section we study in detail the eigenvalue
+density ρ(λ) of the fermions in 2 + 1 flavor QCD by
+performing a continuum extrapolation of the parame-
+ters characterizing the eigenspectrum calculated on the
+lattice with Highly Improved Staggered Quarks (HISQ)
+discretization.
+We first study the eigenvalue spectrum
+for four different temperatures above Tc in order to un-
+derstand whether the flavor singlet and non-singlet parts
+of the chiral symmetry is effectively and simultaneously
+restored or not.
+At zero temperature it is known from chiral perturba-
+tion theory [45] that the bulk eigenvalue density is
+ρ(λ) = ⟨0| ¯ψψ|0⟩
+π
++ |λ|⟨0| ¯ψψ|0⟩2
+N 2
+f − 4
+32π2NfF 4π
++ ..
+(1)
+The intercept of the eigenvalue density gives the chiral
+condensate. The ratio of the slope and the intercept of
+the density as a function of λ should be proportional
+to the chiral condensate. We first focus on the intercept
+and the slope (linear in λ) of the eigenvalue density at the
+lowest temperature T = 145 MeV, shown in the left panel
+of Fig. 1, and compare with the expectations from Eq. 1.
+At this temperature we could only obtain a continuum
+estimate of the slope and intercept as we have data for
+two lattice spacings. From the continuum estimate of the
+intercept we obtain a chiral condensate ⟨0| ¯ψψ|0⟩/T 3 =
+18.4. From the slope we could similarly extract its square
+and hence the chiral condensate (normalized by T 3) to be
+17.3 which is consistent with the one extracted from the
+intercept. Thus leading features of the eigenvalue density
+of QCD at 145 MeV are indeed very well represented
+within chiral perturbation theory.
+The bulk eigenvalue density in the chirally symmetric
+phase has been studied very recently [46]. Most generally,
+it can be expressed as a function of λ as
+ρ(λ)
+T 3
+= ρ0
+T 3 + λ
+T .c1(T, m)
+T 2
++ λ2
+T 2 .c2(T, m)
+T
++ λ3
+T 3 c3(T, m) .
+(2)
+Here c1 is the coefficient that characterizes the leading-
+order growth of the eigenvalue spectrum in the deep infra-
+red and c2 is its next-to leading order coefficient which
+eventually has a λ3-dependence predicted from perturba-
+tion theory. The intercept ρ0 gives the the chiral conden-
+sate. The coefficients c1,2,3 can in general be a function
+of the temperature T and the light-quark mass m.
+The results of the eigenvalue density ρ(λ)/T 3 as a func-
+tion of λ for T > Tc are shown in the middle and right
+panel of Fig. 1. On the finest available Nτ = 16 lattice,
+we observe two distinct features in the eigenvalue spec-
+trum, a peak of near-zero eigenvalues and the linearly
+rising part, which we call as bulk modes. For T ≲ Tc, the
+near-zero and the bulk eigenvalues overlap strongly mak-
+ing it impossible to distinguish them apart. At higher
+temperatures, the bulk eigenvalues separate out from the
+deep-infrared part of the spectrum allowing for near-zero
+modes to be distinctly visible.
+Comparing the results
+of different lattice spacings, we observe the same trend
+at each temperature above Tc i.e. near-zero peak gets
+smeared with the bulk for coarser lattices and becomes
+more prominent in the continuum limit. This is thus a
+physical feature of the eigen spectrum and not a lattice
+artifact. In order to interpret its origin we recall that in
+the instanton liquid model (ILM) at zero temperature,
+the scaled eigenvalue (cλ) density of the Dirac operator
+for Nf flavors and zero topological charge sector is dis-
+tributed according to [47],
+ρ(cλ) = cλ
+2
+�
+J2
+Nf (cλ) − JNf +1(cλ)JNf −1(cλ)
+�
+.
+(3)
+
+3
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25
+ρ(λ)/Τ3
+λ/T
+145 MeV
+Nτ= 12
+= 16
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ρ(λ)/Τ3
+λ/T
+166 MeV
+Nτ = 8
+= 12
+= 16
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ρ(λ)/Τ3
+λ/T
+171 MeV
+Nτ = 8
+= 12
+= 16
+Fig. 1: Eigenvalue spectrum for HISQ Dirac operator for 3 different lattice spacings corresponding to Nτ = 8, 12, 16 at
+T = 166, 171 MeV (center, right) and for two different lattice spacings, Nτ = 12, 16 respectively at T = 145 MeV (left).
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 18
+ρ(cλ)
+cλ
+Nτ=16
+ILM prediction
+T = 162 MeV
+ = 166 MeV
+= 171 MeV
+Fig. 2: Near-zero (scaled) eigenvalue density for HISQ Dirac
+operator at T = 162, 166, 171 MeV for the finest lattice spac-
+ing corresponding to Nτ = 16 and its comparison with ILM
+prediction available at T = 0.
+To compare our data with the above formula, we take
+c = V ⟨0| ¯ψψ|0⟩/T, where V is the spatial volume of the
+system and Nf = 3. A comparison of near zero modes for
+three different temperatures, T = 162, 166, 171 MeV, is
+shown in Fig. 2 by removing the contribution of the bulk
+intercept ρ0. We observe a good agreement with ILM for
+T = 171 MeV, in particular, the initial few oscillations
+of the small eigenvalue density as a function of cλ.
+Now focusing on the bulk modes, it was shown us-
+ing chiral Ward identities that in the symmetry restored
+phase, the sufficient condition for UA(1) restoration ev-
+ident from the degeneracy of up to 6-point correlation
+functions in the scalar-pseudo-scalar sector are c1 =
+O(m2) +... and c3 = c30 +O(m2)+.... The perturbative
+λ3-growth in Eq. 2 can have a mass-independent coeffi-
+cient which however does not lead to UA(1) breaking. We
+verify whether indeed it is true even non-perturbatively
+by performing a fit to the bulk part i.e. all eigenvalues
+λ > λ0 with ρ(λ)
+T 3 = λ
+T . c1(T,m)
+T 2
++ ρ0
+T 3 . This ansatz neglects
+higher powers in λ which is well justified since we are
+in the deep infrared of the eigen spectrum, represented
+by O(100) eigenvalues out of a total million available on
+such lattice sizes. The results of the fit are discussed in
+Table II. The extracted slope c1 for each temperature
+T > Tc, at three different values of Nτ then allows us
+to perform a continuum (∼ 1/N 2
+τ ) extrapolation of this
+coefficient. We next study the m-dependence of this con-
+tinuum extrapolated coefficient c1(m, T). The results of
+the fits are shown in Fig. 3. It is evident from the fit
+that it is more favorable that c1 is proportional to T 2
+(χ2/d.o.f=0.6) to leading order rather than c1 is propor-
+tional to m2 (χ2/d.o.f=0.1). From the fit we obtain the
+value of c1(m, T)/T 2 = 16.8(4).
+ 12
+ 14
+ 16
+ 18
+ 20
+ 22
+ 24
+ 1.02
+ 1.04
+ 1.06
+ 1.08
+ 1.1
+ 1.12
+ 1.14
+c1(m,T)/T2
+T/Tc
+Fig. 3: Continuum estimates for c1(m, T)/T 2 for T > Tc ob-
+tained after fitting the points with a m-independent constant
+(orange band) and a sum of quadratic (m2/T 2) and quartic
+(m4/T 4) dependence (gray band).
+This result for the slope in the continuum limit has
+a very important consequence, i.e. the m-independent
+term in c1 ensures that the UA(1) part of the chiral sym-
+metry will remain effectively broken in the chiral limit in
+the symmetry-restored phase. The coefficient of linear-
+in-λ term at finite temperature is significantly larger than
+its zero temperature value of 0.63 in units of T 2
+c obtained
+from Eq. 1. For extracting the later we have used the
+latest data for the chiral condensate and Fπ from the
+FLAG review [48], for Nf = 3. A significant thermal en-
+hancement in the slope of the eigen spectrum is observed
+above Tc.
+Moreover the slope of the eigen density for
+T ≲ 1.12 Tc is distinctly different from the perturbative
+
+4
+λ3 rise implying significant non-perturbative effects.
+The fate of UA(1) breaking in the continuum
+limit Since the flavor singlet part of the chiral sym-
+metry is anomalous it has no corresponding order pa-
+rameter. Hence to measure whether this singlet part of
+the chiral symmetry is simultaneously (and effectively)
+restored along with the non-singlet part, it has been
+suggested [49] to look at the degeneracies of the in-
+tegrated correlators of mesons i.e., χπ − χδ.
+In the
+continuum, the integrated meson correlators are related
+to each others through the following relations, χδ =
+χσ − 4χdisc and χπ = χη + 4χ5disc. These integrated me-
+son correlators are defined as χπ =
+�
+d4x ⟨πi(x)πi(0)⟩,
+χσ =
+�
+d4x ⟨σ(x)σ(0)⟩, χδ =
+�
+d4x ⟨δi(x)δi(0)⟩ and
+χη =
+�
+d4x ⟨η(x)η(0)⟩ where i = 1, 2, 3.
+We measure
+(χπ − χδ)/T 2 at the four different temperatures above
+Tc, and perform a ∼ 1/N 2
+τ continuum extrapolation at
+each temperature, results of which are shown in Fig. 4.
+For the highest temperature we have only two data points
+available corresponding to Nτ = 8, 12 for continuum ex-
+trapolation hence assigned a 40% and 20% error in slope
+and the intercept obtained from the fit, similar to that
+obtained for the previous temperature. It is evident that
+the continuum extrapolated values of this integrated cor-
+relator drops to 1/6 when T/Tc changes from 1.04-1.12
+and a naive linear extrapolation of the intercept gives a
+temperature around 1.14 Tc when this observable goes to
+zero. In fact the values of this observable increase when
+the lattice spacings are made finer. Performing contin-
+uum estimates with finer lattice sizes Nτ = 16, 12 at
+each temperature, gives a higher intercept than the cor-
+responding extrapolation considering all three Nτ-values.
+Hence the finiteness of this observable is quite robust and
+we conclude that UA(1) does not get effectively restored
+at Tc.
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+ 160
+ 180
+ 200
+ 0
+ 0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016
+(χπ - χδ)/T2
+1/Nτ
+2
+162 MeV
+166 MeV
+171 MeV
+176 MeV
+Fig. 4: The continuum estimates for χπ − χδ normalized by
+the square of temperature for HISQ fermions from 3 dif-
+ferent lattice spacings corresponding to Nτ = 8, 12, 16 at
+T = 162, 166, 171 MeV respectively and from Nτ = 12, 16
+data at T = 176 MeV.
+In the chiral symmetry restored phase, χσ = χπ and
+χδ = χη hence one obtaines χπ − χδ = 4χ5,disc. Using
+chiral Ward identities it is known that χ5,disc = χt/m2
+where χt is the topological susceptibility of QCD. This
+allows relating the UA(1) breaking parameter to the
+topological susceptibility through the relation, 1/4(χπ −
+χδ)m2
+l /T 4 = χt/T 4. A comparison of these two observ-
+ables is shown in Fig. 5. From the figure it is evident
+that for T > 1.05 Tc, when chiral symmetry is effectively
+restored, the two quantities agree with each other within
+errors. This is particularly interesting since for staggered
+quarks, even though the chiral and taste symmetries are
+intermixed at finite lattice spacing, the symmetries of
+QCD and related chiral Ward identities are recovered in
+the continuum limit.
+ 0
+ 0.005
+ 0.01
+ 0.015
+ 0.02
+ 0.025
+ 1.03  1.04  1.05  1.06  1.07  1.08  1.09  1.1  1.11  1.12  1.13
+T/Tc
+(χπ-χδ)ml
+2/4T4
+χt/T4
+Fig. 5: A comparison of the integrated renormalized correla-
+tor (χπ −χδ)m2
+l /4T 4 with the topological susceptibility (mea-
+sured independently using gradient flow in Ref. [50]) for tem-
+peratures > Tc.
+Distribution of the smallest eigenvalue at finite
+temperature The probability distribution of the small-
+est eigenvalue of the QCD Dirac operator λmin has in-
+herent information about the microscopic degrees of free-
+dom. For a random matrix ensemble (at zero tempera-
+ture) the smallest eigenvalue is distributed according to,
+P(cλmin) =
+�π
+2 (cλmin)3/2I3/2(cλmin)e− 1
+2 (cλmin)2 ,
+(4)
+At the lowest temperature T = 145 MeV, we calcu-
+late the probability distribution of the smallest eigen-
+value λmin at different lattice spacings and perform a
+continuum estimate of the distributions, details of which
+are given in Appendix B. The final outcome of the fit is
+given in Fig. 6. The continuum extrapolation of the dis-
+tribution shown as the orange band agrees well with the
+distribution of a chiral Gaussian unitary random matrix
+ensemble. In contrast, we also plot the distribution of
+the lowest eigenvalue at T = 171 MeV whose continuum
+extrapolation is shown as a blue band in Fig. 6. It is
+evident that the lowest eigenvalue which is a part of the
+near-zero peak follows a very different statistics rather
+than known from a chiral RMT.
+
+5
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+P(cλmin)
+cλmin
+T=145 MeV
+=171 MeV
+RMT prediction
+Fig. 6: The continuum extrapolated probability distribution
+of smallest eigenvalue for T = 145, 171 MeV shown as orange
+and blue bands respectively and its comparison with the RMT
+prediction.
+Why is UA(1) effectively restored at tempera-
+ture above Tc? The next question we ask is whether the
+near-zero modes which arise due to interactions among
+instantons can distinctly disentangle out of the bulk
+modes. A similar phenomena occurs in disordered semi-
+metals leading to an Anderson-like transition. In such
+systems, with increasing strength of the disorder poten-
+tial, there is a dynamical transition from a phase of delo-
+calized electron states to that of localized states, with a
+certain energy threshold i.e., the mobility edge separating
+them. It is also known that near such an Anderson-like
+transition, the eigenvalue spacing distribution of the dis-
+ordered states follows a similar behavior as RMTs for all
+spacing values except at the tails of the distribution due
+to the effects of the localized states. We observe the same
+features for the QCD Dirac eigen spacing distribution for
+our finest Nτ = 16 lattices, detailed in Appendix C. In
+addition we have performed a systematic measurement
+of level-spacing distributions at different temperatures
+above Tc for different lattice spacings and extracted the
+parameters that characterize its functional dependence
+in the same Appendix. We find that the bulk modes (ex-
+cept at its higher tails) agree very well with the results
+obtained for random matrices belonging to Gaussian Uni-
+tary ensemble (GUE). Having shown the distinct features
+of near-zero and bulk modes, we have elaborated on how
+reliably we can estimate the temperature at which these
+modes separate in Appendix D. We obtain the tempera-
+ture of ∼ 1.15(3) Tc, which is similar to a mobility edge
+that separates the near-zero from the bulk modes.
+In order to interpret these results, one could visualize
+the quarks moving in the background of an interacting
+ensemble of instantons, where the strength of the inter-
+actions changes as a function of temperature.
+At the
+microscopic level it is conjectured that the instantons
+remain strongly correlated below Tc, subsequently tran-
+sitioning to a liquid-like phase with a finite correlation
+length [51] just above Tc, and eventually to a gas-like
+phase at 2 Tc [13, 15]. Below Tc the intercept of the in-
+frared eigenvalue density quantifies the chiral condensate
+which corresponds to the breaking of the non-singlet part
+of the chiral symmetry. Due to very strong correlations
+the microscopic details of the interactions are lost and the
+eigenvalues repel strongly similar to random matrices of
+a GU ensemble.
+As the temperature is increased, the
+interactions weaken and indeed at ∼ 171 MeV, the near-
+zero eigenvalues with an oscillating behavior, as predicted
+from instanton liquid model, start to become prominent.
+These eventually separate from the bulk at ∼ 1.15 Tc
+analogous to opening of a mobility edge. Earlier studies
+have observed screening of inter-instanton interactions
+and build-up of local pockets of Polyakov loop fluctua-
+tions [38, 52] above such temperatures. This is also the
+region where the constituent dyons of the closely-spaced
+instantons interact semi-classically and thus start to be-
+come detectable [53–56].
+Incidentally this suppression of long range instanton
+interactions also weakens the strength of UA(1) breaking,
+allowing for its effective restoration at T ≲ 1.15 Tc. Lat-
+tice studies [57, 58] have reported a jump in the electrical
+conductivity around this temperature. This also suggests
+that the strength of the attractive potential due to in-
+stantons changes from liquid-like correlations to sparse
+local hot-spots, leaving most of the quark momentum
+states beyond the mobility edge to be delocalized thus
+enhancing the electrical charge transport.
+Conclusions In this letter we have addressed a long-
+standing question of whether the flavor singlet UA(1) sub-
+group of the chiral symmetry gets effectively restored si-
+multaneously with the non-singlet part for QCD with two
+light quark flavors at Tc. The effective restoration of the
+anomalous UA(1) symmetry is a non-perturbative phe-
+nomenon driven by the deep infra-red part of the QCD
+Dirac eigenvalue spectrum. By carefully performing the
+continuum extrapolation of the staggered Dirac spectrum
+on the lattice and studying in detail its properties, we ex-
+plicitly demonstrate that UA(1) remains effectively bro-
+ken in the chirally symmetric phase for T ≲ 1.15 Tc. We
+also provide arguments for why this conclusion should
+remain unchanged even in the chiral limit.
+With the increase in temperature the strength of in-
+teractions between the instantons starts weakening due
+to which the deep infrared part of the spectrum is sepa-
+rated out of the bulk modes which happens to be around
+T ∼ 1.15 Tc. The tunneling probability due to instantons
+also decreases with increasing temperature which results
+in lowering of the height of near-zero peak of eigenvalue
+density. We show for the first time that both these phe-
+nomena are possibly the reason behind the UA(1) restora-
+tion, which also surprisingly happens to be around the
+same temperature. Observations of such rich interplay of
+phenomena in QCD matter above Tc should be quite ro-
+bust, since these are made after performing a continuum
+extrapolation. It will be interesting to observe further
+finer details of chiral transition in the massless limit with
+
+6
+QCD Dirac operators which have exact chiral symmetry
+on the lattice.
+Acknowledgements The authors acknowledge sup-
+port by the Deutsche Forschungsgemeinschaft (DFG,
+German Research Foundation) through the CRC-TR 211
+’Strong-interaction matter under extreme conditions’–
+Project no. 315477589 – TRR 211. S.S. acknowledges
+support by the Department of Science and Technology,
+Govt. of India through a Ramanujan Fellowship. The
+numerical computations in this work were performed on
+the GPU cluster at Bielefeld University. We thank the
+Bielefeld HPC.NRW team for their support. We thank
+the HotQCD Collaboration, specially Christian Schmidt
+for sharing the gauge configurations and software with us.
+We also acknowledge the contribution of Hiroshi Ohno
+who was involved during the early stages of the project.
+S.S. is grateful to Frithjof Karsch for helpful discussions
+and his kind hospitality when this work was finalized. A
+part of this work is based on the MILC collaboration’s
+public lattice gauge theory code [59].
+Appendix A: Details of the lattice calculations
+of the eigenvalue spectrum
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  0.5
+ρ(λ)/Τ3
+λ/T
+162 MeV
+Nτ = 8
+= 12
+= 16
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ρ(λ)/Τ3
+λ/T
+176 MeV
+Nτ = 8
+= 12
+Fig. 7: The eigenvalue spectrum for HISQ Dirac operator at
+three different lattice spacings corresponding to Nτ = 8, 12, 16
+for T = 162 MeV and at Nτ = 8, 12 for T = 176 MeV.
+We first tabulate the lattice sizes, gauge couplings and
+the number of configurations that we have studied for
+each temperature value from 145-176 MeV in Table I. As
+mentioned earlier, it is important that we take the contin-
+uum limit appropriately hence for each temperature we
+performed calculations with three different lattice extents
+Nτ = 8, 12, 16 in order to perform continuum extrapola-
+tion of the parameters characterizing the eigenvalue den-
+sity. We then calculated the first 60, 100, 200 eigenvalues
+of the massless HISQ Dirac matrix for Nτ = 16, 12, 8 re-
+spectively. We have fixed the bin size λa = 0.001 for each
+Nτ for measuring the eigenvalue density and performed a
+jack-knife analysis to remove any auto-correlation effects
+among the data in the bins. We then fit the bulk part
+i.e. all eigenvalues above an infrared cut-off λ > λ0 with
+the fit ansatz ρ(λ)
+T 3 = λ
+T . c1(T,m)
+T 2
++ ρ0
+T 3 . The results of the
+fit and the choice of cut-off at different temperatures are
+mentioned in Table II.
+T (MeV)
+β Ns Nτ Nconfs
+145
+6.285 48 12
+1530
+145
+7.010 64 16
+2860
+162
+6.423 32
+8
+250
+162
+6.825 48 12
+1960
+162
+7.130 64 16
+3390
+166
+6.445 32
+8
+400
+166
+6.850 48 12
+2100
+166
+7.156 64 16
+2190
+171
+6.474 32
+8
+280
+171
+6.880 48 12
+1980
+171
+7.188 64 16
+1040
+176
+6.500 32
+8
+240
+176
+6.910 48 12
+330
+Tab. I: The parameters for the lattice calculations
+T [MeV] Nτ λ0/T
+c1
+T 2
+ρ0/T 3
+145
+12
+0.1
+9.0(5) 7.30(7)
+145
+16
+0.05
+9(1)
+6.67(9)
+162
+8
+0.2
+8.8(3)
+4.1(1)
+162
+12
+0.15 13.2(2) 2.69(5)
+162
+16
+0.1
+17.5(5) 1.93(7)
+166
+8
+0.2
+8.9(1) 3.31(5)
+166
+12
+0.15 13.3(3) 1.92(6)
+166
+16
+0.1
+16.6(8) 1.4(1)
+171
+8
+0.2
+9.3(1) 2.38(5)
+171
+12
+0.15 12.9(1) 1.19(3)
+171
+16
+0.1
+17.0(5) 0.45(8)
+176
+8
+0.2
+9.5(1) 1.67(4)
+176
+12
+0.15 13.0(2) 0.36(6)
+Tab. II: Lattice size (N 3
+σ × Nτ), temperature (T), the esti-
+mated values of c1/T 2 and ρ0/T 3 after the fit of the bulk
+modes by taking the lower cutoff at λ0/T.
+
+7
+We have shown the eigenvalue distributions for three
+different temperatures at 145, 166, 171 MeV in Fig. 1. We
+also have measured the eigenvalue densities at two other
+temperatures at 166, 176 MeV which we show in Fig. 7.
+Appendix B: Details of the calculation per-
+formed for the smallest eigenvalues for T < Tc
+First we have extracted the smallest eigenvalue from
+each configuration for Nτ = 12, 16 and later re-scaled
+to the dimensionless quantity cλmin, where the value of
+⟨ ¯ψψ⟩ at finite temperature is obtained from Ref. [60].
+Keeping the bin size constant we obtained the probabil-
+ity distribution of cλmin for each Nτ and then performed
+a spline interpolation by taking appropriate weights pro-
+portional to the errors for each data point in order to
+have a smoother interpolating curve. Next we performed
+a continuum extrapolation at each value of cλmin of the
+interpolating function with the ansatz c + d/N 2
+τ . We as-
+signed a 15% error for T = 145 MeV, as we only had
+two points while performing the continuum extrapola-
+tion. In Fig. 6 we find a good agreement between the
+continuum extrapolated distribution of the lowest eigen-
+value at T = 145 MeV and the RMT predictions from
+a Gaussian Unitary ensemble. A slight discrepancy exist
+for lower and higher values of cλmin. This can be due to
+the fact that we use a very low but finite convergence cri-
+terion while calculating the eigenvalue spectrum. Hence
+we do not have any data for cλmin < 0.6. Since we are
+plotting a probability distribution (of the smallest eigen-
+value), the area under the curve must be unity. To pre-
+serve this criterion the values of the probability densities
+along the higher end of the tail lie above the RMT curve
+in order to compensate for the relatively lower values in
+the lower portion of the tail.
+Appendix C: The level spacing distribution for
+bulk modes
+Next we look at the level spacing distribution of the
+bulk modes.
+To study the universal properties of the
+eigenvalue level spacing fluctuations one has to remove
+the system dependent mean. This is done by a method
+called unfolding. Let λ represent eigenvalues in the as-
+cending sequence for any particular gauge configuration.
+The average density of the eigenvalues in the sequence i.e.
+the reciprocal of the average spacing as a function of λ
+is represented as ¯ρ(λ). The eigenvalue sequence can then
+be unfolded using the average level-staircase function,
+¯η(λ) =
+� λ
+λ0 dλ′¯ρ(λ′) which tells us how many eigenvalues
+in this sequence are less than λ on an average. Here λ0
+labels the eigenvalue beyond which all the higher eigen-
+values are bulk modes and below which are the near-zero
+modes. The unfolded sequence is labeled by λuf
+i
+= ¯η(λi),
+where the index i labels the original eigenvalue whose un-
+folding is performed.
+When appropriately normalized,
+the average spacing between the unfolded eigenvalues
+equals unity. The nearest neighbor spacing distribution
+is constructed by calculating the differences between con-
+secutive unfolded eigenvalues λuf
+i+1 − λuf
+i
+and organizing
+them into histogram bins. This gives us a picture of how
+the eigenvalue spacings fluctuate about the average which
+we have plotted in Fig. 8 for four different temperatures
+T = 162, 166, 171, 176 MeV and at each temperature, for
+the three different lattice sizes Nτ = 8, 12, 16 except for
+T = 176 MeV. We have then estimated the functional de-
+pendence of these nearest neighbor spacing distributions
+by two different fit ansatz, shown as solid and dotted
+lines in Fig. 8. The dotted curves were obtained after
+performing a fit to the lattice data points with the func-
+tion f(s) = asbe−cs2, motivated by the Wigner surmise.
+The solid curves on the other hand, were obtained after
+fitting the points to an ansatz function f(s) = ps2e−qs2.
+The values of these parameters a, b, c, p, q after perform-
+ing the fits are given in Table III. It is evident that the
+level repulsion between the bulk modes is quadratic simi-
+lar to that of random matrices belonging to the Gaussian
+unitary ensemble (GuE). However for the Nτ = 16 lat-
+tices, due to the contamination with the near-zero modes
+the fit of the tail is not good and can not be explained by
+RMT prediction. In order to account for the long tail of
+the spacing distribution we fit it to a semi-Poisson distri-
+bution P(s) ∼ s2 exp (−αs) which shows strong repulsion
+at small values of s but falls off slowly at large values of
+s parameterized by a fit parameter α. After performing
+the fit of the level separation with this ansatz, we obtain
+the value of α = 3.02(7), 3.17(9), 3.3(1) for temperatures
+T = 162, 166, 171 MeV respectively.
+The lattice data
+now do agree to this new fit ansatz reasonably well for
+Nτ = 16 at all temperatures above Tc, which is evident
+in Fig. 9.
+T (MeV) Nτ
+a
+b
+c
+p
+q
+162
+8 2.91(5) 1.85(3) 1.19(1) 3.16(7) 1.26(2)
+162
+12
+2.6(1) 1.69(6) 1.13(3)
+3.2(1) 1.29(3)
+162
+16
+2.1(4)
+1.2(2)
+1.0(1)
+4.0(6)
+1.6(1)
+166
+8 2.78(5) 1.78(2) 1.16(1) 3.13(9) 1.26(2)
+166
+12
+2.6(2) 1.66(7) 1.12(4)
+3.2(2) 1.30(4)
+166
+16
+2.1(5)
+1.2(2)
+1.1(2)
+4.5(8)
+1.8(2)
+171
+8 2.74(7) 1.76(3) 1.15(2)
+3.2(1) 1.27(2)
+171
+12
+2.5(2)
+1.6(1) 1.11(6)
+3.4(3) 1.35(6)
+171
+16
+1.6(4)
+0.8(2)
+1.0(2)
+5(1)
+2.0(3)
+176
+8 2.77(7) 1.77(4) 1.16(2) 3.15(9) 1.27(2)
+176
+12
+2.3(3)
+1.4(1) 1.07(8)
+3.5(3) 1.39(7)
+Tab. III: The estimated values of the parameters after the fit
+to different unfolded level spacing distributions.
+Appendix D: Details of extraction of the mobil-
+ity edge
+Next, in order to estimate when these bulk modes sep-
+arate from the deep-infrared peak of eigenvalues, we cal-
+culate at what temperature the functional fit of the bulk
+eigenvalue spectrum has a non-zero intercept along the
+λ-axis which is larger than the typical width of the near-
+zero peak.
+In the continuum, we have already calcu-
+lated the slope of the bulk eigenvalue density, which is
+c1(m, T)/T 2 = 16.8(4). Looking at the eigenvalue distri-
+butions in Fig.1, we can choose a typical value of λ at
+
+8
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+P(s)
+Spacing s
+T=162 MeV
+Nτ=16 
+=12 
+=8
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+P(s)
+Spacing s
+T=166 MeV
+Nτ=16
+=12
+=8
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+P(s)
+Spacing s
+T=171 MeV
+Nτ=16
+= 12
+=8
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+P(s)
+Spacing s
+T=176 MeV
+Nτ=12
+=8
+Fig. 8: Unfolded level spacing distribution of bulk eigenvalues modes for different temperatures shown as a function of different
+lattice spacings or equivalently, Nτ. The solid lines in each plot correspond to the two-parameter fit and the dotted curves for
+three-parameter fits inspired from the Wigner surmise for Gaussian unitary random matrix ensembles.
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+P(s)
+Spacing s
+Nτ=16
+T=162 MeV
+=166 MeV
+=171 MeV
+Fig. 9: A fit to the eigenvalue level spacing distribution using
+a mixed ansatz for Nτ = 16 at T = 162, 166, 171 MeV.
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 1
+ 1.02  1.04  1.06  1.08  1.1  1.12  1.14  1.16  1.18
+ρ0/T3
+T/Tc
+Fig. 10: Continuum extrapolation of the bulk intercept for
+eigenvalue densities at different temperatures above Tc. The
+horizontal line corresponds to ρ0/T 3 = −1.34 for the bulk
+spectrum when it is completely separates from near zero
+modes.
+which the near-zero and bulk modes separate out, which
+is most evident for the Nτ = 16 lattices at λ0/T ∼ 0.08.
+Using these inputs and that the bulk modes have a linear-
+in-λ dependence we can calculate the value of bulk inter-
+cept ρ0/T 3 = −1.34 at λ = 0. Next we take the values
+of the intercept of bulk mode density for all T > Tc
+from Table II and perform a continuum extrapolation
+with the function ρ0/T 3 + d/N 2
+τ . The continuum values,
+ρ0/T 3 so-obtained are shown in Fig. 10 for all T > Tc.
+At the highest temperature T = 176 MeV a 10% error is
+assigned to the data point since we could perform a con-
+tinuum estimate with the data available only for two Nτ
+values. Now fitting the continuum extrapolated values,
+ρ0/T 3 with the ansatz ρ0/T 3 = d1(T/Tc) + d2 we obtain
+d1 = −23.1(3) and d2 = 25.3(3). Using this parametric
+dependence of the continuum value of the intercept as
+a function of temperature, we extract a T/Tc = 1.15(3)
+when the value of ρ0/T 3 = −1.34 i.e., when the near-zero
+modes distinctly emerge out from the bulk spectrum.
+[1] T. Banks and A. Casher, Nucl. Phys. B 169, 103 (1980).
+[2] H. Leutwyler and A. V. Smilga, Phys. Rev. D 46, 5607
+(1992).
+[3] A. Bazavov et al. (HotQCD), Phys. Lett. B 795, 15
+(2019), 1812.08235.
+[4] R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338
+
+9
+(1984).
+[5] A. Butti, A. Pelissetto, and E. Vicari, JHEP 08, 029
+(2003), hep-ph/0307036.
+[6] A. Pelissetto and E. Vicari, Phys. Rev. D 88, 105018
+(2013), 1309.5446.
+[7] S. Sharma (HotQCD), PoS CPOD2017, 086 (2018).
+[8] M. P. Lombardo and A. Trunin, Int. J. Mod. Phys. A 35,
+2030010 (2020), 2005.06547.
+[9] A. Bazavov et al. (HotQCD), Phys. Rev. D 86, 094503
+(2012), 1205.3535.
+[10] M. I. Buchoff et al., Phys. Rev. D 89, 054514 (2014),
+1309.4149.
+[11] T. Bhattacharya et al., Phys. Rev. Lett. 113, 082001
+(2014), 1402.5175.
+[12] V. Dick, F. Karsch, E. Laermann, S. Mukherjee, and
+S. Sharma, Phys. Rev. D 91, 094504 (2015), 1502.06190.
+[13] P. Petreczky, H.-P. Schadler, and S. Sharma, Phys. Lett.
+B 762, 498 (2016), 1606.03145.
+[14] A. Bazavov et al., Phys. Rev. D 100, 094510 (2019),
+1908.09552.
+[15] H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya, X. D.
+Wang, and Y. Zhang, Phys. Rev. Lett. 126, 082001
+(2021), 2010.14836.
+[16] O. Kaczmarek, L. Mazur, and S. Sharma, Phys. Rev. D
+104, 094518 (2021), 2102.06136.
+[17] G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko,
+H. Matsufuru, and J.-I. Noaki, Phys. Rev. D 87, 114514
+(2013),
+[Erratum:
+Phys.Rev.D 88,
+019901 (2013)],
+1304.6145.
+[18] T.-W. Chiu, W.-P. Chen, Y.-C. Chen, H.-Y. Chou,
+and T.-H. Hsieh (TWQCD), PoS LATTICE2013, 165
+(2014), 1311.6220.
+[19] A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto,
+T. Kaneko, and J. Noaki, Phys. Rev. D 96, 034509
+(2017), [Addendum:
+Phys.Rev.D 96, 079902 (2017)],
+1612.01908.
+[20] B. B. Brandt, A. Francis, H. B. Meyer, O. Philipsen,
+D. Robaina, and H. Wittig, JHEP 12, 158 (2016),
+1608.06882.
+[21] S. Aoki, Y. Aoki, G. Cossu, H. Fukaya, S. Hashimoto,
+T. Kaneko, C. Rohrhofer, and K. Suzuki (JLQCD), Phys.
+Rev. D 103, 074506 (2021), 2011.01499.
+[22] S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, C. Rohrhofer,
+and K. Suzuki (JLQCD), PTEP 2022, 023B05 (2022),
+2103.05954.
+[23] E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A
+560, 306 (1993), hep-th/9212088.
+[24] J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70,
+3852 (1993), hep-th/9303012.
+[25] J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994),
+hep-th/9401059.
+[26] M. E. Berbenni-Bitsch, S. Meyer, A. Schafer, J. J. M.
+Verbaarschot, and T. Wettig, Phys. Rev. Lett. 80, 1146
+(1998), hep-lat/9704018.
+[27] G. Akemann, P. H. Damgaard, U. Magnea, and S. Nishi-
+gaki, Nucl. Phys. B 487, 721 (1997), hep-th/9609174.
+[28] A. D. Jackson, M. K. Sener, and J. J. M. Verbaarschot,
+Nucl. Phys. B 506, 612 (1997), hep-th/9704056.
+[29] A. M. Garcia-Garcia and J. J. M. Verbaarschot, Nucl.
+Phys. B 586, 668 (2000), hep-th/0003159.
+[30] G. Akemann and T. R. W¨urfel, JHEP 12, 128 (2021),
+2110.03617.
+[31] M. Giordano and T. G. Kovacs, Universe 7, 194 (2021),
+2104.14388.
+[32] J. J. M. Verbaarschot, Nucl. Phys. B 427, 534 (1994),
+hep-lat/9402006.
+[33] A. M. Garcia-Garcia and J. C. Osborn, Phys. Rev. D 75,
+034503 (2007), hep-lat/0611019.
+[34] T. G. Kovacs and F. Pittler, Phys. Rev. Lett. 105, 192001
+(2010), 1006.1205.
+[35] T. G. Kovacs and F. Pittler, Phys. Rev. D 86, 114515
+(2012), 1208.3475.
+[36] M. Giordano, T. G. Kovacs, and F. Pittler, Phys. Rev.
+Lett. 112, 102002 (2014), 1312.1179.
+[37] M. Giordano, S. D. Katz, T. G. Kovacs, and F. Pittler,
+JHEP 02, 055 (2017), 1611.03284.
+[38] L. Holicki, E.-M. Ilgenfritz, and L. von Smekal, PoS
+LATTICE2018, 180 (2018), 1810.01130.
+[39] T. G. Kovacs and R. A. Vig, Phys. Rev. D 97, 014502
+(2018), 1706.03562.
+[40] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
+[41] A. Alexandru and I. Horv´ath, Phys. Lett. B 833, 137370
+(2022), 2110.04833.
+[42] A. Alexandru and I. Horv´ath, Phys. Rev. D 100, 094507
+(2019), 1906.08047.
+[43] A. Alexandru and I. Horv´ath, Phys. Rev. Lett. 127,
+052303 (2021), 2103.05607.
+[44] A. Bazavov et al., Phys. Rev. D 95, 054504 (2017),
+1701.04325.
+[45] A. V. Smilga and J. Stern, Phys. Lett. B 318, 531 (1993).
+[46] S. Aoki, H. Fukaya, and Y. Taniguchi, Phys. Rev. D 86,
+114512 (2012), 1209.2061.
+[47] J. J. M. Verbaarschot, Nucl. Phys. B Proc. Suppl. 53, 88
+(1997), hep-lat/9607086.
+[48] Y.
+Aoki
+et
+al.
+(Flavour
+Lattice
+Averaging
+Group
+(FLAG)), Eur. Phys. J. C 82, 869 (2022), 2111.09849.
+[49] E. V. Shuryak, Comments Nucl. Part. Phys. 21, 235
+(1994), hep-ph/9310253.
+[50] L. Mazur, Ph.D. thesis, Bielefeld U. (2021).
+[51] T. Sch¨afer and E. V. Shuryak, Rev. Mod. Phys. 70, 323
+(1998), hep-ph/9610451.
+[52] F. Bruckmann, T. G. Kovacs, and S. Schierenberg, Phys.
+Rev. D 84, 034505 (2011), 1105.5336.
+[53] V. G. Bornyakov, E. M. Ilgenfritz, B. V. Martemyanov,
+and M. M¨uller-Preussker, Phys. Rev. D 93, 074508
+(2016), 1512.03217.
+[54] R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Lett. B
+794, 14 (2019), 1811.07914.
+[55] R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Rev. D
+102, 034501 (2020), 1912.09141.
+[56] R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Rev. D
+105, L071501 (2022), 2112.04537.
+[57] A. Amato, G. Aarts, C. Allton, P. Giudice, S. Hands,
+and J.-I. Skullerud, Phys. Rev. Lett. 111, 172001 (2013),
+1307.6763.
+[58] G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands,
+and J.-I. Skullerud, JHEP 02, 186 (2015), 1412.6411.
+[59] http://physics.utah.edu/~detar/milc.html.
+[60] P. Steinbrecher, Ph.D. thesis, U. Bielefeld (main) (2018).
+

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@@ -0,0 +1,1531 @@
+TunesFormer: Forming Tunes with Control Codes
+Shangda Wu
+Music AI and Information Technology
+Central Conservatory of Music
+Beijing, China
+[email protected]
+Maosong Sun
+Computer Science and Technology
+Tsinghua University
+Beijing, China
+[email protected]
+ABSTRACT
+In recent years, deep learning techniques have been applied
+to music generation systems with promising results. How-
+ever, one of the main challenges in this field has been the
+lack of annotated datasets, making it difficult for models to
+learn musical forms in compositions. To address this issue,
+we present TunesFormer1, a Transformer-based melody gen-
+eration system that is trained on a large dataset of 285,449
+ABC tunes. By utilizing specific symbols commonly found
+in ABC notation to indicate section boundaries, Tunes-
+Former can understand and generate melodies with given
+musical forms based on control codes. Our objective evalu-
+ations demonstrate the effectiveness of the control codes in
+achieving controlled musical forms, and subjective experi-
+ments show that the generated melodies are of comparable
+quality to human compositions. Our results also provide in-
+sights into the optimal placement of control codes and their
+impact on the generated melodies. TunesFormer presents
+a promising approach for generating melodies with desired
+musical forms through the use of deep learning techniques.
+Author Keywords
+Transformer, controllable melody generation, musical form,
+control codes, ABC notation
+CCS Concepts
+•Computing methodologies → Neural networks; •Applied
+computing → Sound and music computing;
+1.
+INTRODUCTION
+Musical form plays a crucial role in shaping the aesthetic
+and expressive qualities of music.
+Examples of musical
+forms include verse-chorus, ABAB, and sonata forms, which
+are commonly found in popular and classical music. The
+ability to generate melodies with specific musical forms is a
+highly sought-after feature in music generation systems, as
+it allows users to create music that adheres to specific mu-
+sical conventions and styles. This can be particularly useful
+for music producers, composers, and educators seeking to
+generate music that follows a specific form.
+Deep learning techniques have gained widespread atten-
+tion in recent years as a means of generating music with di-
+verse styles and properties. Various approaches have been
+proposed, including the use of recurrent neural networks
+(RNNs) [9, 8, 22, 25], generative adversarial networks (GANs)
+1https://github.com/sander-wood/tunesformer
+[5, 24, 26], and Transformer models [4, 6, 13]. Among these
+approaches, Transformer models [21] have proven particu-
+larly effective for music generation due to their ability to
+model long-range dependencies, handle variable-length in-
+put sequences, and generate coherent and consistent output.
+While deep learning techniques have shown promising re-
+sults in generating music, a major challenge faced by mu-
+sic generation systems is the ability to generate melodies
+with predefined musical forms. Previous related work has
+achieved some level of melody generation based on struc-
+tural information, but there are limitations such as a focus
+only on harmony or section length [1, 17, 28], considera-
+tion of only bar-level structure [23, 29], or reliance on rules
+to generate phrases and sections [3, 16]. Accurately iden-
+tifying specific musical forms can be difficult for rules or
+algorithms, and manually labelling this type of data is ex-
+pensive due to the time and resources required, as well as
+the high level of musical knowledge and understanding re-
+quired. Thus, the problem of effectively teaching models to
+learn musical forms from datasets remains largely unsolved.
+To address the challenge of melody generation conditioned
+on musical forms, we introduce TunesFormer, a Transformer-
+based melody generation system trained on a large dataset
+of 285,449 ABC tunes. ABC notation is a widely used text-
+based representation of music that is more comprehensive
+and expressive than MIDI. In addition to the symbols used
+to represent pitches and rhythms, ABC notation also in-
+cludes symbols to represent section boundaries and other
+structural elements.
+Based on these symbols, we design
+several control codes that allow TunesFormer to generate
+melodies with specific musical forms based on user input.
+We present a thorough evaluation of TunesFormer through
+both objective and subjective experiments. Our objective
+evaluations demonstrate the effectiveness of the control codes
+in achieving controlled musical forms, while subjective ex-
+periments show that the control codes for edit distance sim-
+ilarity are relevant to human subjective perception, and the
+quality of the generated melodies is comparable to that of
+human compositions as evaluated by professional musicians.
+These experimental results also provide insight into the op-
+timal placement of control codes and their impact on the
+generated melodies. Overall, our results highlight the po-
+tential of TunesFormer as a powerful and flexible tool for
+generating melodies with desired musical forms.
+The main contributions of this paper are:
+• The introduction of TunesFormer, a Transformer-based
+melody generation system that generates melodies with
+specific musical forms using control codes.
+• TunesFormer is trained on a large ABC notation dataset,
+allowing it to learn a more comprehensive representa-
+tion of music notation compared to systems trained
+on MIDI datasets.
+arXiv:2301.02884v1  [cs.SD]  7 Jan 2023
+
+• We conduct both objective and subjective experiments
+to comprehensively evaluate TunesFormer, demonstrat-
+ing the effectiveness of the control codes.
+2.
+RELATED WORK
+There has been a significant amount of research dedicated
+to music generation using deep learning techniques. Much
+of this work has focused on the use of deep neural networks
+to model complex patterns in symbolic music generation,
+but a significant challenge remains in generating full-length
+music with consistent long-term structure.
+Chen et al. explored the use of explicit structure encod-
+ing in neural networks for symbolic music generation [1].
+They found that incorporating explicit structure encoding
+significantly improved the quality and structure of the gen-
+erated music. However, this approach relies on harmony to
+guide the model in generating melodies with good structure,
+without considering the actual musical form.
+PopMNet [23], a model for generating structured pop mu-
+sic melodies, consists of a Structure Generation Net (SGN)
+and a Melody Generation Net (MGN), with the SGN gen-
+erating melody structures based on pairwise relations be-
+tween bars (repetition and sequence) and the MGN gener-
+ating melodies based on these structures and chord progres-
+sions. MELONS [29], a framework based on Transformer
+for generating melodies with long-term structures, also con-
+sists of a structure generation net and a melody generation
+net, which are used to factor the melody generation process
+into two sub-problems: structure generation and structure-
+conditional melody generation. While these approaches are
+able to generate melodies with clearer structures compared
+to other models, they are limited to generating melodies
+with pairwise relations between bars, rather than more com-
+plex structural patterns.
+MusicFrameworks [3] is a hierarchical music structure
+representation and a multi-step generative process for cre-
+ating full-length melodies guided by long-term repetitive
+structure, chord, melodic contour, and rhythm constraints.
+This approach allows for the customization of chords, ba-
+sic melody, and rhythm structure, providing more control
+over the generated melodies. However, this method requires
+structural information to be extracted from existing songs
+to generate new ones, which relies on hand-crafted rules and
+may not always be available.
+MeloForm [16] utilizes an expert system to generate a
+melody by developing musical elements from motifs to phrases,
+and then to sections with repetitions and variations ac-
+cording to a given musical form. However, the generated
+melodies may lack musical richness, so the approach also
+utilizes a Transformer-based refinement model to improve
+the melody without altering its musical form. While this ap-
+proach allows for precise control of musical form, the model
+does not learn the concept of musical form from the data
+and relies on an expert system in the generation process.
+A predictive deep network [2] models polyphonic music
+using a novel graphical representation, inspired by tonnetz
+from music theory, in a deep neural network. This tonnetz-
+inspired representation is evaluated using a dataset of clas-
+sical music and is found to produce musical sequences that
+are more tonally stable and contain more repeated patterns
+than sequences generated by pianoroll-based models. CM-
+HRNN [8], a conditional melody generation model based on
+a hierarchical recurrent neural network, generates melodies
+with long-term structures based on given chord accompani-
+ments. Both approaches learn long-term dependencies, re-
+sulting in the implicit generation of melodies with repetitive
+patterns, although these patterns do not represent specific
+musical forms.
+MorpheuS [10] is a music generation system that can gen-
+erate polyphonic pieces with a given tension profile and
+long- and short-term repeated pattern structures. A math-
+ematical model for tonal tension is used to quantify the
+tension profile and state-of-the-art pattern detection algo-
+rithms are utilized to extract repeated patterns in a tem-
+plate piece. These patterns are then used to constrain long-
+term structure in the generated pieces. However, this ap-
+proach is limited to the generation of music with predefined
+tension profiles and does not consider the incorporation of
+additional constraints or variables in the music generation
+process.
+Zhang et al.
+proposed a harmony-aware learning ap-
+proach [28] for generating structured pop music, which can
+improve the structure and quality of the generated music.
+Naruse et al. developed a method for generating pop mu-
+sic with controllable phrase lengths [17] using a deep neu-
+ral network and adding PHRASE and BAR COUNTDOWN events.
+However, neither of these approaches explicitly captures the
+relationships between sections.
+3.
+METHODOLOGY
+3.1
+Data Representation
+In this research, we aim to generate score information for
+music [7, 20], rather than performance information [12, 18].
+Thus, the data representation used must effectively encode
+sheet music.
+The three most commonly used symbolic music formats
+are ABC notation, MusicXML, and MIDI. ABC notation is
+designed for simplicity and was originally intended for use
+with folk music, while MusicXML is geared towards the ex-
+change of musical notation. MIDI, on the other hand, is
+focused on the sequencing of instrument sounds at a low
+level, rather than higher-level musical concepts. Most pre-
+vious works on symbolic music information retrieval and
+generation [11, 13, 27] utilize MIDI as the data represen-
+tation due to its popularity.
+However, in this study, we
+adopt ABC notation as our data representation due to its
+advantages for score-oriented music generation over MIDI.
+One advantage of ABC notation is that it can distinguish
+enharmonic notes (e.g., B#3 and C4), while MIDI assigns
+numerical codes to specific pitches without considering note
+names.
+This means that MIDI is unable to differentiate
+between enharmonic notes.
+Additionally, for music generation tasks, ABC notation
+can accurately represent complex durations, while MIDI re-
+quires a trade-off between accuracy and sequence length
+or vocabulary size. This can result in quantization errors
+where certain notes cannot be accurately represented due
+to pre-defined time resolution (e.g., 16th notes).
+Furthermore, ABC notation includes a comprehensive set
+of musical symbols found in sheet music, including impor-
+tant elements like ornamentation and articulation that are
+not explicitly represented in MIDI, as shown in Fig.
+1.
+More importantly, some symbols used to indicate section
+boundaries in ABC notation can serve as the basis for con-
+trol codes. While MusicXML also has these advantages over
+MIDI, it is based on XML, which can be more complex and
+time-consuming to work with compared to ABC notation,
+which is based on ASCII and therefore easier to use with
+fewer errors.
+Overall, the use of ABC notation for score-oriented mu-
+sic generation allows for a more accurate and comprehen-
+sive representation of music while maintaining simplicity,
+enabling the generation of more complex and musically co-
+
+(a) ABC notation
+(b) MusicXML
+(c) MIDI
+Figure 1: Excerpts from Nocturne Op. 9 No. 2 (E Flat Major) rendered by MuseScore 4 in different formats. While (a) and
+(b) are essentially the same, (c) does not distinguish between enharmonic notes and loses many musical symbols.
+herent melodies. This makes ABC notation a better choice
+for music generation systems compared to MIDI, partic-
+ularly for tasks that require a greater level of detail and
+control over the generated music.
+3.2
+Control Codes
+Control codes are symbols that are added to the ABC no-
+tation representation to indicate the desired musical form
+of the generated melodies.
+The most important information in musical forms lies in
+the number of sections and the similarity between the indi-
+vidual sections. For example, the musical form ABA’ refers
+to a structure with three sections, where there is a main
+section A followed by a contrasting section B (dissimilar)
+and then the main section reappears as the recapitulation
+A’ (similar) but with some slight variation.
+Incorporating control codes that specify the number of
+bars in each section can provide an additional level of con-
+trol. These control codes can effectively influence the pacing
+and flow of the music, as the number of bars in each sec-
+tion can significantly impact the overall structure and form
+of the piece. For instance, melodies with the same struc-
+ture but different numbers of bars in each section, such as
+A8B8A8 and A4B8A4, exhibit distinct musical characteristics
+due to the varied length of their sections.
+Based on the above reasons, we add the following control
+codes to each ABC tune in the dataset through an auto-
+mated process to indicate its musical form:
+• Number of Bars (NB): controls the number of bars
+in a section of the melody. For example, users could
+specify that they want a section to contain 8 bars, and
+TunesFormer would generate a section that fits within
+that structure. It counts on the bar symbol |.
+• Number of Sections (NS): controls the number of sec-
+tions in the entire melody. This can be used to create a
+sense of structure and coherence within the melody, as
+different sections can be used to create musical themes
+or motifs. It counts on several symbols that are com-
+monly used in ABC notation and can be used to rep-
+resent section boundaries: [|,||,|],|:,::, and :|.
+• Edit Distance Similarity (EDS): controls the similar-
+ity level between the current section c and a previous
+section p in the melody.
+It is based on the Leven-
+shtein distance [14] lev(c, p), and can be formalised as
+follows:
+eds(c, p) = 1 −
+lev(c, p)
+max(|c|, |p|)
+(1)
+where |c| and |p| are the string length of two sections.
+The EDS control code is discretized into 11 levels,
+ranging from 0 (no match at all) to 10 (exact match).
+To investigate the impact of different placements of these
+control codes on generated melodies, we designed the fol-
+lowing five placements:
+• Global Placement (GP): all control codes are placed at
+the beginning of the ABC notation.
+• Section-based Placement (SP): NB and EDS control
+codes are placed at the beginning of each section to
+indicate the number of bars and the similarity of the
+edit distances in that section.
+• Section Countdown Placement (SCP): similar to section-
+based placement, but NS control codes are also placed
+at the beginning of each section to indicate the num-
+ber of sections remaining in the piece.
+• Bar Countdown Placement (BCP): similar to section-
+based placement, but NB control codes are placed at
+the beginning of each bar to indicate the number of
+bars remaining in the section.
+• Section & Bar Countdown Placement (SBCP): a com-
+bination of SCP and BCP, with NS control codes placed
+at the beginning of each section and NB control codes
+placed at the beginning of each bar. This placement
+allows for both the countdown of sections and bars to
+be presented in the piece.
+Fig. 2 shows an example of an ABC tune with control
+codes using the GP. Other placements of control codes can
+be found in Appendix A. The tune header includes the time
+signature and key signature, and the tune body consists of
+three sections, each with 8 bars.
+The first control code
+[SECS_3] specifies there are 3 sections in the tune, and
+
+a tempo
+fpatempo
+fp3Tune Body I
+[SECS_3][BARS_8][SIM_3][BARS_8][SIM_10][SIM_3][BARS_8]
+L:1/4
+M:4/4
+K:C
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 ||
+“C” e e“G” d d/d/ |“Am” c A“Em” G E | “F” F3/2 G/ A F |“C” E/E/G/G/ c G |
+e e“G” d d/d/ |“Am” c A“Em” G E |“F” F3/2 G/“G” A B | “C” d c c2 ||
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 |]
+Control Codes
+Tune Header
+Tune Body II
+Tune Body III
+Figure 2: An example of the GP. For the purpose of demon-
+stration, it is separated into several sections.
+the following control code [BARS_8] indicates the first sec-
+tion has 8 bars. The next two control codes [SIM_3] and
+[BAR_8] indicate that the EDS between tune body II and
+tune body I is approximately 0.3, and tune body II has 8
+bars. The last three control codes [SIM_10], [SIM_3] and
+[BARS_8] specify that tune body III is identical to tune
+body I while dissimilar to tune body II, and has 8 bars.
+3.3
+Model Architecture
+TunesFormer is a Transformer-based language model that
+utilizes the GPT-2 small [19] architecture as its basis, which
+is a decoder-only, unidirectional Transformer. The GPT-2
+small architecture is a deep learning model that consists of
+12 layers, each with a hidden size of 768 and 12 attention
+heads.
+This allows TunesFormer to effectively learn and
+recognize complex patterns and structures in ABC notation.
+To accurately represent the independent semantics of each
+character in ABC notation, we employ character-level tok-
+enization. In addition, we also include control codes as spe-
+cial tokens. During inference, these control codes can either
+be provided by users as prompts or generated by Tunes-
+Former itself, allowing for a high degree of flexibility in the
+music generation process.
+We trained TunesFormer from scratch using the learning
+rate α = 10−4, with a 1,000-step linear warmup and learning
+rate decay. We trained a total of 30 epochs with a batch
+size of 32, using the AdamW [15] optimizer with β1 = 0.9,
+β2 = 0.999, ϵ = 10−8, and a weight decay coefficient of
+0.01.
+We also use automatic mixed precision to improve
+the efficiency of the training process.
+4.
+EXPERIMENTS
+4.1
+Dataset
+The dataset used to train and evaluate TunesFormer is
+collected from two sources: The Session2 and ABCnota-
+tion.com3. The Session is a community website focused on
+Irish traditional music, while ABCnotation.com is a website
+that provides a standard for folk and traditional music nota-
+tion in the form of ASCII text files. The combined dataset
+consists of 285,449 ABC tunes, with 99% (282,595) of the
+tunes used as the training set and the remaining 1% (2854)
+used as the evaluation set.
+To ensure consistency and standardization among the ABC
+tunes in the dataset, we first converted them all into Mu-
+sicXML format and then re-converted them back into ABC
+notation. In order to focus solely on the musical content,
+we removed any natural language elements (such as titles,
+composers, and lyrics) and unnecessary information (such
+as reference numbers and sources).
+2https://thesession.org
+3https://abcnotation.com
+0.0000
+0.1000
+0.2000
+0.3000
+0.4000
+0.5000
+0.6000
+0.7000
+0.8000
+0.9000
+1.0000
+GP
+SP
+SCP
+BCP
+SBCP
+Eval Set
+Figure 3: Results of bar length accuracy at different settings.
+As depicted in Fig. 4, in this dataset, 99.4% of the pieces
+have no more than 8 sections and 99.1% of the sections have
+no more than 32 bars. Therefore, we set an upper limit of
+8 for the number of sections and 32 for the number of bars.
+4.2
+Objective Experiments
+We present the objective experimental results to evaluate
+the effectiveness of TunesFormer in generating controlled
+musical forms. We measured the bar length accuracy, sec-
+tion number accuracy, bar number accuracy, and Edit Dis-
+tance Similarity (EDS) of the generated tunes in each of
+the five placements: Global Placement (GP), Section-based
+Placement (SP), Section Countdown Placement (SCP), Bar
+Countdown Placement (BCP), and Section & Bar Count-
+down Placement (SBCP). The evaluation set consisted of
+2854 tunes, which were used as a benchmark for compari-
+son. To provide context for our evaluation, we analyzed the
+distribution of the number of sections, number of bars per
+section, and EDS of the tunes in the dataset.
+We first measure the bar length accuracy at different set-
+tings to evaluate the grammatical correctness of the tunes
+generated by TunesFormer. Bar length accuracy refers to
+the correctness of the number of beats in each bar in a tune,
+as defined by the time signature. For example, in a 4/4 time
+signature, there are 4 beats per bar and the quarter note re-
+ceives one beat. To maintain grammatical correctness, the
+total number of beats in each bar must match the time sig-
+nature. Bar length accuracy is therefore a measure of how
+well TunesFormer can generate melodies that adhere to the
+specified time signature.
+In order to evaluate the bar length accuracy of Tunes-
+Former, we generated 100 tunes in each setting and com-
+pared them to 2854 tunes from the evaluation set. Upon
+manual examination, we found that almost all inaccuracies
+in the generated tunes were due to incomplete bars at the
+beginning and end of sections, which are still grammati-
+cally correct.
+We conducted independent samples t-tests
+and found that there were no statistically significant dif-
+ferences between the accuracy of the generated tunes in
+each setting and the evaluation set, with the exception of
+the SCP and BCP settings which had slightly lower accu-
+racy. However, this difference was not statistically signifi-
+cant as indicated by a p-value> 0.05. These results suggest
+that TunesFormer is able to generate grammatically correct
+tunes under all settings.
+To verify the effectiveness of the control codes for different
+placements, we conducted three separate experiments:
+• Bar number accuracy: we used the NS and NB control
+codes to specify the number of sections (1 section) and
+the number of bars (1-32 bars), while the EDS control
+codes were generated by TunesFormer itself. To de-
+termine the accuracy of the bar number, we compared
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1
+2
+3
+4
+5
+6
+7
+8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1
+2
+3
+4
+5
+6
+7
+8
+GP
+SP
+SCP
+BCP
+SBCP
+(b) Section Number Accuracy
+(e) Section Number Distribution
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+GP
+SP
+SCP
+BCP
+SBCP
+EDS
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1
+3
+5
+7
+9
+11
+13
+15
+17
+19
+21
+23
+25
+27
+29
+31
+GP
+SP
+SCP
+BCP
+SBCP
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+1
+3
+5
+7
+9
+11 13 15 17 19 21 23 25 27 29 31
+(a) Bar Number Accuracy
+(c) Edit Distance Similarity Comparison
+(d) Bar Number Distribution
+(f) Edit Distance Similarity Distribution
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Figure 4: Evaluating the effectiveness of control codes in TunesFormer for generating controlled musical forms. The green
+dotted line in (c) is the theoretical EDS values at each level.
+the actual number of bars generated to the NB con-
+trol codes. For each setting, we generated 100 tunes,
+resulting in a total of 5 placements × 32 bar numbers
+× 100 tunes = 16,000 tunes.
+• Section number accuracy: we used the NS control code
+to specify the number of sections (1-8 sections), while
+the NB and EDS control codes were generated by
+TunesFormer itself. To determine the accuracy of the
+section number, we compared the actual number of
+sections generated to the NS control code. For each
+setting, we generated 100 tunes, resulting in a total
+of 5 placements × 8 section numbers × 100 tunes =
+4000 tunes.
+• Edit distance similarity comparison: we used the NS
+and EDS control codes to specify the number of sec-
+tions (2 sections) and the similarity level (0-10 levels),
+while the NB control codes were generated by Tunes-
+Former itself. To compare the average edit distance
+similarity values at each EDS level, we compared them
+to the theoretical EDS values. For each setting, we
+generated 100 tunes, resulting in a total of 5 place-
+ments × 11 levels × 100 tunes = 5500 tunes.
+The results are provided in Fig. 4, which includes plots
+for bar number accuracy (Fig. 4a), section number accuracy
+(Fig. 4b), and edit distance similarity comparison (Fig. 4c).
+In Fig. 4a, it is shown that TunesFormer generally has high
+accuracy in generating the correct number of bars when the
+number specified is 17 or less for all placements. However,
+when the number of bars specified exceeds 17, there is a
+noticeable decrease in accuracy for the GP, SP, and SCP.
+This decrease in accuracy is likely due to the distribution of
+the number of bars in the dataset, as shown in Fig. 4d. A
+higher proportion of a certain number of bars corresponds
+to more of its NB control codes being learned by Tunes-
+Former, resulting in a more robust representation of those
+control codes. Both the BCP and SBCP, which insert NB
+control codes before each bar, have higher accuracy in gen-
+erating the correct number of bars regardless of the number
+specified.
+Fig. 4b demonstrates a similar trend in section number
+accuracy: TunesFormer can generate the correct number of
+sections almost 100% of the time for all placements, except
+the GP, SP, and BCP when the number of specified sec-
+tions is greater than 6. This is also due to the distribution
+of the number of sections in the dataset, as shown in Fig.
+4e. Both the SCP and SBCP, which insert NS control codes
+before each section, have higher accuracy in generating the
+correct number of sections regardless of the number speci-
+fied. However, because the distribution of section numbers
+is not as concentrated as bar numbers, not using the section
+countdown does not have as much of an impact on accuracy
+as the bar countdown.
+Fig. 4c presents the results of the EDS comparison. The
+ability of TunesFormer to generate sections with specified
+levels of similarity to the reference sections was evaluated by
+comparing the average EDS values of the generated tunes
+to the specified EDS levels. Overall, TunesFormer performs
+well at most EDS levels for all placements, with the av-
+erage EDS values consistently close to the specified levels.
+However, for EDS levels less than 2, all placements except
+for GP exhibit a statistically significant difference from the
+theoretical EDS values. This deviation from the expected
+results is not due to the distribution of EDS levels in the
+dataset, as levels 0 and 1 outnumber level 2. Rather, it is
+likely caused by the fact that when the EDS between two
+sections is at a low level, their bar lengths are often signif-
+icantly different. The model is more likely to capture this
+pattern when all control codes are placed at the beginning
+(GP). This suggests that the placement of control codes
+has a significant impact on the ability of TunesFormer to
+generate sections with a low level of EDS similarity.
+Overall, the SBCP performs well in both bar and section
+number accuracy, while the GP performs best in EDS.
+4.3
+Subjective Experiments
+In our subjective experiments, we sought to assess the qual-
+ity of generated tunes in various settings and evaluate the
+relevance of the EDS control code to human subjective per-
+
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Subjective Similarity Score
+Similarity Level
+Figure 5: Subjective similarity scores of selected tunes from
+the evaluation set.
+ception. We recruited music school students who majored
+in music as participants.
+We randomly selected 100 tunes with two sections from
+our evaluation set, with 10 tunes at each level of similarity
+ranging from 0 to 9. We excluded tunes with a similarity
+level of 10, as two identical sections would be the same in
+terms of subjective perception. For each tune, participants
+were asked to rate its similarity to the control code on a scale
+of 1 (completely dissimilar) to 5 (exact match), resulting in
+a total of 100 ratings.
+Participants were presented with
+sheet music for the selected tunes, with section boundaries
+marked, as well as audio of the tunes. They were asked to
+select the most appropriate description of the tune from five
+options based on their subjective perception:
+• Completely dissimilar: the two sections have no simi-
+larity in terms of melody, rhythm, or structure, or the
+two sections are too far apart in length.
+• Mildly dissimilar: the two sections do not share the
+motif or theme and are significantly different in the
+overall structure and melody.
+• Moderately similar: the two sections have a similar
+structure and some shared motifs, but there are still
+significant differences in terms of rhythm and pitch.
+• Highly similar: the two sections have a very similar
+structure and many shared motifs, but with noticeable
+differences in rhythm or pitch.
+• Exact match: the two sections are identical in every
+aspect, including melody, rhythm, and structure.
+As shown in Fig. 5, the subjective similarity scores ob-
+tained from our study participants were strongly correlated
+with the calculated similarity levels.
+The Pearson corre-
+lation coefficient for this relationship was 0.948, indicating
+that EDS can be used as a reliable measure of similarity in
+melody generation, as it is closely related to the subjective
+perception of similarity.
+Furthermore, Fig. 5 shows that when the EDS similarity
+level is below 4, the two sections are perceived as dissimilar
+by our participants, and vice versa. Based on these findings,
+we can conclude that setting the EDS control codes to a
+similarity level above 4 will result in the target section being
+perceived as similar to the reference section, while setting
+the control codes to a level below 4 will result in the target
+section being perceived as dissimilar to the reference section.
+To evaluate the quality of the tunes generated by Tunes-
+Former under different settings, we conducted a subjective
+evaluation in which participants rated 10 randomly selected
+tunes from the evaluation set and 10 tunes generated from
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+GP
+SP
+SCP
+BCP
+SBCP
+Eval Set
+Figure 6: Results of subjective ratings for generated tunes
+quality compared to the evaluation set.
+scratch for each placement on a scale ranging from 1 (poor
+quality) to 5 (excellent quality).
+The results, depicted in Fig. 6, show that the mean rat-
+ings of the generated tunes were generally similar across
+all placement settings, with values ranging from 3.03 to
+3.47. However, the insertion of control codes before each bar
+(BCP) resulted in a statistically significantly lower mean
+rating compared to the evaluation set, with a p-value <
+0.05. This suggests that the BCP may negatively impact
+the perceived quality of the generated tunes. In contrast,
+when the section countdown was introduced (SBCP), the
+ratings increased.
+This may be because the insertion of
+too many NB control codes can reduce the quality of the
+generation, while NS control codes enhance TunesFormer’s
+understanding of the relationships between sections while
+only slightly increasing the sequence length.
+The evalu-
+ation set had a higher mean rating compared to all other
+placements, although the difference was not statistically sig-
+nificant.
+These results demonstrate that TunesFormer is
+capable of generating tunes of comparable quality to those
+in the evaluation set under all settings (except BCP).
+5.
+CONCLUSIONS
+In this paper, we present TunesFormer, a melody gener-
+ation system that leverages the power of Transformer and
+is trained on a large dataset of 282,595 ABC notation tunes.
+By utilizing control codes, TunesFormer can generate melodies
+that match a given musical form. Our results indicate that
+TunesFormer can generate high-quality melodies that are
+comparable to those in the evaluation set.
+Through objective experiments, we demonstrate the ef-
+fectiveness of these control codes in achieving the desired
+number of sections and bars, as well as in achieving a spe-
+cific level of edit distance similarity. Subjective experiments
+also show that edit distance similarity is highly relevant to
+the human subjective perception of similarity.
+However,
+we also find that the insertion of control codes before ev-
+ery bar may negatively impact the perceived quality of the
+generated melodies. On the other hand, the introduction
+of a small number of NS control codes can enhance Tunes-
+Former’s understanding of the relationships between sec-
+tions and improve the quality of the generated melodies.
+These findings have important implications for the design
+and development of melody generation systems, and have
+the potential to facilitate the creation of more controlled
+and expressive musical forms.
+6.
+REFERENCES
+
+[1] K. Chen, W. Zhang, S. Dubnov, and G. Xia. The
+effect of explicit structure encoding of deep neural
+networks for symbolic music generation. CoRR,
+abs/1811.08380, 2018.
+[2] C. Chuan and D. Herremans. Modeling temporal
+tonal relations in polyphonic music through deep
+networks with a novel image-based representation. In
+S. A. McIlraith and K. Q. Weinberger, editors,
+Proceedings of the Thirty-Second AAAI Conference
+on Artificial Intelligence, (AAAI-18), the 30th
+innovative Applications of Artificial Intelligence
+(IAAI-18), and the 8th AAAI Symposium on
+Educational Advances in Artificial Intelligence
+(EAAI-18), New Orleans, Louisiana, USA, February
+2-7, 2018, pages 2159–2166. AAAI Press, 2018.
+[3] S. Dai, Z. Jin, C. Gomes, and R. B. Dannenberg.
+Controllable deep melody generation via hierarchical
+music structure representation. In J. H. Lee, A. Lerch,
+Z. Duan, J. Nam, P. Rao, P. van Kranenburg, and
+A. Srinivasamurthy, editors, Proceedings of the 22nd
+International Society for Music Information Retrieval
+Conference, ISMIR 2021, Online, November 7-12,
+2021, pages 143–150, 2021.
+[4] S. Di, Z. Jiang, S. Liu, Z. Wang, L. Zhu, Z. He,
+H. Liu, and S. Yan. Video background music
+generation with controllable music transformer. In
+H. T. Shen, Y. Zhuang, J. R. Smith, Y. Yang,
+P. C´esar, F. Metze, and B. Prabhakaran, editors, MM
+’21: ACM Multimedia Conference, Virtual Event,
+China, October 20 - 24, 2021, pages 2037–2045.
+ACM, 2021.
+[5] H. Dong, W. Hsiao, L. Yang, and Y. Yang. Musegan:
+Multi-track sequential generative adversarial networks
+for symbolic music generation and accompaniment. In
+S. A. McIlraith and K. Q. Weinberger, editors,
+Proceedings of the Thirty-Second AAAI Conference
+on Artificial Intelligence, (AAAI-18), the 30th
+innovative Applications of Artificial Intelligence
+(IAAI-18), and the 8th AAAI Symposium on
+Educational Advances in Artificial Intelligence
+(EAAI-18), New Orleans, Louisiana, USA, February
+2-7, 2018, pages 34–41. AAAI Press, 2018.
+[6] J. Ens and P. Pasquier. MMM : Exploring conditional
+multi-track music generation with the transformer.
+CoRR, abs/2008.06048, 2020.
+[7] C. Geerlings and A. Merono-Penuela. Interacting with
+gpt-2 to generate controlled and believable musical
+sequences in abc notation. In Proceedings of the 1st
+Workshop on NLP for Music and Audio
+(NLP4MusA), pages 49–53, 2020.
+[8] Z. Guo, D. Makris, and D. Herremans. Hierarchical
+recurrent neural networks for conditional melody
+generation with long-term structure. In International
+Joint Conference on Neural Networks, IJCNN 2021,
+Shenzhen, China, July 18-22, 2021, pages 1–8. IEEE,
+2021.
+[9] G. Hadjeres, F. Pachet, and F. Nielsen. Deepbach: a
+steerable model for bach chorales generation. In
+D. Precup and Y. W. Teh, editors, Proceedings of the
+34th International Conference on Machine Learning,
+ICML 2017, Sydney, NSW, Australia, 6-11 August
+2017, volume 70 of Proceedings of Machine Learning
+Research, pages 1362–1371. PMLR, 2017.
+[10] D. Herremans and E. Chew. Morpheus: generating
+structured music with constrained patterns and
+tension. CoRR, abs/1812.04832, 2018.
+[11] W. Hsiao, J. Liu, Y. Yeh, and Y. Yang. Compound
+word transformer: Learning to compose full-song
+music over dynamic directed hypergraphs. In
+Thirty-Fifth AAAI Conference on Artificial
+Intelligence, AAAI 2021, Thirty-Third Conference on
+Innovative Applications of Artificial Intelligence,
+IAAI 2021, The Eleventh Symposium on Educational
+Advances in Artificial Intelligence, EAAI 2021,
+Virtual Event, February 2-9, 2021, pages 178–186.
+AAAI Press, 2021.
+[12] C. A. Huang, A. Vaswani, J. Uszkoreit, I. Simon,
+C. Hawthorne, N. Shazeer, A. M. Dai, M. D.
+Hoffman, M. Dinculescu, and D. Eck. Music
+transformer: Generating music with long-term
+structure. In 7th International Conference on
+Learning Representations, ICLR 2019, New Orleans,
+LA, USA, May 6-9, 2019. OpenReview.net, 2019.
+[13] Y. Huang and Y. Yang. Pop music transformer:
+Beat-based modeling and generation of expressive pop
+piano compositions. In C. W. Chen, R. Cucchiara,
+X. Hua, G. Qi, E. Ricci, Z. Zhang, and
+R. Zimmermann, editors, MM ’20: The 28th ACM
+International Conference on Multimedia, Virtual
+Event / Seattle, WA, USA, October 12-16, 2020,
+pages 1180–1188. ACM, 2020.
+[14] V. I. Levenshtein et al. Binary codes capable of
+correcting deletions, insertions, and reversals. In
+Soviet physics doklady, volume 10, pages 707–710.
+Soviet Union, 1966.
+[15] I. Loshchilov and F. Hutter. Decoupled weight decay
+regularization. In 7th International Conference on
+Learning Representations, ICLR 2019, New Orleans,
+LA, USA, May 6-9, 2019. OpenReview.net, 2019.
+[16] P. Lu, X. Tan, B. Yu, T. Qin, S. Zhao, and T. Liu.
+Meloform: Generating melody with musical form
+based on expert systems and neural networks. CoRR,
+abs/2208.14345, 2022.
+[17] D. Naruse, T. Takahata, Y. Mukuta, and T. Harada.
+Pop music generation with controllable phrase
+lengths. In Proc. of the 23rd Int. Society for Music
+Information Retrieval Conf., Bengaluru, India, 2022.
+[18] S. Oore, I. Simon, S. Dieleman, D. Eck, and
+K. Simonyan. This time with feeling: learning
+expressive musical performance. Neural Comput.
+Appl., 32(4):955–967, 2020.
+[19] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei,
+I. Sutskever, et al. Language models are unsupervised
+multitask learners. OpenAI blog, 1(8):9, 2019.
+[20] B. L. Sturm, J. F. Santos, O. Ben-Tal, and
+I. Korshunova. Music transcription modelling and
+composition using deep learning. CoRR,
+abs/1604.08723, 2016.
+[21] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit,
+L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin.
+Attention is all you need. In I. Guyon, U. von
+Luxburg, S. Bengio, H. M. Wallach, R. Fergus,
+S. V. N. Vishwanathan, and R. Garnett, editors,
+Advances in Neural Information Processing Systems
+30: Annual Conference on Neural Information
+Processing Systems 2017, December 4-9, 2017, Long
+Beach, CA, USA, pages 5998–6008, 2017.
+[22] J. Wu, C. Hu, Y. Wang, X. Hu, and J. Zhu. A
+hierarchical recurrent neural network for symbolic
+melody generation. CoRR, abs/1712.05274, 2017.
+[23] J. Wu, X. Liu, X. Hu, and J. Zhu. Popmnet:
+Generating structured pop music melodies using
+neural networks. Artif. Intell., 286:103303, 2020.
+[24] L. Yang, S. Chou, and Y. Yang. Midinet: A
+
+convolutional generative adversarial network for
+symbolic-domain music generation. In S. J.
+Cunningham, Z. Duan, X. Hu, and D. Turnbull,
+editors, Proceedings of the 18th International Society
+for Music Information Retrieval Conference, ISMIR
+2017, Suzhou, China, October 23-27, 2017, pages
+324–331, 2017.
+[25] R. Yang, D. Wang, Z. Wang, T. Chen, J. Jiang, and
+G. Xia. Deep music analogy via latent representation
+disentanglement. In A. Flexer, G. Peeters, J. Urbano,
+and A. Volk, editors, Proceedings of the 20th
+International Society for Music Information Retrieval
+Conference, ISMIR 2019, Delft, The Netherlands,
+November 4-8, 2019, pages 596–603, 2019.
+[26] Y. Yu, F. Harsco¨et, S. Canales, G. R. M, S. Tang, and
+J. Jiang. Lyrics-conditioned neural melody generation.
+In Y. M. Ro, W. Cheng, J. Kim, W. Chu, P. Cui,
+J. Choi, M. Hu, and W. D. Neve, editors, MultiMedia
+Modeling - 26th International Conference, MMM
+2020, Daejeon, South Korea, January 5-8, 2020,
+Proceedings, Part II, volume 11962 of Lecture Notes
+in Computer Science, pages 709–714. Springer, 2020.
+[27] M. Zeng, X. Tan, R. Wang, Z. Ju, T. Qin, and T. Liu.
+Musicbert: Symbolic music understanding with
+large-scale pre-training. In C. Zong, F. Xia, W. Li,
+and R. Navigli, editors, Findings of the Association
+for Computational Linguistics: ACL/IJCNLP 2021,
+Online Event, August 1-6, 2021, volume
+ACL/IJCNLP 2021 of Findings of ACL, pages
+791–800. Association for Computational Linguistics,
+2021.
+[28] X. Zhang, J. Zhang, Y. Qiu, L. Wang, and J. Zhou.
+Structure-enhanced pop music generation via
+harmony-aware learning. In J. Magalh˜aes, A. D.
+Bimbo, S. Satoh, N. Sebe, X. Alameda-Pineda,
+Q. Jin, V. Oria, and L. Toni, editors, MM ’22: The
+30th ACM International Conference on Multimedia,
+Lisboa, Portugal, October 10 - 14, 2022, pages
+1204–1213. ACM, 2022.
+[29] Y. Zou, P. Zou, Y. Zhao, K. Zhang, R. Zhang, and
+X. Wang. Melons: generating melody with long-term
+structure using transformers and structure graph,
+2021.
+APPENDIX
+A.
+EXAMPLES OF VARIOUS PLACEMENTS
+[SECS_3]
+L:1/4
+M:4/4
+K:C
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 ||
+“C” e e“G” d d/d/ |“Am” c A“Em” G E | “F” F3/2 G/ A F |“C” E/E/G/G/ c G |
+e e“G” d d/d/ |“Am” c A“Em” G E |“F” F3/2 G/“G” A B | “C” d c c2 ||
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 |]
+Control Codes
+Tune Header
+Tune Body II
+[SIM_10][SIM_3][BARS_8]
+[BARS_8] 
+[SIM_3][BARS_8] 
+Control Codes
+Control Codes
+Control Codes
+Tune Body I
+Tune Body III
+Figure 7: Section-based Placement (SP): NB and EDS con-
+trol codes are inserted before each section of the tune.
+L:1/4
+M:4/4
+K:C
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 ||
+“C” e e“G” d d/d/ |“Am” c A“Em” G E | “F” F3/2 G/ A F |“C” E/E/G/G/ c G |
+e e“G” d d/d/ |“Am” c A“Em” G E |“F” F3/2 G/“G” A B | “C” d c c2 ||
+“C” E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/ F A |“G” A G“C” C2 |
+E3/2 D/“G” G3/2“C” E/ | c G E G |“G” D3/2 E/“D” F D |“G” A G“C” C2 |]
+Tune Header
+Tune Body II
+[SECS_1][SIM_10][SIM_3][BARS_8]
+[SECS_3][BARS_8] 
+[SECS_2][SIM_3][BARS_8] 
+Control Codes
+Control Codes
+Control Codes
+Tune Body I
+Tune Body III
+Figure 8: Section Countdown Placement (SCP): NS control
+codes are inserted before each section of the tune as a count-
+down of the number of sections remaining in the tune.
+[SECS_3]
+L:1/4
+M:4/4
+K:C
+[BARS_8]  “C” E3/2 D/“G” G3/2“C” E/ |
+[BARS_7]  c G E G |
+[BARS_6]  “G” D3/2 E/ F A |
+[BARS_5]  “G” A G“C” C2 |
+[BARS_4]  E3/2 D/“G” G3/2“C” E/ |
+[BARS_3]  c G E G |
+[BARS_2]  “G” D3/2 E/“D” F D |
+[BARS_1]  “G” A G“C” C2 ||
+Control Codes
+Tune Header
+[SIM_3]
+Control Codes
+Tune Body I
+&
+Control Codes
+Tune Body II
+&
+Control Codes
+[BARS_8]  “C” e e“G” d d/d/ |
+[BARS_7]  “Am” c A“Em” G E |
+[BARS_6]  “F” F3/2 G/ A F |
+[BARS_5]  “C” E/E/G/G/ c G |
+[BARS_4]  e e“G” d d/d/ |
+[BARS_3]  “Am” c A“Em” G E |
+[BARS_2]  “F” F3/2 G/“G” A B |
+[BARS_1]  “C” d c c2 ||
+[SIM_10][SIM_3]
+Control Codes
+[BARS_8]  “C” E3/2 D/“G” G3/2“C” E/ |
+[BARS_7]  c G E G |
+[BARS_6]  “G” D3/2 E/ F A |
+Tune Body III
+&
+Control Codes
+…
+Figure 9:
+Bar Countdown Placement (BCP): NB control
+codes are inserted before each bar of the tune as a count-
+down of the number of bars remained in the section.
+[SECS_3]
+L:1/4
+M:4/4
+K:C
+[BARS_8]  “C” E3/2 D/“G” G3/2“C” E/ |
+[BARS_7]  c G E G |
+[BARS_6]  “G” D3/2 E/ F A |
+[BARS_5]  “G” A G“C” C2 |
+[BARS_4]  E3/2 D/“G” G3/2“C” E/ |
+[BARS_3]  c G E G |
+[BARS_2]  “G” D3/2 E/“D” F D |
+[BARS_1]  “G” A G“C” C2 ||
+Control Codes
+Tune Header
+[SECS_2][SIM_3] 
+Control Codes
+Tune Body I
+&
+Control Codes
+Tune Body II
+&
+Control Codes
+[BARS_8]  “C” e e“G” d d/d/ |
+[BARS_7]  “Am” c A“Em” G E |
+[BARS_6]  “F” F3/2 G/ A F |
+[BARS_5]  “C” E/E/G/G/ c G |
+[BARS_4]  e e“G” d d/d/ |
+[BARS_3]  “Am” c A“Em” G E |
+[BARS_2]  “F” F3/2 G/“G” A B |
+[BARS_1]  “C” d c c2 ||
+[SECS_1][SIM_10][SIM_3]
+Control Codes
+[BARS_8]  “C” E3/2 D/“G” G3/2“C” E/ |
+[BARS_7]  c G E G |
+[BARS_6]  “G” D3/2 E/ F A |
+Tune Body III
+&
+Control Codes
+…
+Figure 10: Section & Bar Countdown Placement (SBCP):
+NS and NB control codes are inserted before each section
+and bar of the tune respectively, which allows for both the
+countdown of sections and bars to be presented in the piece.
+
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+(a) Hey Jude - NB Control Codes Only
+(b) Hey Jude - NB and NS Control Codes Only
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+(c) Hey Jude - All Control Codes
+(d) Hey Jude - Original
+Figure 11: Visualisations of the self-similarity matrices of “Hey Jude” with form A9A’9B12A"9. (a), (b) and (c) are generated
+by TunesFormer with A9 (the first 9 bars) from the original composition (d) as the prompt.
+B.
+CASE STUDY OF CONTROL CODES
+Fig. 11 presents visualizations of the self-similarity matrices of several melodies. Fig. 11a-c were generated by TunesFormer-
+GP using the first nine bars of the original tune “Hey Jude” (Fig. 11d) as the prompt with different control codes specified.
+Fig. 11a was generated using only the NB control codes, which indicate the number of bars in the melody. The resulting
+melody exhibits a less cohesive structure than the original tune, with fewer clear phrase boundaries and a less distinct musical
+form. This suggests that while the NB control codes are important in generating melodies with a certain number of bars,
+they are not sufficient in achieving the same level of structural cohesiveness as the original tune.
+Fig. 11b specifies the NB and NS control codes, which indicate the number of sections and the number of bars within
+each section, respectively.
+The EDS control codes, which indicate the relationships between sections, are generated by
+TunesFormer itself. This generation strategy is similar to the approach used in [17], but the resulting self-similarity matrix
+is significantly different from the original tune as TunesFormer is not specified in terms of the relationships between sections.
+Fig. 11c uses all control codes from the original tune to form the structure of the generated tune. It is clear that Fig. 11c
+is very close to Fig. 11d, demonstrating the importance of EDS control codes for constructing well-structured melodies. It
+should be noted that the use of the same musical form does not mean that the content of the original tune is also copied.
+Overall, Fig. 11a-c show that while the NB control codes are important in generating melodies with a certain number
+of bars, they are not sufficient in achieving the same level of structural cohesiveness as the original tune. The introduction
+of NS control codes improves the structure of the generated melodies, but the EDS control codes are crucial in achieving a
+melody with a similar structure to the original tune.
+

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+Time-optimal universal quantum gates on superconducting circuits
+Ze Li,1 Ming-Jie Liang,1 and Zheng-Yuan Xue1, 2, ∗
+1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physics
+and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
+2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics,
+South China Normal University, Guangzhou 510006, China
+(Dated: January 10, 2023)
+Decoherence is an inevitable problem when targeting to increase the fidelity of quantum gates, and thus
+is one of the main obstacles for large-scale quantum computation. The longer a gate operation is, the more
+decoherence-induced gate infidelity will be. Therefore, how to shorten the gate time becomes an urgent prob-
+lem to be solved. To this end, time-optimal control based on solving the quantum brachistochron equation
+is a straightforward solution. Here, based on time-optimal control, we propose a scheme to realize universal
+quantum gates on superconducting qubits, in a two-dimensional square lattice configuration, and the two-qubit
+gate fidelity can be higher than 99.7%. Meanwhile, we can further accelerate the z-axis gate considerably by
+adjusting the time-independent detuning. Finally, in order to reduce the influence of the dephasing error, deco-
+herence free subspace is also incorporated in our physical implementation. Therefore, we present a promising
+fast scheme for large-scale quantum computation.
+I.
+INTRODUCTION
+Due to the intrinsic superposition nature, quantum compu-
+tation can not only greatly shorten the calculation time of cer-
+tain problems, but also deal with some hard problems that are
+hard for classical computers. Recently, quantum computation
+has been implemented in a variety of systems [1–4], among
+which, the superconducting quantum circuits system is one
+of the most promising candidates [5–9]. However, besides
+the existence of operational errors, a quantum system will in-
+evitably couple to its surrounding environment, and thus lead
+to an increase in the distortion of quantum states or opera-
+tions. Therefore, how to achieve high fidelity quantum gates
+in quantum systems is an urgent problem to be solved.
+In the presence of noise, precise quantum control can be
+realized by the fastest possible evolution.
+Therefore, find-
+ing a shorter gate evolution path to shorten the gate-time
+has become an effective means to achieve high fidelity quan-
+tum gates. Time-optimal control (TOC) based on solving the
+quantum brachistochrone equation (QBE) [10] is an effective
+scheme to shorten the evolution time [11]. Recently, TOC
+based schemes for unitary operations have been proposed
+[11–17] and experimental demonstrated [18–23], where the
+needed time for specific quantum gate operations has been re-
+duced significantly. However, universal quantum control with
+analytical solution can only be possible for specific cases [12].
+Here, based on TOC, we propose a scheme to realize uni-
+versal quantum gates on superconducting transmon qubits, ar-
+ranged in a two-dimensional (2D) square lattice configuration,
+which is capable for large-scale universal quantum computa-
+tion. In our scheme, controlling the time-dependent frequency
+of the qubits, we can achieve the tunable coupling between
+two transmon qubits [24, 25].
+Meanwhile, we can further
+shorten the evolution time of the Z-axis gate by adjusting the
+time-independent detuning. Furthermore, to eliminate the ef-
+∗ [email protected]
+fect of dephasing, which is another important factor affecting
+the quantum gate fidelity, decoherence-free subspaces (DFS)
+encoding [26–28] has been incorporated and the robustness of
+our gates with respect to the decoherence is presented. There-
+fore, our work realized high fidelity universal quantum gate
+on superconducting circuits, which is a promising alternation
+for future large-scale quantum computation.
+II.
+THE GENERAL THEORY
+For a general two-level system, denoted by {|0⟩
+=
+(1, 0)† , |1⟩ = (0, 1)†}, assuming ℏ = 1 hereafter, when under
+the driving of an external field, its general interaction Hamil-
+tonian is
+H(t) = 1
+2
+�
+δ(t)
+Ω(t)e−iφ(t)
+Ω(t)eiφ(t)
+−δ(t)
+�
+,
+(1)
+where Ω(t) and φ(t) is the time-dependent coupling strength
+and phase of the driving field, δ(t) is the time-dependent
+detuning between the qubit frequency and the driving field
+frequency.
+Assuming there are two mutually orthogo-
+nal evolution states |Ψ±(t)⟩ that satisfy the time-dependent
+Schr¨odinger equation of Hamiltonian in Eq. (1). The evolu-
+tion operator can be written as
+U(t) = Tei
+�
+H(t)dt
+= |Ψ+(t)⟩ ⟨Ψ+(0)| + |Ψ−(t)⟩ ⟨Ψ−(0)| ,
+(2)
+where T is the time-ordering operator.
+In order to con-
+struct a particular evolution operator, we need to define a set
+of auxiliary basis vectors |ψ±(t)⟩ = e−iγ±(t) |Ψ±(t)⟩ with
+γ±(0) = 0 and γ+(t) = −γ−(t), which satisfy the boundary
+condition of |ψ±(τ)⟩ = |ψ±(0)⟩ = |Ψ±(0)⟩. We select a pair
+of dressed states
+|ψ+(t)⟩ = cos χ(t)
+2 |0⟩ + sin χ(t)
+2 eiξ(t)|1⟩,
+|ψ−(t)⟩ = sin χ(t)
+2 e−iξ(t)|0⟩ − cos χ(t)
+2 |1⟩,
+(3)
+arXiv:2301.03334v1  [quant-ph]  9 Jan 2023
+
+2
+as a set of auxiliary basis vectors, which are the eigenstates of
+the Lewis-Riesenfeld invariant [29] of Eq. (1),
+I(t) = µ
+2
+�
+cos χ(t)
+sin χ(t)e−iξ(t)
+sin χ(t)eiξ(t)
+− cos χ(t)
+�
+,
+(4)
+where µ is an arbitrary constant. The auxiliary basis vectors
+|ψ±(t)⟩ shows their evolutionary details on the Bloch sphere
+through the time-dependent parameters ξ(t) and χ(t).
+Then,
+by
+solving
+dynamic
+invariant
+equation
+of
+i∂I(t)/∂t − [H(t), I(t)] = 0, the parameter {ξ(t), χ(t)} of
+|ψ±(t)⟩ are decided by the parameters {Ω(t), φ(t), δ(t)} of
+the Hamiltonian in Eq. (1) as [29–31]
+˙ξ(t) = δ(t) − Ω(t) cot χ(t) cos[φ(t) − ξ(t)],
+˙χ(t) = Ω(t) sin[φ(t) − ξ(t)].
+(5)
+After an evolution time τ, by solving the Schr¨odinger equa-
+tion H(t) |Ψ±(t)⟩ = iℏ ∂
+∂t |Ψ±(t)⟩, we can obtain the overall
+phase as
+γ(τ) =
+� τ
+0
+2 ˙ξ(t) sin2 [χ(t)/2] − δ(t)
+2 cos χ(t)
+dt.
+(6)
+By setting χ(t) to be a constant, we can get a general evolution
+operator of the process as
+U(τ) = cos γ′
+�
+e−iξ−
+0
+0
+eiξ−
+�
++i sin γ′
+�
+cos χe−iξ−
+sin χe−iξ+
+sin χeiξ+
+− cos χeiξ−
+�
+,
+(7)
+where γ′ = γ(τ) + ξ− and ξ± = [ξ(τ) − ξ(0)]/2. We can get
+the appropriate parameters γ, χ and ξ(t) to construct target
+quantum gates by controlling the coupling strength Ω(t) and
+the phase φ(t).
+For realistic physical implementation, the interaction term
+in Eq. (1), Hc(t) = Ω(t) [cos φ(t)σx + sin φ(t)σy] /2, needs
+to satisfy the following two conditions.
+Firstly, the cou-
+pling strength is limited, here we set the maximum of Ω(t)
+as Ω = [Ω(t)]max, that is, we need to satisfy f1(Hc(t)) =
+[Tr(Hc(t))2 − Ω2/2]/2 = 0. Then, the form of the inter-
+action Hamiltonian is usually not arbitrary. Here, indepen-
+dent σz operator can’t be achieved, so it is necessary to satisfy
+f2(Hc(t)) = Tr(Hc(t)σz) = 0. Considering the these two
+conditions, based on the QBE of
+dF
+dt = −i[H(t), F],
+(8)
+where
+F=
+∂LC
+∂Hc(t) =
+∂
+��
+j=1,2 λjfj(Hc(t))
+�
+∂Hc(t)
+= λ1Hc(t) + λ2σz,
+(9)
+and Lagrange multiplier λj is defined as λ1 = 1/Ω and
+λ2 = −c/2, we can obtain φ(t) = φ0 + φ′(t), φ′(t) =
+� t
+0 [cΩ + δ] dt′ = ηt, where η and c are the constant. Fi-
+nally, in order to implement TOC based quantum operations,
+(a)
+(b)
+(c)
+1
+2
+3
+4
+FIG. 1. Illustration of our proposed scheme. (a) A scalable 2D square
+lattice consists of transmon qubits, where adjacent qubits are capac-
+itively coupled. Two physical qubits of the same color encoded as
+a DFS logical qubit. (b) The energy levels of two adjacent coupled
+qubits Ti and Tj, where different excitation subspaces can be used
+to implement different quantum gate. (c) Illustration of the evolution
+path of the TOC based scheme (red line) on the Bloch sphere, where
+χ is the angle between the direction of the auxiliary basis vector and
+the vertical axis, and ξ(τ) − ξ(0) is the horizontal angle shift of the
+auxiliary basis vector at a specific time τ.
+the coupling strength, detuning and phase need to satisfy con-
+ditions of ˙Ω = 0, ˙δ = 0 and ˙φ = η, respectively.
+Now, we calculated the gate operation time τ by solving the
+Eq. (5) and Eq. (6). The gate time of H, S, and T gates can
+be expressed as
+τH =
+√
+2π
+2Ω ,
+τS =
+π
+2(Ω2 + δ2)
+��
+16δ2 + 7Ω2 − 3δ
+�
+,
+τT =
+π
+4(Ω2 + δ2)
+��
+64δ2 + 15Ω2 − 7δ
+�
+,
+(10)
+which indicate that δ can be adjusted to further accelerate the
+S and T gates, due to the extra freedom on χ.
+III.
+PHYSICAL IMPLEMENTATION
+Here, in this section, we implemented our TOC-based
+scheme in the 2D square superconducting qubit lattice, as
+shown in Fig. 1(a), where the coupling strength between two
+adjacent transmons is fixed.
+A.
+Parametric Tunable Coupling
+In order to control single-logical-qubit units and two-
+logical-qubits units independently and construct the targeted
+quantum gates exactly, tunable interactions between any two
+transmon qubits should be implemented. For two adjacent
+transmon qubits Ti and Tj, the interaction Hamiltonian can be
+expressed as
+H0
+ij =
+�
+k=i,j
+[ωk|1⟩k ⟨1 |+ (2ωk − αk)| 2⟩k ⟨2|]
+(11)
++ gij(|10⟩ij⟨01| +
+√
+2|11⟩ij⟨02| +
+√
+2|20⟩ij⟨11| + H.c.),
+
+<10/:(0S
+100)
+0
+2.0
+0'4
+bobnjgfiou
+24.0
+2.0
+.0
+a.0100)
+0
+2.0
+0'4
+bobnjgfiou
+24.0
+2.0
+.0
+a.0s(t)-(0)1
+HH
+m
+HH
+2
+HH
+4
+H[0] L1/1(T)S(O)3
+where ωi,j and αi,j are the frequency and anharmonicity of
+the i,j-th transmon qubit Ti and Tj, respectively, |CD⟩ij =
+|C⟩i ⊗ |D⟩j. To achieve tunable coupling between Ti and Tj,
+we added a frequency modulation in the form of ωj = ωj0 +
+ϵj cos[νjt+φj(t)] for qubit Tj, with the driving frequency and
+phase being νj and φj(t), respectively. Meanwhile, frequency
+of Ti is fixed, which is written as ωi = ωi0 for the same layout
+as ωj. Moving into the interaction picture with respect to
+U I
+ij = U I
+i × U I
+j ,
+(12)
+with
+U I
+i = exp[−i(ωi0b+
+i bi − αi
+2 b+
+i b+
+i bibi)t],
+(13)
+U I
+j = exp{−i[ωj0t + Γj sin(vjt + φj(t))]b+
+j bj
+−αj
+2 b+
+j b+
+j bjbjt},
+(14)
+with bi,j = (|0⟩i,j⟨1| +
+√
+2|1⟩i,j⟨2|), Γj = ϵj/[νj + ˙φj(t)],
+and then using Jacobi-Anger identity, exp(−iΓ sin θ)
+=
+�
+n Jn(Γ) exp(−inθ), with Jn is the n−th Bessel function,
+the transformed Hamiltonian can be written as
+HI
+ij = gijei∆ij
++∞
+�
+n=−∞
+Jn(Γj)e[−in(νjt+φj(t))]{|10⟩ij⟨01|
++
+√
+2eiαjt|11⟩ij⟨02|
++
+√
+2e−iαit|20⟩ij⟨11|} + H.c.,
+(15)
+where ∆ij = −∆ji = ωi0 − ωj0 is the frequency difference
+between Ti and Tj. The energy level diagram is shown in
+Fig. 1(b), in which any two adjacent levels can be used to
+realize quantum computation. In addition, Γj can be tuned
+to achieve adjustable coupling between Ti and Tj, then we
+can select appropriate parameters of the modulation filed to
+construct target quantum gates.
+B.
+Single-Logical-Qubit gate based on TOC
+In order to decrease dephasing which is a type of the deco-
+herence, the DFS encoding can be incorporated in our scheme.
+Set two adjacent transmon qubits in Eq. (15) to be qubits
+T1 and T2 as shown in Fig. 1(a), where the logical qubits
+{|0⟩L, |1⟩L} can be encoded in their single excitation sub-
+space, i.e., S1 = Span{|0⟩L = |10⟩12, |1⟩L = |01⟩12}. In
+order to obtain a Hamiltonian in the form of Eq. (1) for con-
+structing universal quantum gates. It is natural to go into the
+rotating frame with respect to
+UA = exp
+�
+iδ
+2t(|0⟩L⟨0| − |1⟩L⟨1|)
+�
+,
+(16)
+the transformed Hamiltonian of Eq. (15) can be written as
+H′
+12 = δ
+2(|0⟩L⟨0| − |1⟩L⟨1|),
++ g12[K12ei(∆12−δ)t|0⟩L⟨1| + H.c.]
+(17)
+(a)
+(b)
+(c)
+(d)
+(f)
+(e)
+FIG. 2. The gate fidelity as a function of the qubits’ frequency dif-
+ferences ∆12 and the coupling strength g12. The numerical result of
+H, S and T gates are shown in (a), (c) and (e), respectively. The
+dynamics of the state population and fidelity of H, S and T gates are
+shown in (b), (d) and (f), respectively.
+where K12 = �+∞
+n=−∞ Jn (Γ2) exp{−in(ν2t+φ2)}. Choose
+the modulating frequency to meet ν2 = ∆12 − δ in Eq. (17),
+after the rotational wave approximation, we obtain the Hamil-
+tonian in the logical basis S1 as
+Heff
+12 = 1
+2
+�
+δ
+Ωe−iφ2
+Ωeiφ2
+−δ
+�
+,
+(18)
+where Ω = 2g12J1(Γ2). Therefore, according to the general
+theory in the last section, we can use TOC based scheme to
+construct arbitrary single-logical-qubit quantum gates. We set
+different parameters of the physical qubits for H, S and T
+gates. For H gate, which is correspond to γ′H = π
+2 , ξ+
+H =
+ξ−
+H = π and χH = π
+4 . For S and T gates, which is correspond
+to γ′S = γ′T = π, ξ−
+S = −3π/4 and ξ−
+T = −7π/8. Based on
+TOC and solving Eq. (15), here, ξ(t) is in the form of a linear
+function and χ is a constant, whose path in Bloch sphere is
+intuitive shown in Fig. 1(c).
+Next, we use the Lindblad master equation of
+˙ρ = −i [H(t), ρ] + r−
+2 A (b−) + rz
+2 A (bz) ,
+(19)
+to simulate the performance of our scheme for the single-
+logical-qubit gates, where ρ is density operator of the quan-
+tum system, A(b) = 2bρb+ − ρb+b − b+bρ is the Lindblad
+operator, r1
+− = r2
+− = r− and r1
+z = r2
+z = rz are the decay and
+dephasing rates of the two transmons qubits T1 and T2, re-
+spectively, with b− = �
+k=1,2(|0⟩k⟨1|+
+√
+2|1⟩k⟨2|) and bz =
+
+(sHM).
+J0
+J2
+SO
+300
+400
+sHM)
+200
+Q00JO
+J2
+SO
+300
+400
+(≤HM)
+200
+e00Tt
+0
+2.0
+0
+bobjsou
+2.0Tt
+0
+2.0
+5.0
+04
+bobjsou
+0.0
+8.0T.2.H
+0
+2.0
+0'4
+oitsluqo
+0.0
+8.0Tt
+0
+.0
+5.0
+04
+bobjsrlou
+0.0
+8.0Qe.0
+see.0
+te.0
+Qee.0
+8ee.0
+J
+avs
+10
+12
+SO
+500
+(sHM)
+400
+eo0o(sHM)sr
+J0
+J2
+SO
+300
+400
+(sHM)
+200
+Q004
+(a)
+(b)
+(c)
+(d)
+FIG. 3. (a) Comparative results for the gate robustness. Frequency drift error of TOC based (solid line) and S-L based gates (dot line). (b) The
+operation time τ2 in unit of 1/Ω with respect to the rotation angle γ(τ2) and δ2/Ω. (c) State fidelity as the function of the qubits frequency
+differences ∆24 and capacitive coupling strength g24. (d) Considering the adjacent interactions from T1 and T3, state population and fidelity
+dynamics of the CP-gate process with prescribed parameters as presented in the maintext.
+�
+k=1,2(|1⟩k⟨1| + 2|2⟩k⟨2|). Setting r = r− = rz = 2π × 4
+kHz [8], as shown in Fig. 2, taking g12 and ∆12 as variables,
+we numerically obtain the fidelity of the H, S and T gates,
+which are defined as F = Tr(U †U ′)/Tr(U †U), where U ′ rep-
+resents the evolution matrix under decoherence. For typical
+examples, we consider the parameters of the physical qubits as
+follow. The qubit frequency difference ∆12 = 2π×520 MHz,
+the capacitive coupling strength g12 = 2π×14.5 MHz, the de-
+tuning of H, S and T gates are modulated to δH = 2π×29.58
+MHz, δS = 2π × 25 MHz, δT = 2π × 15 MHz, Γ2 is set as
+1.5, and Ω = 2π × 16.18 MHz. With those settings, fidelities
+of the H, S, and T gates can reach FH=99.89%, FS=99.97%,
+and FT =99.97%, respectively.
+Next, to test the gate robustness of our scheme, we con-
+sider the frequency drift error of the two transmons qubits T1
+and T2, which is the main error source of the superconduct-
+ing qubit lattice and is in the form of ω1,β = ω1 + βΩ and
+ω2,β = ω2 − βΩ. Under the interaction picture, the interac-
+tion Hamiltonian with error can be expressed as
+H′
+12,β = HI
+12 + βΩ
+�
+b+
+1 b1 − b+
+2 b2
+�
+(20)
+As shown in Fig. 3(a), we found that under the effect of qubit
+frequency drift, our scheme exhibits a better resistance than
+the single-loop (S-L) based gate scheme [32].
+C.
+Two-Logical-Qubit gate based on TOC
+We next consider the implementation of the controlled
+phase gate (CP-gate), which is an important element for the
+universal quantum gates. As shown in Fig. 1(a), we consider
+a two-logical qubits unit with two pairs of transmon qubits, T1
+and T2, T3 and T4. Assuming |CDEF⟩ = |C⟩i ⊗|D⟩j ⊗|E⟩k ⊗
+|F⟩l, there exists a four-dimensional DFS S2 = Span{|00⟩L =
+|1010⟩, |01⟩L = |1001⟩, |10⟩L = |0110⟩, |11⟩L = |0101⟩}.
+In addition, an auxiliary state |a⟩ = |0200⟩ is needed to assist
+the implementation of the CP-gate. We consider the interac-
+tion between two adjacent physical qubits T2 and T4. Similar
+to the single-logical-qubit case, frequency of the T2 qubit ω2
+needs to be modulated as ω2 = ω20 + ϵ2 cos(ν2t + φ2) to
+achieve tunable coupling between qubits T2 and T4, in the
+subspace of {|a⟩, |11⟩L}, the interacting Hamiltonian can be
+written as
+H′
+42 = δ
+2(|a⟩⟨a| − |11⟩L⟨11|)
++ [
+√
+2g42K42ei(∆42+α2+δ)t|11⟩L⟨a| + H.c. ], (21)
+where K42 = �+∞
+n=−∞ Jn (Γ′
+2) exp [−in (ν2t + φ2)]. When
+we choose the resonance frequency ν2 = ∆24 − α2 − δ2, see
+Eq. (15), assume Ω = 2g42J1(Γ2), Γ′
+2 = 1.6 and φ = φ2 +π,
+then the Hamiltonian in Eq. (21) reduces to
+Heff
+42 = 1
+2
+�
+δ
+Ωe−iφ
+Ωeiφ
+−δ
+�
+,
+(22)
+where |a⟩ and |11⟩L form the set of orthogonal basis vectors,
+and φ2 = ηt is a linear function according to TOC solution.
+The evolution operator is shown in Eq. (7). When we set
+γ′ = π, we can obtain the evolution operator in the subspace
+S2 as
+U(τ2) =
+�
+�
+�
+�
+1 0 0
+0
+0 1 0
+0
+0 0 1
+0
+0 0 0 eiγ(τ2)
+�
+�
+�
+� ,
+(23)
+where γ(τ2) = ξ−
+2 + π. In this way, the CP gate can be ob-
+tained. The gate time can be solved as
+τ2 =
+2
+Ω2 + δ2 {δ[γ(τ2) − π]
++
+�
+π2δ2 − Ω2[γ(τ2)2 − 2πγ(τ2)]
+�
+.
+(24)
+Similar to the S and T gates, the detuning δ2 can also be used
+to further accelerate the gate time, as shown in Fig. 3(b),
+where we have set γ(τ2) = π/2, δ2 = 2π × 27 MHz and
+δ2/Ω = 2.3929.
+In order to properly evaluate the performance of the CP-
+gate, with the initial state |ψin⟩ = (|10⟩L + |11⟩L)/
+√
+2, the
+effect of frequency difference ∆24 and coupling strength g12
+on the gate fidelity is shown in Fig.
+3(c).
+When the pa-
+rameters are set as ∆24 = 2π × 600 MHz, g24 = 2π × 7
+MHz, α2 = 2π × 210 MHz and α4 = 2π × 230 MHz,
+the fidelity of CP-gate can reach 99.88%. Actually, the leak-
+age about two adjacent qubits T1 and T3 should be consid-
+ered as well, when we set ∆12 = ∆34 = 2π × 900 MHz,
+
+0
+-3
+0
+-J
+216
+0
+J
+3(sHM)
+1
+2
+JO
+J2
+300
+ce.0
+e.0
+400
+re.0
+(≤HM)
+200
+8e.0
+Q00
+Qe.0
+1000
+.0
+0
+bobjsIou
+[10)[
+.0
+-lo1)7
+-100)F
+VilsbiH-B
+1.0-
+0
+O'J
+8e.0
+I-2 T- - -
+28Q.0
+1-2 2--
+ --H2-1
+OOT T
+Qe.0
+OOT 2
+OOT H-
+Qe.0
+125
+α1 = 2π × 200 MHz, α3 = 2π × 220 MHz, the fidelity
+of CP gate can reach 99.72%. The state evolution process
+is shown in Fig. 3(d), Lindblad master equation in Eq. (19)
+should be considered as b− = �4
+k=1(|0⟩k⟨1| +
+√
+2|1⟩k⟨2|),
+bz = �4
+k=1(|1⟩k⟨1| + 2|2⟩k⟨2|), and the rates of decay and
+dephasing for each transmon qubit is set as r = rk
+− = rk
+z =
+2π × 4 kHz.
+IV.
+CONCLUSION
+In conclusion, we propose a scheme with TOC combined
+with DFS, and implement a universal quantum gate set on
+superconducting transmon qubits. For S, T and CP gates,
+by adjusting the detuning, the gate operations can be com-
+pleted in an extremely short time. Thus, our scheme provides
+a promising way towards the practical realization of fast quan-
+tum gates.
+ACKNOWLEDGMENTS
+This work was supported by the Key-Area Research and
+Development Program of GuangDong Province (Grant No.
+2018B030326001), the National Natural Science Foundation
+of China (Grant No. 12275090), and Guangdong Provincial
+Key Laboratory (Grant No. 2020B1212060066).
+[1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).
+[2] Q. A. Turchette, C. J. Hood and W. Lange, Phys. Rev. Lett. 75,
+4710 (1995).
+[3] G. K. Brennen, C. M. Caves and P. S. Jessen, Phys. Rev. Lett.
+82, 1060 (1999).
+[4] D. Jaksch, H. J. Briegel and J. I. Cirac, Phys. Rev. Lett. 82, 1975
+(1999).
+[5] Y. Makhlin, G. Sch¨on and A. Shrman, Rev. Mod. Phys. 73, 357
+(2001).
+[6] J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008).
+[7] J. Q. You and F. Nori, Nature 474, 589 (2011).
+[8] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013).
+[9] X. Gu, A. F. Kockum and A. Miranowicz, Phys. Rep. 718, 1102
+(2017).
+[10] H. J. Sussmann and J. C. Willems, IEEE Control Syst. Mag. 17,
+32 (1997).
+[11] A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Phys. Rev.
+Lett. 96, 060503 (2006).
+[12] A. Carlini and T. Koike, J. Phys. A. 46, 045307 (2013).
+[13] X.-T. Wang, M. Allegra, K. Jacobs, S. Lloyd, C. Lupo, and M.
+Mohseni, Phys. Rev. Lett. 114, 170501 (2015).
+[14] T. Chen and Z.-Y. Xue, Phys. Rev. Appl. 14, 064009 (2020).
+[15] T. Chen, P. Shen and Z.-Y. Xue, Phys. Rev. Appl. 14, 034038
+(2020).
+[16] B.-J. Liu, Z.-Y. Xue, and M.-H. Yung, arXiv:2001.05182.
+[17] G. O. Alves and E. Sj¨oqvist, Phys. Rev. A 106, 032406 (2022).
+[18] M. Lapert, Y. Zhang, M. Braun, S. J. Glaser, and D. Sugny,
+Phys. Rev. Lett. 104, 083001 (2010).
+[19] M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo,
+D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O.
+Morsch, Nat. Phys. 8, 147 (2012).
+[20] C. Avinadav, R. Fischer, P. London, and D. Gershoni, Phys.
+Rev. B 89, 245311 (2014).
+[21] J. Geng, Y. Wu, X.-T. Wang, K. Xu, F. Shi, Y. Xie, X. Rong,
+and J. Du, Phys. Rev. Lett. 117, 170501 (2016).
+[22] Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J.
+Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D.
+Lan, M.-H. Yung, and Y. Yu arXiv:2004.10364
+[23] Y. Dong, C. Feng, Y. Zheng, X.-D. Chen, G.-C. Guo, and F.-W.
+Sun, Phys. Rev. Res. 3, 043177 (2021).
+[24] X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. P.
+Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, Phys. Rev. Appl. 10,
+054009 (2018).
+[25] J. Chu, D. Li, X. Yang, S. Song, Z. Han, Z. Yang, Y. Dong, W.
+Zheng, Z. Wang, X. Yu, D. Lan, X. Tan, and Y. Yu, Phys. Rev.
+Appl. 13, 064012 (2020).
+[26] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997).
+[27] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997).
+[28] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett.
+81, 2594 (1998).
+[29] H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458
+(1969).
+[30] X. Chen, E. Torrontegui, and J. G. Muga, Phys. Rev. A. 83,
+062116 (2011).
+[31] A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J.
+Phys. 14, 093040 (2012).
+[32] T. Chen and Z.-Y. Xue, Phys. Rev. Appl. 10, 054051 (2018).
+

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+page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' China  2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' and Frontier Research Institute for Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' China  (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 2023)  Decoherence is an inevitable problem when targeting to increase the fidelity of quantum gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' and thus  is one of the main obstacles for large-scale quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The longer a gate operation is, the more  decoherence-induced gate infidelity will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Therefore, how to shorten the gate time becomes an urgent prob-  lem to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' To this end, time-optimal control based on solving the quantum brachistochron equation  is a straightforward solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Here, based on time-optimal control, we propose a scheme to realize universal  quantum gates on superconducting qubits, in a two-dimensional square lattice configuration, and the two-qubit  gate fidelity can be higher than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Meanwhile, we can further accelerate the z-axis gate considerably by  adjusting the time-independent detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Finally, in order to reduce the influence of the dephasing error, deco-  herence free subspace is also incorporated in our physical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Therefore, we present a promising  fast scheme for large-scale quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  INTRODUCTION  Due to the intrinsic superposition nature, quantum compu-  tation can not only greatly shorten the calculation time of cer-  tain problems, but also deal with some hard problems that are  hard for classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Recently, quantum computation  has been implemented in a variety of systems [1–4], among  which, the superconducting quantum circuits system is one  of the most promising candidates [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' However, besides  the existence of operational errors, a quantum system will in-  evitably couple to its surrounding environment, and thus lead  to an increase in the distortion of quantum states or opera-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Therefore, how to achieve high fidelity quantum gates  in quantum systems is an urgent problem to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  In the presence of noise, precise quantum control can be  realized by the fastest possible evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Therefore, find-  ing a shorter gate evolution path to shorten the gate-time  has become an effective means to achieve high fidelity quan-  tum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Time-optimal control (TOC) based on solving the  quantum brachistochrone equation (QBE) [10] is an effective  scheme to shorten the evolution time [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Recently, TOC  based schemes for unitary operations have been proposed  [11–17] and experimental demonstrated [18–23], where the  needed time for specific quantum gate operations has been re-  duced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' However, universal quantum control with  analytical solution can only be possible for specific cases [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Here, based on TOC, we propose a scheme to realize uni-  versal quantum gates on superconducting transmon qubits, ar-  ranged in a two-dimensional (2D) square lattice configuration,  which is capable for large-scale universal quantum computa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' In our scheme, controlling the time-dependent frequency  of the qubits, we can achieve the tunable coupling between  two transmon qubits [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Meanwhile, we can further  shorten the evolution time of the Z-axis gate by adjusting the  time-independent detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Furthermore, to eliminate the ef-  ∗ zyxue83@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='com  fect of dephasing, which is another important factor affecting  the quantum gate fidelity, decoherence-free subspaces (DFS)  encoding [26–28] has been incorporated and the robustness of  our gates with respect to the decoherence is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' There-  fore, our work realized high fidelity universal quantum gate  on superconducting circuits, which is a promising alternation  for future large-scale quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  THE GENERAL THEORY  For a general two-level system, denoted by {|0⟩  =  (1, 0)† , |1⟩ = (0, 1)†}, assuming ℏ = 1 hereafter, when under  the driving of an external field, its general interaction Hamil-  tonian is  H(t) = 1  2  �  δ(t)  Ω(t)e−iφ(t)  Ω(t)eiφ(t)  −δ(t)  �  ,  (1)  where Ω(t) and φ(t) is the time-dependent coupling strength  and phase of the driving field, δ(t) is the time-dependent  detuning between the qubit frequency and the driving field  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Assuming there are two mutually orthogo-  nal evolution states |Ψ±(t)⟩ that satisfy the time-dependent  Schr¨odinger equation of Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The evolu-  tion operator can be written as  U(t) = Tei  �  H(t)dt  = |Ψ+(t)⟩ ⟨Ψ+(0)| + |Ψ−(t)⟩ ⟨Ψ−(0)| ,  (2)  where T is the time-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  In order to con-  struct a particular evolution operator, we need to define a set  of auxiliary basis vectors |ψ±(t)⟩ = e−iγ±(t) |Ψ±(t)⟩ with  γ±(0) = 0 and γ+(t) = −γ−(t), which satisfy the boundary  condition of |ψ±(τ)⟩ = |ψ±(0)⟩ = |Ψ±(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' We select a pair  of dressed states  |ψ+(t)⟩ = cos χ(t)  2 |0⟩ + sin χ(t)  2 eiξ(t)|1⟩,  |ψ−(t)⟩ = sin χ(t)  2 e−iξ(t)|0⟩ − cos χ(t)  2 |1⟩,  (3)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='03334v1  [quant-ph]  9 Jan 2023  2  as a set of auxiliary basis vectors, which are the eigenstates of  the Lewis-Riesenfeld invariant [29] of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (1),  I(t) = µ  2  �  cos χ(t)  sin χ(t)e−iξ(t)  sin χ(t)eiξ(t)  − cos χ(t)  �  ,  (4)  where µ is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The auxiliary basis vectors  |ψ±(t)⟩ shows their evolutionary details on the Bloch sphere  through the time-dependent parameters ξ(t) and χ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Then,  by  solving  dynamic  invariant  equation  of  i∂I(t)/∂t − [H(t), I(t)] = 0, the parameter {ξ(t), χ(t)} of  |ψ±(t)⟩ are decided by the parameters {Ω(t), φ(t), δ(t)} of  the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (1) as [29–31]  ˙ξ(t) = δ(t) − Ω(t) cot χ(t) cos[φ(t) − ξ(t)],  ˙χ(t) = Ω(t) sin[φ(t) − ξ(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (5)  After an evolution time τ, by solving the Schr¨odinger equa-  tion H(t) |Ψ±(t)⟩ = iℏ ∂  ∂t |Ψ±(t)⟩, we can obtain the overall  phase as  γ(τ) =  � τ  0  2 ˙ξ(t) sin2 [χ(t)/2] − δ(t)  2 cos χ(t)  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (6)  By setting χ(t) to be a constant, we can get a general evolution  operator of the process as  U(τ) = cos γ′  �  e−iξ−  0  0  eiξ−  �  +i sin γ′  �  cos χe−iξ−  sin χe−iξ+  sin χeiξ+  − cos χeiξ−  �  ,  (7)  where γ′ = γ(τ) + ξ− and ξ± = [ξ(τ) − ξ(0)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' We can get  the appropriate parameters γ, χ and ξ(t) to construct target  quantum gates by controlling the coupling strength Ω(t) and  the phase φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  For realistic physical implementation, the interaction term  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (1), Hc(t) = Ω(t) [cos φ(t)σx + sin φ(t)σy] /2, needs  to satisfy the following two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Firstly, the cou-  pling strength is limited, here we set the maximum of Ω(t)  as Ω = [Ω(t)]max, that is, we need to satisfy f1(Hc(t)) =  [Tr(Hc(t))2 − Ω2/2]/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Then, the form of the inter-  action Hamiltonian is usually not arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Here, indepen-  dent σz operator can’t be achieved, so it is necessary to satisfy  f2(Hc(t)) = Tr(Hc(t)σz) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Considering the these two  conditions, based on the QBE of  dF  dt = −i[H(t), F],  (8)  where  F=  ∂LC  ∂Hc(t) =  ∂  ��  j=1,2 λjfj(Hc(t))  �  ∂Hc(t)  = λ1Hc(t) + λ2σz,  (9)  and Lagrange multiplier λj is defined as λ1 = 1/Ω and  λ2 = −c/2, we can obtain φ(t) = φ0 + φ′(t), φ′(t) =  � t  0 [cΩ + δ] dt′ = ηt, where η and c are the constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Fi-  nally, in order to implement TOC based quantum operations,  (a)  (b)  (c)  1  2  3  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Illustration of our proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (a) A scalable 2D square  lattice consists of transmon qubits, where adjacent qubits are capac-  itively coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Two physical qubits of the same color encoded as  a DFS logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (b) The energy levels of two adjacent coupled  qubits Ti and Tj, where different excitation subspaces can be used  to implement different quantum gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (c) Illustration of the evolution  path of the TOC based scheme (red line) on the Bloch sphere, where  χ is the angle between the direction of the auxiliary basis vector and  the vertical axis, and ξ(τ) − ξ(0) is the horizontal angle shift of the  auxiliary basis vector at a specific time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  the coupling strength, detuning and phase need to satisfy con-  ditions of ˙Ω = 0, ˙δ = 0 and ˙φ = η, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Now, we calculated the gate operation time τ by solving the  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The gate time of H, S, and T gates can  be expressed as  τH =  √  2π  2Ω ,  τS =  π  2(Ω2 + δ2)  ��  16δ2 + 7Ω2 − 3δ  �  ,  τT =  π  4(Ω2 + δ2)  ��  64δ2 + 15Ω2 − 7δ  �  ,  (10)  which indicate that δ can be adjusted to further accelerate the  S and T gates, due to the extra freedom on χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  PHYSICAL IMPLEMENTATION  Here, in this section, we implemented our TOC-based  scheme in the 2D square superconducting qubit lattice, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1(a), where the coupling strength between two  adjacent transmons is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Parametric Tunable Coupling  In order to control single-logical-qubit units and two-  logical-qubits units independently and construct the targeted  quantum gates exactly, tunable interactions between any two  transmon qubits should be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' For two adjacent  transmon qubits Ti and Tj, the interaction Hamiltonian can be  expressed as  H0  ij =  �  k=i,j  [ωk|1⟩k ⟨1 |+ (2ωk − αk)| 2⟩k ⟨2|]  (11)  + gij(|10⟩ij⟨01| +  √  2|11⟩ij⟨02| +  √  2|20⟩ij⟨11| + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='),  <10/:(0S  100)  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content="0  0'4  bobnjgfiou  24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0100)  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content="0  0'4  bobnjgfiou  24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0s(t)-(0)1  HH  m  HH  2  HH  4  H[0] L1/1(T)S(O)3  where ωi,j and αi,j are the frequency and anharmonicity of  the i,j-th transmon qubit Ti and Tj, respectively, |CD⟩ij =  |C⟩i ⊗ |D⟩j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' To achieve tunable coupling between Ti and Tj,  we added a frequency modulation in the form of ωj = ωj0 +  ϵj cos[νjt+φj(t)] for qubit Tj, with the driving frequency and  phase being νj and φj(t), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Meanwhile, frequency  of Ti is fixed, which is written as ωi = ωi0 for the same layout  as ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Moving into the interaction picture with respect to  U I  ij = U I  i × U I  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (12)  with  U I  i = exp[−i(ωi0b+  i bi − αi  2 b+  i b+  i bibi)t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (13)  U I  j = exp{−i[ωj0t + Γj sin(vjt + φj(t))]b+  j bj  −αj  2 b+  j b+  j bjbjt},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (14)  with bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='j = (|0⟩i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='j⟨1| +  √  2|1⟩i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='j⟨2|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Γj = ϵj/[νj + ˙φj(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  and then using Jacobi-Anger identity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' exp(−iΓ sin θ)  =  �  n Jn(Γ) exp(−inθ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' with Jn is the n−th Bessel function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  the transformed Hamiltonian can be written as  HI  ij = gijei∆ij  +∞  �  n=−∞  Jn(Γj)e[−in(νjt+φj(t))]{|10⟩ij⟨01|  +  √  2eiαjt|11⟩ij⟨02|  +  √  2e−iαit|20⟩ij⟨11|} + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=',  (15)  where ∆ij = −∆ji = ωi0 − ωj0 is the frequency difference  between Ti and Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The energy level diagram is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1(b), in which any two adjacent levels can be used to  realize quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' In addition, Γj can be tuned  to achieve adjustable coupling between Ti and Tj, then we  can select appropriate parameters of the modulation filed to  construct target quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Single-Logical-Qubit gate based on TOC  In order to decrease dephasing which is a type of the deco-  herence, the DFS encoding can be incorporated in our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Set two adjacent transmon qubits in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (15) to be qubits  T1 and T2 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1(a), where the logical qubits  {|0⟩L, |1⟩L} can be encoded in their single excitation sub-  space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=', S1 = Span{|0⟩L = |10⟩12, |1⟩L = |01⟩12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' In  order to obtain a Hamiltonian in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (1) for con-  structing universal quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' It is natural to go into the  rotating frame with respect to  UA = exp  �  iδ  2t(|0⟩L⟨0| − |1⟩L⟨1|)  �  ,  (16)  the transformed Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (15) can be written as  H′  12 = δ  2(|0⟩L⟨0| − |1⟩L⟨1|),  + g12[K12ei(∆12−δ)t|0⟩L⟨1| + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=']  (17)  (a)  (b)  (c)  (d)  (f)  (e)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The gate fidelity as a function of the qubits’ frequency dif-  ferences ∆12 and the coupling strength g12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The numerical result of  H, S and T gates are shown in (a), (c) and (e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The  dynamics of the state population and fidelity of H, S and T gates are  shown in (b), (d) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  where K12 = �+∞  n=−∞ Jn (Γ2) exp{−in(ν2t+φ2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Choose  the modulating frequency to meet ν2 = ∆12 − δ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (17),  after the rotational wave approximation, we obtain the Hamil-  tonian in the logical basis S1 as  Heff  12 = 1  2  �  δ  Ωe−iφ2  Ωeiφ2  −δ  �  ,  (18)  where Ω = 2g12J1(Γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Therefore, according to the general  theory in the last section, we can use TOC based scheme to  construct arbitrary single-logical-qubit quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' We set  different parameters of the physical qubits for H, S and T  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' For H gate, which is correspond to γ′H = π  2 , ξ+  H =  ξ−  H = π and χH = π  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' For S and T gates, which is correspond  to γ′S = γ′T = π, ξ−  S = −3π/4 and ξ−  T = −7π/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Based on  TOC and solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (15), here, ξ(t) is in the form of a linear  function and χ is a constant, whose path in Bloch sphere is  intuitive shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Next, we use the Lindblad master equation of  ˙ρ = −i [H(t), ρ] + r−  2 A (b−) + rz  2 A (bz) ,  (19)  to simulate the performance of our scheme for the single-  logical-qubit gates, where ρ is density operator of the quan-  tum system, A(b) = 2bρb+ − ρb+b − b+bρ is the Lindblad  operator, r1  − = r2  − = r− and r1  z = r2  z = rz are the decay and  dephasing rates of the two transmons qubits T1 and T2, re-  spectively, with b− = �  k=1,2(|0⟩k⟨1|+  √  2|1⟩k⟨2|) and bz =  (sHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  J0  J2  SO  300  400  sHM)  200  Q00JO  J2  SO  300  400  (≤HM)  200  e00Tt  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  0  bobjsou  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0Tt  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  04  bobjsou  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='H  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content="0  0'4  oitsluqo  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0Tt  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  04  bobjsrlou  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  Qee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  8ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  J  avs  10  12  SO  500  (sHM)  400  eo0o(sHM)sr  J0  J2  SO  300  400  (sHM)  200  Q004  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (a) Comparative results for the gate robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Frequency drift error of TOC based (solid line) and S-L based gates (dot line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (b) The  operation time τ2 in unit of 1/Ω with respect to the rotation angle γ(τ2) and δ2/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (c) State fidelity as the function of the qubits frequency  differences ∆24 and capacitive coupling strength g24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (d) Considering the adjacent interactions from T1 and T3, state population and fidelity  dynamics of the CP-gate process with prescribed parameters as presented in the maintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  �  k=1,2(|1⟩k⟨1| + 2|2⟩k⟨2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Setting r = r− = rz = 2π × 4  kHz [8], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 2, taking g12 and ∆12 as variables,  we numerically obtain the fidelity of the H, S and T gates,  which are defined as F = Tr(U †U ′)/Tr(U †U), where U ′ rep-  resents the evolution matrix under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' For typical  examples, we consider the parameters of the physical qubits as  follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The qubit frequency difference ∆12 = 2π×520 MHz,  the capacitive coupling strength g12 = 2π×14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='5 MHz, the de-  tuning of H, S and T gates are modulated to δH = 2π×29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='58  MHz, δS = 2π × 25 MHz, δT = 2π × 15 MHz, Γ2 is set as  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='5, and Ω = 2π × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='18 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' With those settings, fidelities  of the H, S, and T gates can reach FH=99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='89%, FS=99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='97%,  and FT =99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='97%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Next, to test the gate robustness of our scheme, we con-  sider the frequency drift error of the two transmons qubits T1  and T2, which is the main error source of the superconduct-  ing qubit lattice and is in the form of ω1,β = ω1 + βΩ and  ω2,β = ω2 − βΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Under the interaction picture, the interac-  tion Hamiltonian with error can be expressed as  H′  12,β = HI  12 + βΩ  �  b+  1 b1 − b+  2 b2  �  (20)  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 3(a), we found that under the effect of qubit  frequency drift, our scheme exhibits a better resistance than  the single-loop (S-L) based gate scheme [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Two-Logical-Qubit gate based on TOC  We next consider the implementation of the controlled  phase gate (CP-gate), which is an important element for the  universal quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 1(a), we consider  a two-logical qubits unit with two pairs of transmon qubits, T1  and T2, T3 and T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Assuming |CDEF⟩ = |C⟩i ⊗|D⟩j ⊗|E⟩k ⊗  |F⟩l, there exists a four-dimensional DFS S2 = Span{|00⟩L =  |1010⟩, |01⟩L = |1001⟩, |10⟩L = |0110⟩, |11⟩L = |0101⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  In addition, an auxiliary state |a⟩ = |0200⟩ is needed to assist  the implementation of the CP-gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' We consider the interac-  tion between two adjacent physical qubits T2 and T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Similar  to the single-logical-qubit case, frequency of the T2 qubit ω2  needs to be modulated as ω2 = ω20 + ϵ2 cos(ν2t + φ2) to  achieve tunable coupling between qubits T2 and T4, in the  subspace of {|a⟩, |11⟩L}, the interacting Hamiltonian can be  written as  H′  42 = δ  2(|a⟩⟨a| − |11⟩L⟨11|)  + [  √  2g42K42ei(∆42+α2+δ)t|11⟩L⟨a| + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' ], (21)  where K42 = �+∞  n=−∞ Jn (Γ′  2) exp [−in (ν2t + φ2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' When  we choose the resonance frequency ν2 = ∆24 − α2 − δ2, see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (15), assume Ω = 2g42J1(Γ2), Γ′  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='6 and φ = φ2 +π,  then the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (21) reduces to  Heff  42 = 1  2  �  δ  Ωe−iφ  Ωeiφ  −δ  �  ,  (22)  where |a⟩ and |11⟩L form the set of orthogonal basis vectors,  and φ2 = ηt is a linear function according to TOC solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  The evolution operator is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' When we set  γ′ = π, we can obtain the evolution operator in the subspace  S2 as  U(τ2) =  �  �  �  �  1 0 0  0  0 1 0  0  0 0 1  0  0 0 0 eiγ(τ2)  �  �  �  � ,  (23)  where γ(τ2) = ξ−  2 + π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' In this way, the CP gate can be ob-  tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The gate time can be solved as  τ2 =  2  Ω2 + δ2 {δ[γ(τ2) − π]  +  �  π2δ2 − Ω2[γ(τ2)2 − 2πγ(τ2)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  (24)  Similar to the S and T gates, the detuning δ2 can also be used  to further accelerate the gate time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 3(b),  where we have set γ(τ2) = π/2, δ2 = 2π × 27 MHz and  δ2/Ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='3929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  In order to properly evaluate the performance of the CP-  gate, with the initial state |ψin⟩ = (|10⟩L + |11⟩L)/  √  2, the  effect of frequency difference ∆24 and coupling strength g12  on the gate fidelity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  When the pa-  rameters are set as ∆24 = 2π × 600 MHz, g24 = 2π × 7  MHz, α2 = 2π × 210 MHz and α4 = 2π × 230 MHz,  the fidelity of CP-gate can reach 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='88%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Actually, the leak-  age about two adjacent qubits T1 and T3 should be consid-  ered as well, when we set ∆12 = ∆34 = 2π × 900 MHz,  0  3  0  J  216  0  J  3(sHM)  1  2  JO  J2  300  ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  400  re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  (≤HM)  200  8e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  Q00  Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  1000  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  0  bobjsIou  [10)[  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  lo1)7  100)F  VilsbiH-B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content="0-  0  O'J  8e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  I-2 T- - -  28Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  1-2 2--  --H2-1  OOT T  Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  OOT 2  OOT H-  Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='0  125  α1 = 2π × 200 MHz, α3 = 2π × 220 MHz, the fidelity  of CP gate can reach 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='72%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' The state evolution process  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 3(d), Lindblad master equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' (19)  should be considered as b− = �4  k=1(|0⟩k⟨1| +  √  2|1⟩k⟨2|),  bz = �4  k=1(|1⟩k⟨1| + 2|2⟩k⟨2|), and the rates of decay and  dephasing for each transmon qubit is set as r = rk  − = rk  z =  2π × 4 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  CONCLUSION  In conclusion, we propose a scheme with TOC combined  with DFS, and implement a universal quantum gate set on  superconducting transmon qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' For S, T and CP gates,  by adjusting the detuning, the gate operations can be com-  pleted in an extremely short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Thus, our scheme provides  a promising way towards the practical realization of fast quan-  tum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by the Key-Area Research and  Development Program of GuangDong Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  2018B030326001), the National Natural Science Foundation  of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 12275090), and Guangdong Provincial  Key Laboratory (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 2020B1212060066).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Cirac and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Zoller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 74, 4091 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [2] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Turchette, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Hood and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lange, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 75,  4710 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Brennen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Caves and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Jessen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  82, 1060 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Jaksch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Briegel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Cirac, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 82, 1975  (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Makhlin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Sch¨on and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Shrman, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 73, 357  (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Clarke and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wilhelm, Nature 453, 1031 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' You and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Nori, Nature 474, 589 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Devoret and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Schoelkopf, Science 339, 1169 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Gu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Kockum and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Miranowicz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 718, 1102  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Sussmann and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Willems, IEEE Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 17,  32 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
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+page_content=' Hosoya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Koike, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Okudaira, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 96, 060503 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Carlini and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Koike, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 46, 045307 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Allegra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Jacobs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lloyd, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lupo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Mohseni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 114, 170501 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xue, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 14, 064009 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Shen and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xue, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 14, 034038  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xue, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yung, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='05182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Alves and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Sj¨oqvist, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' A 106, 032406 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lapert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Braun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Glaser, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Sugny,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 104, 083001 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Bason, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Viteau, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Malossi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Huillery, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Arimondo,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Ciampini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Fazio, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Giovannetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Mannella, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Morsch, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 8, 147 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Avinadav, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' London, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Gershoni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' B 89, 245311 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Geng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rong,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Du, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 117, 170501 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [22] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Dong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
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+page_content=' Qiu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
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+page_content=' Xu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
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+page_content='  Lan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yung, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yu arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='10364  [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Guo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Sun, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 3, 043177 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Han, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Cai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Song, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xue, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Sun, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 10,  054009 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Song, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Dong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Tan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Yu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 13, 064012 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Duan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 79, 1953 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Zanardi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rasetti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 79, 3306 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lidar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chuang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Whaley, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  81, 2594 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Lewis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Riesenfeld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 10, 1458  (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [30] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Torrontegui, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Muga, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 83,  062116 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Ruschhaupt, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Alonso, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Muga, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 14, 093040 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='  [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Chen and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Xue, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}
+page_content=' 10, 054051 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf'}

CdE5T4oBgHgl3EQfTw8s/content/tmp_files/2301.05538v1.pdf.txt

@@ -0,0 +1,1387 @@
+PMFault: Faulting and Bricking Server CPUs
+through Management Interfaces
+Or: A Modern Example of Halt and Catch Fire
+Zitai Chen1 and David Oswald2
+1 University of Birmingham, Birmingham, UK, [email protected]
+2 University of Birmingham, Birmingham, UK, [email protected]
+Abstract. Apart from the actual CPU, modern server motherboards contain other
+auxiliary components, for example voltage regulators for power management. Those
+are connected to the CPU and the separate Baseboard Management Controller
+(BMC) via the I2C-based PMBus. In this paper, using the case study of the widely
+used Supermicro X11SSL motherboard, we show how remotely exploitable software
+weaknesses in the BMC (or other processors with PMBus access) can be used to access
+the PMBus and then perform hardware-based fault injection attacks on the main
+CPU. The underlying weaknesses include insecure firmware encryption and signing
+mechanisms, a lack of authentication for the firmware upgrade process and the IPMI
+KCS control interface, as well as the motherboard design (with the PMBus connected
+to the BMC and SMBus by default). First, we show that undervolting through the
+PMBus allows breaking the integrity guarantees of SGX enclaves, bypassing Intel’s
+countermeasures against previous undervolting attacks like Plundervolt/V0ltPwn.
+Second, we experimentally show that overvolting outside the specified range has the
+potential of permanently damaging Intel Xeon CPUs, rendering the server inoperable.
+We assess the impact of our findings on other server motherboards made by Supermicro
+and ASRock. Our attacks, dubbed PMFault, can be carried out by a privileged
+software adversary and do not require physical access to the server motherboard or
+knowledge of the BMC login credentials. We responsibly disclosed the issues reported
+in this paper to Supermicro and discuss possible countermeasures at different levels.
+To the best of our knowledge, the 12th generation of Supermicro motherboards, which
+was designed before we reported PMFault to Supermicro, is not vulnerable.
+Keywords: fault injection · software-based faults · Intel SGX · under/overvolting
+1
+Introduction
+In recent years, the security implications of software-exposed power and clock manage-
+ment features have received substantial attention by the research community. Several
+attacks including CLKSCREW [TSS17], Plundervolt [MOG+20], V0ltPwn [KFG+20],
+and VoltJockey [QWLQ19] showed that undervolting or overclocking from software can
+be used to inject faults (e.g., bitflips) into computations and break Trusted Execution
+Environments (TEEs) like Intel Software Guard Extensions (SGX) and ARM TrustZone.
+Subsequent attacks like VoltPillager [CVM+21] and the work by Buhren et al. [BJKS21]
+showed that similar attacks can be mounted with direct access to the computer hardware,
+physically connecting to the control interface of the Voltage Regulator (VR).
+In particular, Chen et al. targeted the Serial Voltage Identification (SVID) interface
+used by Intel CPUs to set the desired supply voltage. However, apart from SVID, many
+systems, in particular servers, support a second interface, the so-called Power Management
+Bus (PMBus), to control the Voltage Regulator Module (VRM). PMBus is an open
+arXiv:2301.05538v1  [cs.CR]  13 Jan 2023
+
+2
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+standard for digital power management [pmb] and has been adopted by more than 40
+companies. It is based on the Inter-Integrated Circuit (I2C) bus and offers monitoring
+features apart from voltage and current control.
+Another component usually presents on server motherboards is the Baseboard Manage-
+ment Controller (BMC). This chip, intended to remotely manage the server even if e.g.,
+the main CPU has crashed or is powered down, has connections to several buses and chips
+on the motherboard, including the I2C bus on which the VRM resides.
+Previous research on x86 platforms has focused on the software-hardware interface
+provided by the Central Processing Unit (CPU) itself and on the security within the
+perimeter of each individual component, e.g., the BMC [PGC18] or Intel Management
+Engine (Intel ME) [TW09, MIT17, GE17]. There is a lack of board-level security analysis
+that reviews the system and motherboard design and interactions between the different
+components: even if an individual part of the system is secure within its individual threat
+model, the combination of it with other parts can cause security risks. In our PMFault
+attacks, the privileged position of the BMC, combined with its large attack surface, makes
+it interesting from an adversary’s perspective to exploit vulnerabilities of the system via
+power management features.
+1.1
+Our Contribution
+Our main contributions in this paper are:
+PMBus-based under/overvolting against server platforms:
+We first analyse the VRM
+management interface at the hardware level. We discovered that the semi-standardised
+PMBus can be used to control the CPU voltage. Using the case study of a widely-used
+server motherboard, the Supermicro X11SSL-CF, we explore this attack surface and
+show that software vulnerabilities in the BMC (or another programmable chip connected
+to the PMBus) can have severe consequences for the security and safety of the server
+platform. To determine if the vulnerabilities can affect other server motherboards, we
+also investigated the PMBus connections and usage on an ASRock E3C246D4I-2T and a
+Supermicro X12DPi-NT6.
+PMBus access through BMC exploits: We then study the BMC firmware and—based
+on prior work in [Ecl18, Rak15, Nie20]—found that it can indeed be exploited to send
+arbitrary PMBus commands to control the voltage of the CPU. More precisely, several
+software vulnerabilities in the BMC, including incorrect firmware encryption and signing
+mechanisms, a lack of authentication for firmware upgrades and control interfaces, an
+attacker can manipulate the CPU voltage remotely because the PMBus is connected to
+the BMC and the System Management Bus (SMBus) by default.
+PMBus-based undervolting against SGX enclaves: With this, we observed the same
+faults as with Plundervolt/V0ltPwn (CVE-2019-11157), including for code running inside
+an SGX enclave. As the BMC has an independent, external flash chip for its firmware,
+SGX attestation currently does not have the ability to verify its status. Crucially, because
+the software voltage-control interface in Model Specific Register (MSR) 0x150 is not used,
+Intel’s fix for CVE-2019-11157 does not address this attack.
+Permanent denial-of-service through overvolting:
+We also discovered a novel overvolting
+attack: by sending a certain sequence of PMBus commands, we can set the CPU voltage
+outside the specification (as high as 2.84 V) and permanently brick the Xeon CPU used in
+our experiments.
+Countermeasures and mitigations: Finally, we develop the PMBusDetect tool for
+detecting if the VRM is connected to the PMBus, and then discuss countermeasures and
+challenges in securing server platforms. Importantly, we point out that TEEs like SGX
+must not only rely on the security of the CPU itself, but also of that of management
+components the hardware design of the platform.
+
+Zitai Chen and David Oswald
+3
+The details of our experiments and source code can be found at: https://github.com/
+zt-chen/PMFault. CVE number CVE-2022-43309 has been reserved for PMFault.
+1.2
+Adversary Model
+In this paper, we assume a privileged software attacker, i.e., who has obtained root on the
+host CPU. This is the standard adversary model in the case of TEEs like SGX, and is also
+realistic in the case of overvolting to permanently destroy the CPU, which could be e.g.,
+exploited by ransomware with root rights. Notably, our attacks do not require physical
+access (for additional hardware to be added to the system) and can thus be conducted
+remotely e.g., over SSH.
+1.3
+Responsible Disclosure
+We have responsibly disclosed our findings to Intel and Supermicro in April 2022. We
+discussed the details of our methods in several calls with Supermicro, and they acknowledge
+the existence of the issue and are looking into deploying fixes for their 11th generation
+products like the Supermicro X11SSL-CF. Supermicro highlighted that the attacks do
+not replicate on their 12th generation, which e.g., include secure boot and update for
+the BMC and filtering on PMBus commands. Both of these features break the attack
+chains described in the paper. Intel considered the issue in the context of their own server
+motherboards and did not find them vulnerable. Intel did not comment on the impact on
+SGX.
+1.4
+Related Work
+Since Boneh et al.’s seminal work on fault injection [BDL97], the research community
+has devoted substantial efforts to investigating fault attacks and developing according
+countermeasures (cf. e.g., [BECN+06] for an overview).
+Software-based Fault Injection
+Often, fault injection was considered a technique limited
+to attacks with physical access to the target. However, with the discovery of the Rowhammer
+effect [KDK+14], it was shown that faults can also be injected from software (through
+specific memory access patterns in the case of Rowhammer). Then, in 2017, Tang et al.
+showed that the clock management features of ARM processors can be exploited to inject
+faults into computations shielded inside a TEE like ARM TrustZone [TSS17]. Similarly,
+Plundervolt, V0ltPwn, and VoltJockey [MOG+20, KFG+20, QWLQ19] (all tracked via
+CVE-2019-11157) use the software-exposed voltage control MSR in Intel processors to
+break the integrity guarantees of SGX enclaves. In response, Intel deployed a microcode
+update that disables the undervolting interface in MSR 0x150 and allows remote parties to
+verify correct configuration through SGX’s remote attestation. Thus, purely software-based
+undervolting attacks against Intel processors were considered no longer possible.
+Hardware-based Fault Injection on TEEs
+The second generation of undervolting attacks
+on TEEs like SGX and AMD Secure Encrypted Virtualization (SEV) require physical
+access to the target motherboard. In the case of VoltPillager [CVM+21], the adversary
+attaches two wires to the data and clock lines of the SVID bus and can then control the
+VRM external to the CPU, enabling undervolting even if Intel’s microcode fixes for CVE-
+2019-11157 are installed. For AMD SEV, the adversary does not glitch the actual CPU,
+but the separate security co-processor, the AMD Secure Processor (SP) [BJKS21]. The
+adversary then proceeds to upload custom firmware to the SP to leak memory encryption
+keys and also endorsement secrets, which ultimately enable attacks without permanent
+physical access.
+
+4
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+Security of servers and BMCs
+Independent of hardware-based attacks, the security of
+server platforms has received attention in the research community and wider society. In
+2018, Bloomberg published a—since widely disproven—article that incorrectly claimed
+the inclusion of small backdoor chips on Supermicro motherboards [RR18]. However, at
+the same time, researchers at Eclypsium showed that it is indeed possible to maliciously
+manipulate the BMC firmware of Supermicro motherboards from 8th to 11th genera-
+tion [Ecl18], without the need to add a hardware implant. They also demonstrated how
+flashing corrupted BMC firmware can “brick” the server system by preventing it to boot.
+Niewöhner [Nie20] subsequently published a tool to exploit the (weak) firmware en-
+cryption of Supermicro BMCs. Other work, for example by Waisman et al. [WS18] and
+Périgaud et al. [PGC18], has shown that software weaknesses in BMCs are not limited to
+Supermicro motherboards, but also applied to Dell, HP, and Lenovo systems.
+However, the implications of direct access to the PMBus from a compromised BMC
+have not been deeply studied to our knowledge.
+1.5
+Paper Outline
+The remainder of this paper is structured as follows: in Section 2, we review the PMBus
+protocol and analyse its specific implementation and usage on Supermicro motherboards.
+Then, in Section 3, we describe Supermicro’s BMC implementation and methods to modify
+the firmware. In Section 4, we experimentally investigate how a compromised BMC can
+interact with the VRM through the PMBus. We then use this to develop over/undervolting
+attacks in Section 5, before concluding in Section 7.
+2
+Analysis of Power Management Bus
+We started our work by analysing how the PMBus is used on practical server mother-
+boards. PMBus is an interface that is used to control the VRM, supplying the power
+to the CPU. The most recent public available specification is version 1.3 [pmb]. This
+specification standardises the physical interface, packet structure, and command set of
+the PMBus. However, some commands are left as “manufacturer specified”, so that each
+VRM manufacturer can have a slightly different implementation of the command set. This
+matches what we found during our investigation of the MP2955 VRM on the Supermicro
+X11SSL-CF platform described in the following.
+2.1
+Experimental Setup
+We carried out initial experiments with an Intel Xeon E3-1220 v6 (CPU family: 6, model:
+158, microcode version: 0xea) on a Supermicro X11SSL-CF Rev 1.01 motherboard (BMC
+microcontroller ASPEED AST2400, firmware revision 01.63, BIOS version: 2.4).We used 64-
+bit Ubuntu 18.04.3 LTS with a stock 5.4.0-107-generic kernel, Intel SGX driver V2.11.0, and
+Intel SGX-SDK V2.15.100.3. We refer to this system as E3-1220V6-X11SSL-CF throughout
+the paper. An overview of the server motherboard representative for Supermicro’s 11th
+generation products is shown in Figure 1. The target of the PMFault attack is an Intel CPU
+with SGX technology. As mentioned, our actual attacks do not require additional hardware
+or physical access to the system, though we soldered some wires to the motherboard during
+the analysis phase.
+On Intel platforms, the voltage of the CPU is controlled by an external VRM Integrated
+Circuit (IC). The CPU connects to the VRM via the SVID bus to control the voltage
+supplied by it. This interface for CPU voltage control is present on all desktop and server
+motherboards.
+
+Zitai Chen and David Oswald
+5
+CPU
+Voltage
+Regulator
+(VRM)
+Board
+Management
+Controller (BMC)
+SVID
+Other I2C
+Devices
+SMBus/I2C Bus
+Ethernet 0
+KCS
+PMBus
+Management
+Ethernet
+BMC Flash
+Chip
+Figure 1: Overview of the connections on the server motherboard.
+However, server VRMs—including the Supermicro X11SSL-CF—often have an ad-
+ditional I2C-based communication interface called PMBus. This interface allows e.g.,
+overclocking or fine-tuning of the CPU voltage. One of the crucial steps in the PMFault
+is to get access to this interface and understand the communication protocol, so that we
+gain full control of the CPU voltage.
+One of the design issues we found on our server motherboard is that the PMBus can
+be directly connected to the more general SMBus. There are various components on the
+system on that bus, including the CPU, BMC, and other I2C devices. A compromise of
+any of these components leads to the takeover of PMBus and thus control of the CPU
+voltage.
+In this paper, we use the BMC as the starting point of the attack, as it commonly
+exists on server platforms. In order to analyse the attack surface of the BMC, we further
+investigated its connection and hardware design on the Supermicro X11SSL-CF. First, we
+found that its firmware is stored in a Serial Peripheral Interface (SPI) flash chip, separate
+from the BIOS flash. We also found there are two Ethernet ports on the system for
+communication with the BMC: one is called “Management Ethernet” and is dedicated
+for server management. The other port can be shared between CPU and BMC so that
+devices on this Ethernet port can communicate with both CPU and BMC. Finally, the
+BMC also has a Keyboard Controller Style (KCS) interface that enables direct access from
+the Operating System (OS) running on the CPU. These management interfaces open a
+large attack surface on the BMC, and make remote attacks possible.
+2.2
+Protocol Structure
+To be able to eavesdrop and forge PMBus commands, knowledge of the protocol structure
+shown in Figure 2 is necessary. The PMBus is an I2C-based protocol (with clock speed
+of 100 kHz–1 MHz and an open-drain data pin) and uses a master-slave communication
+mechanism. The master device can query or change the setting of the slave device. Each
+slave device is assigned a unique 7-bit device address.
+The master device first sends a starting bit to initiate a transmission. During transmis-
+sion, every group of 9 bits forms a segment, with the 9th bit indicating ACK (0) or NACK
+(1) for every 8 bits received. The starting bit and the (N)ACK mechanism are handled at
+hardware level and do not need to be handled manually.
+The first segment is always sent by the master. The first 7 bits are the address of the
+target slave, and the 8th bit indicates whether this transmission is a read (1) or write (0).
+
+6
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+S
+R
+/
+W
+A
+C
+K
+A
+C
+K
+A
+C
+K
+Device
+Address
+Command or
+Register Addr
+Data
+A
+C
+K
+. . . . . .
+0
+1
+8
+9
+10
+18
+19
+27
+28
+Master Device -> Slave Device Control bit
+Data bits
+Slave Device -> Master Device
+Figure 2: PMBus protocol structure
+The second segment is the register address to operate on. In the PMBus specification, this
+segment is called the PMBus command. The segments after the second one contain the
+data read from or written to the register.
+Interaction between PMBus and SVID
+Although the functionality of the PMBus pro-
+tocol is similar to SVID, they have different specifications for the digital signal interface
+and command sets. A VRM can have both SVID and PMBus interfaces, with the SVID
+interface directly connected to the CPU and the PMBus interface connected to the SMBus.
+Both interfaces can be used to control the voltage of the CPU, and some implementations
+of the PMBus specification also have commands to override the voltages set through the
+SVID interface.
+2.3
+PMBus Commands
+For an adversary to communicate with the VRM and e.g., configure voltage levels, they
+also need to know the specific PMBus commands. As mentioned, the PMBus specification
+allows manufacturers to have custom implementations of PMBus commands. The E3-
+1220V6-X11SSL-CF motherboard features an Monolithic Power MP2955 voltage regulator.
+To understand the PMBus implementation of this VRM, we first started looking for its
+datasheet, but unfortunately, found that it is not publicly available. However, on the
+Monolithic Power website1, we found the datasheet of an alternative VRM (MP2965) [Mon].
+As both chips are manufactured by the same company, we used this datasheet as a reference
+and starting point to discover the available PMBus commands by analysing the PMBus
+traffic on the Supermicro X11SSL-CF.
+We found the relevant PMBus commands by reading and analysing the response
+(ACK or NACK) of the registers, and validating found commands according to the PMBus
+specification and the MP2965 datasheet : Table 1 gives the command name, command code,
+and description of each commands. The first three commands in the table are implemented
+according to the PMBus 1.3 specification [pmb], while the rest are manufacturer-specific.
+Table 1: Discovered PMBus commands on E3-1220V6-X11SSL-CF.
+Command name
+Command code
+Usage
+CMD_PAGE
+0x00
+Switch between different voltage rails
+CMD_OPERATION
+0x01
+PMBus override
+VOUT_COMMAND
+0x21
+Output voltage settings
+READ_VOUT
+0x8B
+Voltage reading from sensor
+MFR_VR_CONFIG
+0xE4
+Enable overclock mode
+MFR_OCP_TOTAL_SET
+0xEE
+Over-current protection configuration
+1https://www.monolithicpower.com/
+
+Zitai Chen and David Oswald
+7
+With CMD_OPERATION, we can configure the operation mode of the VRM. By setting
+bit 1 of this register, we can enable the PMBus override mode. In this mode, the voltage
+configured in the VOUT_COMMAND register will override the voltage configuration from the
+SVID bus.
+Another command that is useful for PMFault is READ_VOUT, as it allows
+us to read the current voltage of the CPU and establish a baseline for undervolting.
+The MFR_VR_CONFIG register is manufacturer-specific. By setting bit 3 or bit 10 and
+configuring CMD_OPERATION, we could enable the tracking or fixed voltage overclocking
+mode, respectively.
+Bit 8 VID_STEP_SEL of MFR_VR_CONFIG also allow us to use an
+alternative mode of SVID. In this mode, the VRM uses 10 mV Voltage Identifier (VID)
+steps instead of the default of 5 mV. This makes overvolting up to 3 V possible, which is well
+beyond the operating voltage range of the E3-1220 V6 Intel CPU, with a maximum voltage
+of 1.52 V [Cor18]. We also discovered that the VRM has an Over Current Protection (OCP)
+circuit, which can be configured or disabled by another manufacturer-specific register
+(MFR_OCP_TOTAL_SET). Some VRM also support multiple voltage output rails. CMD_PAGE
+command is used to select the target rail to send the commands to.
+With these discovered commands, we can now control the CPU voltage through the
+PMBus. In Section 4.1, we detail how this interface is used as part of attack chains for
+undervolting and overvolting attacks.
+2.4
+Jumper Settings
+On the Supermicro X11SSL-CF motherboard, there are several jumpers that control
+different functionalities, including the connection of the VRM to other parts of the system.
+We kept all jumpers in the default status as delivered by the vendor. To avoid confusion,
+we still list the jumper settings in Table 2. During inspection of the jumper settings, we
+discovered that the SMBDAT_VRM and SMBCLK_VRM jumpers are neither mentioned in the user
+manual [Supb] nor in the quick reference guide [Supa]. Using an oscilloscope while sending
+PMBus commands, we found that these two jumpers can be used for (dis)connecting
+the VRM from/to the PMBus. The experiments and attacks described in this paper are
+conducted under the “connected” setting of both jumpers, which according to Supermicro
+is the default.
+We also found server motherboard without such jumpers, e.g., Supermicro X11SPG-TF
+and ASRock E3C246D4I-2T. For those, the VRM is always connected to the BMC. We
+detail our finding on other motherboards in Section 6. It is worth mentioning that to the
+best of our knowledge, SGX attestation does not have the functionality to include the
+configuration of these (external) jumpers.
+Table 2: Jumper settings on Supermicro X11SSL-CF.
+Jumper name
+Description
+JPME2
+Manufacturer mode normal (Default)
+JPB1
+BMC enabled (Default)
+SMBDAT_VRM
+Connect VRM data line to PMBus
+SMBCLK_VRM
+Connect VRM clock line to PMBus
+3
+Supermicro’s BMC and Server Management Interface
+Having understood the basic PMBus protocol and commands, we next look at different
+ways to gain access to the PMBus and send commands to the VRM. To achieve that, an
+attacker needs access to the SMBus. As described in Section 2.1, on E3-1220V6-X11SSL-
+CF, one of the devices on the SMBus is the ASPEED AST2400 BMC controller. In this
+
+8
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+section, we introduce the functionalities and vulnerabilities in these management interfaces
+that allow us to achieve our main goal—to take control of the SMBus.
+During the initial investigation of the BMC, we found there are mainly three services
+available: there is a web service running on port 80 (HTTP) and 443 (HTTPS), an
+Intelligent Platform Management Interface (IPMI) over LAN service on port 623, and the
+SSH service on port 22. Besides, we also found that the IPMI service can be accessed
+through the KCS interface from the CPU.
+Some of these interfaces require authentication: to use HTTP, HTTPS, SSH, and IPMI
+-over-LAN, all exposed through Ethernet, one has to authenticate to the BMC. The used
+credentials in this authentication process are individual for each Supermicro motherboard.
+However, the IPMI-over-KCS interface does not require any authentication to the BMC.
+Instead, having root privileges on the host OS running on the CPU is sufficient to access
+this interface. One can also use the IPMI-over-KCS interface to add/remove/modify BMC
+credentials to subsequently login to the Ethernet-exposed interfaces.
+3.1
+SSH Shell
+Since SSH is one of the most common interfaces that allows us to get a shell and possibly
+take over the system, we first started our investigation with it. However, the SSH service
+on E3-1220V6-X11SSL-CF provides a custom shell called “ATEN SMASH-CLP System
+Management Shell”. It only provides limited commands that enable server monitoring
+and basic management. Previously, a vulnerability was reported in [Vaz13]: the command
+shell sh allows gaining root access from this shell, however, this command was not
+available on our system-under-investigation.
+3.2
+BMC Firmware Analysis
+To further investigate the services running on the BMC and check if it is possible to
+enable an SSH root shell, we dumped the firmware of the BMC with a CH341A SPI flash
+programmer as shown in Figure 3. This procedure is only used once to assist our analysis,
+and is not necessary to execute the actual attack.
+Figure 3: Dumping BMC firmware with a flash programmer.
+We found that the firmware stored in the SPI flash is neither encrypted nor signed.
+There are five partitions in the firmware, where the second one contains a Linux operating
+system. The SMASH shell is provided by /SMASH/msh and it is possible to change it to a
+different shell by replacing this file.
+The Linux operating system also has an I2C kernel module installed, which provides an
+interface to communicate with the SMBus. However, during our testing in Section 4.1, we
+found that the API provided by this kernel module is not compatible with the commonly
+
+0000
+C
+C
+C
+O
+O
+SOP16
+014
+13
+12
+O
+O
+100
+O
+C
+1.27MM
+O
+90
+C
+D
+GOAET
+25XX24XX
+以
+二
+4683
+S9Zitai Chen and David Oswald
+9
+used libi2c in i2c-tool2. As the result, in Section 4.1, we opted to write a custom
+library to use the I2C interface of the BMC and communicate with the VRM.
+3.3
+Firmware Upgrade
+After analysing the firmware, we conclude that it is possible to enable an SSH shell by
+modifying the firmware. We then started to look for software methods to re-flash the BMC
+SPI flash chip. We found that the firmware upgrade functionality of the BMC provides a
+way to do this. There are two interfaces for firmware upgrade: one is through the web
+interface, the other through the KCS interface.
+Through Web Interface
+The web interface has a firmware upgrade page that can switch
+the BMC into upgrade mode and allows the user to upload a BMC firmware update
+package. To prevents unauthorised user from upgrading the firmware, there is a login
+portal. The user is authenticated by the BMC. As the BMC is a system independent from
+the OS running on the CPU, users do not need to have privileged access to the OS to be
+able to use this method. Besides, this web interface can be accessed remotely through
+Ethernet. The remote BMC firmware upgrade attack chain described in Section 4.3 uses
+this method to upgrade the firmware.
+Through IPMI-over-KCS Interface
+Crucially, the BMC firmware can also be updated
+through the KCS interface, using the following command: AlUpdate -f firmware.bin
+-i kcs -r y. As mentioned, the KCS interface can be accessed from the OS running on
+the CPU, only requiring root access to the OS, but not the BMC credentials.
+Firmware Upgrade Package
+After finding the firmware upgrade interface, the next step
+is to produce an upgrade package that can be uploaded to the BMC. We started with the
+analysis of the structure of the upgrade package. Figure 4 shows the layout of a firmware
+upgrade package. Previous work by [Ecl18] founds that in the firmware upgrade package,
+there is a region that contains a magic value (ATENs_FW), a half-length CRC checksum,
+and the length of each section. We call this part the firmware footer. There is also a
+region containing metadata of the firmware image, including the name of each region and
+their length and CRC, starting with “[img]”. We refer to this region as firmware table.
+In the X11 series, the firmware table, the file system header of the root file system and the
+website files system header are AES-CBC encrypted. However, the files in these regions
+are not encrypted, but only LZMA compressed. As a result, the key of the AES-CBC
+encryption can be recovered from the ipmi.so file on the root file system.
+With this information, we can modify the firmware and construct a valid firmware up-
+grade package for the web interface. We discuss firmware repacking in detail in Section 4.2.
+3.4
+IPMI I2C functionality
+When exploring the functionalities of IPMI, we also found that the interface also allows
+direct sending I2C packets with the ipmitool i2c command. This can be used either
+through the Ethernet or KCS IPMI channel. The authentication requirement for using
+IPMI-controlled I2C is the same as those described in Section 3.3. As shown in Section 4.3,
+we can use this functionality for direct access to the SMBus/PMBus without modifying
+BMC firmware.
+2https://git.kernel.org/pub/scm/utils/i2c-tools/i2c-tools.git/
+
+10
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+Figure 4: Layout of the BMC firmware upgrade package.
+The NVRAM region stores the current
+configuration of the BMC, the rootFS is a LZMA-compressed cramFS file system with only its header
+encrypted. The kernel region stores a Linux kernel image, while the BMC website FS is another compressed
+file system with only the file system header encrypted. The FW Footer starts with a magic value ATENs_FW
+and contain information about the firmware version, checksum, etc. The FW Table is an encrypted region
+and stores a table of the image layout. All encrypted region of the firmware can be decrypted with a key
+extracted from ipmi.so on the rootFS.
+4
+Practical Experiments
+Finally, using the results from the previous sections, we explain how to construct practical
+Proof-of-Concept (PoC) attacks for PMFault. Some of our experiments require physical
+access to the system to understand the hardware configuration (with an overview shown
+in Figure 5). Note however that physical access is not required when performing PMFault
+attacks on a real-world system, as the hardware components and connections are identical
+for a given motherboard model.
+Oscilloscope
+connected to
+PMBus
+Oscilloscope
+to monitor
+CPU voltage
+BMC flash chip
+soldered out
+PMBus connection
+for Raspberry Pi
+Management
+Ethernet
+Connection
+BMC
+micro-controller
+Power Button
+Figure 5: Setup of the E3-1220V6-X11SSL-CF for practical experiments. These connections are for
+experiments only; physical access is not required in the actual attack.
+4.1
+PMBus-based Voltage Control
+To understand the configuration and capabilities of using the PMBus to control the CPU
+voltage, we conducted two experiments. Firstly, we used the “probe and verify” method to
+find the I2C address of the VRM. Then we tried different ways of sending commands to
+VRM to change the voltage.
+
+ipmi.so
+Decompressed
+Files of RootFS
+C
+BMC
+rootFS
+FW
+FW
+NVRAM
+kernel
+WebsiteFS
+(Compressed)
+Footer
+Table
+(Compressed)ROHSZitai Chen and David Oswald
+11
+Discovering the VRM Address
+Finding the I2C address of the VRM is the first step
+of PMFault. The easiest way to explore the I2C buses is to use the interface provided
+by the OS. There are two I2C buses that can be used from the OS running on the CPU:
+i2c-0 is shown by default, while i2c-1 requires the i2c_i801 kernel module to be loaded.
+To find all available devices on both I2C buses, we ran the i2cdetect tool on them. We
+found that there are 12 devices in total connected to the I2C bus. The full list of device
+addresses can be found in Appendix A.
+To then determine which device is a VRM, we use the result of the standard PMBus
+command, READ_VOUT, as an indicator. The Plundervolt [MOG+20] attack showed that
+the normal operating voltage of the CPU should be greater than 0.55 V, thus, if the
+voltage read by READ_VOUT is within this range, it may be a VRM. Of the 12 devices
+detected, only one device with address 0x20 on I2C bus 1 responded with a value in this
+voltage range. We hence suspect this device is the VRM. To verify the result, we also
+used MFR_ADDR_PMBUS (0xE1) command found in the MP2965 datasheet [Mon] to read the
+PMBus address of the device. The result is 0x20, which confirms our finding.
+Changing CPU Voltage with PMBus Commands
+Having identified the VRM, one can
+next attempt to send commands to change the CPU voltage.
+Set target voltage to
+VOUT_COMMAND
+Configure VOUT_OPERATION
+with PMBus Override Mode
+Set Bit 3 of MFR_VR_CONFIG
+Figure 6: Command sequence to change the voltage via PMBus.
+In the datasheet of the MP2965 [Mon], we found an “overclocking” procedure that can
+be used for this purpose. There are two overclocking modes, tracking mode and fix mode.
+In PMFault, we mainly use the fix mode to set a defined voltage.
+In the fix overclocking mode, the VRM uses the VID configured with the PMBus
+command VOUT_COMMAND and ignores the configuration from the SVID bus. Figure 6 shows
+the steps of using this mode to change voltage. First, we need to configure two registers:
+The first one is VOUT_OPERATION; by setting the first bit of this register, we enable PMBus
+override mode. We also have to set bit 3 of MFR_VR_CONFIG to make the VRM act on
+these changes. After this, the voltage supplied to the CPU will be changed according to
+the configuration in VOUT_COMMAND. To send this PMBus command sequence and change
+the CPU voltage, we wrote a PoC with the libi2c. This PoC can be compiled and run
+under Linux.
+“Stalls” caused by PMBus Commands
+The experiments in Section 4.1 also show that
+the VRM responds to the PMBus commands sent from the CPU. One may thus assume
+that it would then be straightforward to directly send PMBus commands to change the
+CPU voltage with this method. However, we found that the CPU stalls after sending the
+MFR_VR_CONFIG command to actually configure the VRM to use the new voltage. This
+will make the CPU voltage being kept at the changed value with no way to change it back.
+This phenomenon raised two questions: Is the CPU stall caused by a crash or a recoverable
+halt? If it is caused by a recoverable halt, will this protect against targeted undervolting
+fault injection?
+To answer this, we connected a Raspberry Pi to the PMBus to directly control the
+VRM. The I2C interface to the VRM is exposed with two pins, SDA and SCL. As shown in
+Figure 5, we connected the I2C interface of the Raspberry Pi to these pins.
+In the first experiment, we sent a command to disable overclocking after the stall
+happens. It appears that with the VRM reconfigured to normal mode, the CPU recovers
+from the stall situation if the undervolting value is not too low. This shows that the stall is
+
+12
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+caused by a recoverable halt and not a crash. The second experiment is used to find out if
+the halt will prevent the fault from happening. In this experiment, we used the CRT-RSA
+PoC of the Plundervolt attack. With the CPU running this PoC, we used Raspberry Pi
+to send PMBus commands to produce voltage glitches. We found that with glitches with
+gradually lower voltage, an exploitable fault happens with the CRT-RSA calculation.
+Hence, in summary, the “stall” phenomenon will prevent the PMBus attack from being
+conducted by the CPU-VRM I2C interface, but it does not prevent the fault caused by
+undervolting from having an impact on CPU calculations.
+Voltage Control with BMC
+Because our attempt of voltage glitching failed with the
+PoC running on the CPU, we started to look into the BMC-VRM I2C interface. In the
+BMC firmware dumped in Section 3.1, we found the i2c.ko kernel module, which provides
+a driver for the I2C interface. However, this module does not implement a standard
+ioctl() for I2C devices, which is required for using libi2c. This means that the above
+PoC, which uses this standard I2C library, cannot be used to communicate with this kernel
+module.
+As the kernel module in the firmware did not implement the standard I2C API, we
+had to find another way to utilize the BMC’s I2C interface. With the help of the I2C
+driver in the latest Linux kernel [astb, asta], we found that there are 14 I2C interfaces
+on the AST2400 BMC controller. Each has a set of memory-mapped registers to control
+the interface. We also found the setup and message sending/receiving sequence of the
+I2C interface. We then created a small library to directly write these registers and send
+I2C bus commands from the BMC CPU to the address of the VRM. By monitoring
+the I2C activity with an oscilloscope (this was only required for debugging and during
+development), we found that the I2C bus 2 (counted from bus 0) of the BMC has the
+VRM connected.
+4.2
+Enabling SSH Access and Firmware Repacking
+Modification of the firmware can be used to obtain a root shell on the BMC. With the
+“Supermicro BMC firmware image decryptor” [Nie20] and a modified version of the “ipmi
+firmware tool” [Rak15] with added support for X11 images, we were able to extract the
+firmware encryption key and decrypt the file system header. With these, we can unpack
+and modify the full root file system.
+As described in Section 3.2, /SMASH/msh provides the shell for SSH service. To enable
+full root shell access, we replaced this file with a shell script with a single line to execute
+/bin/sh.
+Besides, as the SSH service is running with root privileges, with the shell
+redirected to sh, we could obtain a root shell once connected to the SSH.
+To repack the image, we modified the “Supermicro BMC firmware image decryptor”
+tool to add firmware encryption support and constructed a firmware package with a valid
+footer and firmware table. We successfully tested and installed this modified firmware
+package both with the web firmware upgrade interface and the IPMI firmware upgrade
+interface via the AlUpdate tool.
+4.3
+Attack Chains for PMBus Access
+In this section, we discuss three possible attack chains to take over the PMBus with the
+techniques shown in the previous sections. The attacker can use any of these attack chains
+and change the CPU voltage to perform PMFault attacks, i.e., to over/undervolt the CPU.
+Remote BMC Firmware Upgrade
+The first attack chain assumes a malicious insider
+threat model. This attack chain makes use of the web or IPMI interface through the BMC
+Ethernet connection. To use this interface, the attacker needs to have access to the BMC
+
+Zitai Chen and David Oswald
+13
+management Ethernet port or the shared management Ethernet port eth0 on the system.
+Besides, the attacker needs to obtain valid credentials to login to the BMC.
+In detail, the attacker can first use the method described in Section 4.2 to repack
+the SMT_X11_163 firmware upgrade package from [bmc] to enable SSH root access to
+the BMC. Then, they can upload the firmware with the web management interface or
+the IPMI management interface over Ethernet. With the SSH interface enabled, the
+attacker can cross-compile the voltage-changing PoC described in Section 4.1 for the
+BMC, and then upload and execute it to send PMBus commands. We used base64 -d >
+/tmp/i2c-pmbus-send to upload our exploit code due to the unavailability of the SCP
+service on the BMC OS.
+Local BMC Firmware Upgrade
+Similar to the first, this attack chain also involves a
+firmware upgrade for code execution on the BMC. However, we use the KCS interface
+discussed in Section 3.3 to upgrade the firmware. The attacker does not require access to
+the management Ethernet plane, instead, only root privileges on the OS running on the
+CPU is required. This is e.g., relevant for data centers that host bare metal machines for
+customers or for malware/ransomware that has obtained root through other exploits.
+IPMI Interface
+The third attack chain uses the IPMI I2C functionality. An attacker
+with root access on the CPU OS or access to the management port of the BMC can use
+this interface to send commands to any I2C device that is connected to the BMC. The
+command used for sending the raw I2C packets is shown in Listing 1. The I2C mapping
+of this interface is the same as found during the initial investigation in Section 4.1. The
+VRM is at address 0x20 on bus 2. However, since the last bit of the first packet of I2C
+indicates the type of operation (read or write), we need to shift the device address left by
+one bit and set the last bit accordingly when using this interface to control PMBus.
+ipmitool
+i2c bus=2 0x40 <PMBus
+Command > <PMBus Data >
+Listing 1: IPMI command for sending I2C packets.
+5
+Undervolting and Overvolting Attacks
+In this section, we show how under/overvolting through the PMBus leads to attacks on
+SGX and also permanent physical damage to the CPU. The attack requires any flaw that
+gives a software attacker access to the PMBus. As mentioned in Section 4.3, this can
+e.g., be a malicious firmware upgrade or the use of the IPMI-to-I2C functionality. The
+attack is generic in the sense that various flaws can lead to the same outcome: remote
+fault injection attacks on SGX and bricking the CPU. Figure 7 shows an overview of the
+attacks.
+5.1
+Undervolting Attack against Intel SGX
+Adversary Model
+As mentioned in Section 1.2, we assume a threat model where an
+attacker (including a malicious insider) has full software access to the system but no
+(or limited) physical access. More precisely, the attacker has root access to the OS and
+software access to the BMC via the KCS interface or Ethernet. All attack chains described
+in Section 4.3 can generally be used under this threat model. It is worth mentioning that
+the attack that uses ipmitool through the KCS interface does not require knowledge of
+the BMC credentials. A privileged local user on a compromised host CPU can thus use
+ipmitool to inject fault into SGX purely from software.
+
+14
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+BMC
+PMBus
+Overvolting
+Undervolting
+Brick CPU
+Fault Injection to
+SGX
+Firmware Upgrade
+to Enable SSH
+IPMI I2C Command
+Code Execution
+in BMC
+Remotely Executable Action (Management LAN)
+Locally Executable Action on OS (With root)
+Result of Attack
+Voltage Control
+Entity or Connection
+Legend
+Figure 7: Overview of the PMFault attack. With root access to the OS or access to the BMC via Ethernet
+or KCS, the attacker can perform a malicious firmware upgrade of the BMC and then takeover the PMBus.
+The attacker can also use the ipmi i2c command to directly control the PMBus via BMC. With control
+over the CPU voltage, the attacker can overvolt to brick the CPU or undervolt to inject faults into SGX.
+Proof of Concept
+We used the same PoC code as Plundervolt/VoltPillager [MOG+20].
+Before injecting the voltage glitch, we use the attack chain described in Section 4.3 to gain
+control of the PMBus.
+To start with, we used the multiply operation as the first target, as it is a simple target
+to fault. By gradually lowering the CPU voltage with the PMBus commands sent by
+the BMC while running the Plundervolt/VoltPillager PoC on the CPU, we successfully
+injected faults into the multiply operation (in our experiments at voltage 0.845 V with the
+CPU running at 2 GHz.
+To verify the fault injection also works for encryption operations running in SGX, we
+ran the CRT-RSA signature PoC from Plundervolt/VoltPillager, with an RSA signature
+computed inside an enclave using the Intel Integrated Performance Primitives (Intel IPP)
+cryptography library functions [Cor]. Again, we could obtain faulty signatures as shown
+in Listing 2. Furthermore, we confirmed that these faulty values could be used to factor
+the RSA modulus and recover the private RSA key using the Lenstra attack [BDL97].
+// Faulty
+calculation 1
+0x3f , 0xe0 , 0xb8 , 0x74 , 0x04 , 0x18 , 0x9c , 0xed , 0x91 , 0x1a , 0x02 , 0x12 , 0x2a ,
+0xce , 0x89 , 0xf8 , 0x32 , 0x00 , 0xdc , 0x05 , 0x15 , 0x53 , 0x72 , 0x8d , 0x84 , 0x00 ,
+0xd3 , 0x67 , 0xbe , 0xa1 , 0xc2 , 0x40 , 0x76 , 0xbc , 0x8c , 0xd8 , 0xfe , 0xb1 , 0x00 ,
+0xd7 , 0x9e , 0x0e , 0xb6 , 0xac , 0x61 , 0xc0 , 0xec , 0x9c , 0xf7 , 0x7e , 0xbc , 0x4b ,
+0xde , 0x18 , 0xa5 , 0xa4 , 0x1c , 0x74 , 0xc4 , 0xb5 , 0x6a , 0x8d , 0xd3 , 0xb1 , 0x35 ,
+0xf9 , 0xad , 0x0b , 0xe3 , 0x4a , 0x01 , 0x52 , 0xd4 , 0xc6 , 0xb2 , 0x95 , 0xbc , 0xdc ,
+0xad , 0x61 , 0x8e , 0x07 , 0x84 , 0x4d , 0xe3 , 0xa7 , 0xff , 0xf0 , 0xd1 , 0xa0 , 0xd4 ,
+0x58 , 0x9f , 0xbc , 0x37 , 0x0b , 0xa8 , 0x91 , 0x83 , 0x15 , 0x7b , 0xee , 0x28 , 0x83 ,
+0x12 , 0x4a , 0x89 , 0x61 , 0x1e , 0x2c , 0xe1 , 0x02 , 0x2f , 0x08 , 0x4d , 0x5b , 0x04 ,
+0x92 , 0x5e , 0x31 , 0xd0 , 0x7e , 0x94 , 0x85 , 0xd0 , 0xce , 0x75 , 0x4a , 0x00 , 0x00 ,
+0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 , 0x00 ,
+[...
+zeroes
+left
+out
+...]
+Incorrect
+result!
+Listing 2: Faulty CRT-RSA decryptions/signatures generated by the respective ipps functions.
+Reproducibility of CRT-RSA Fault Injection
+To further evaluate the reproducibility of
+the attack, we setup an automated testing environment by connecting a Raspberry Pi to
+an Ethernet port (eth0) and the power button of the motherboard. We ran a Python
+script to repeat the following steps numerous times:
+1. Upload the exploit for controlling the CPU voltage to BMC via an SSH connection.
+
+Zitai Chen and David Oswald
+15
+2. SSH into the OS running on the host CPU and trigger CRT-RSA signing in an SGX
+enclave.
+3. Run the PMFault exploit on the BMC to gradually lower the CPU voltage while the
+signature is computed in the SGX enclave.
+4. Stop lowering the CPU voltage when a fault occurs.
+5. Record the result and cleanup.
+6. If no faulty result is output, the system may have crashed due to too low voltage. In
+this case, we use the connection to the motherboard power button to reboot the system
+and wait to allow the system to boot into a stable status.
+In total, we conducted 253 tests within 545 min. Of those, faults occurred in 194 tests.
+66 of these faulty results could be used to successfully recover the correct RSA private key
+using the Lenstra attack, which translates to a success rate of 26%. On average, a useful
+fault could be obtained within 9 minutes.
+5.2
+Overvolting to Permanently Brick a CPU
+Apart from the undervolting attack to extract keys from an SGX enclave, we also discovered
+another attack, which is an overvolting attack that can permanently destroy the CPU.
+Adversary Model
+In this attack, as described in Section 1.2, we assume an attacker who
+has root privilege on the host CPU. For example, this could be in the case that an attacker
+has placed ransomware on a system and threatens to damage the CPU unless a ransom is
+paid. Clearly, root should have full control of all software running on the CPU, but should
+not be able to cause any physical damage to the system. The attack chain described in
+Section 4.3 using ipmitool with KCS can be used within this threat model.
+Proof of Concept
+To overvolt the CPU, we firstly configure the MFR_VR_CONFIG register
+of the VRM to use the 10 mV SVID table. This allows changing the CPU voltage up to
+3 V. We also disabled the over-current protection by reconfiguring the MFR_OCP_TOTAL_SET
+register. Then we used the voltage changing procedure to change the CPU voltage to a
+value much higher than the normal operating voltage.
+We found that this procedure allows changing the CPU voltage up to ∼2.84 V for
+∼1 ms, which is outside the typical operating range of Intel CPUs. By increasing the
+voltage beyond the specified operating voltage range (0.55 V–1.52 V) [Cor18] of a 7th Gen
+Intel E3-1220V6 CPU two times, we permanently destroyed the CPU and left the system
+in an unbootable state within a few seconds. We successfully repeated the experiment
+with a second, identical CPU. An example of overvolting is shown in Figure 8.
+For environmental and financial reasons, we were satisfied after successfully destroying
+two CPUs and decided to not perform further experiments in that regard.
+6
+Evaluation of other Server Motherboards
+As we found the PMBus to be a common interface present on server motherboard, we
+decided to investigate other manufacturers as well. To facilitate larger-scale testing of
+this, we wrote a tool called PMBusDetect. With this tool, we scan the system for a
+PMBus connection and try to detect the VRM address. We applied this tool to several
+other systems, including an ASRock rack motherboard (ASRock E3C246D4I-2T) and a
+Supermicro X12DPi-NT6 motherboard (kindly provided by Supermicro for testing). We
+then conducted further analysis of these systems to check if they are vulnerable to any
+PMBus-related attack.
+
+16
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+Figure 8: Oscillocope capture of voltage change during overvolting, VOUT_COMMAND set to 0xFF (with 10 mV
+VID table). Yellow: PMBus clock, blue: Vcpu. Vcpu shoots up to 2.84 V during overvolting.
+PMBusDetect Tool for VRM Detection
+Based on the VRM detection process mentioned
+in Section 4.1, we built the PMBusDetect tool to automatically scan all addresses of a
+specified I2C bus for VRMs. During testing, we found that the implementation of PMBus
+and usage of the VRM is different between motherboard, and the most stable command to
+identify a VRM is READ_TEMPERATURE (0x8d). We use the response to this command as
+an initial indicator to identify whether a VRM is present, and then use the VRM detection
+process from Section 4.1 to verify the result.
+Moreover, as the capabilities and voltage changing sequence can differ between VRM
+vendor, we added an additional procedure to detect the vendor of the VRM. For this, we
+use the result of reading ISL_DEVICE_ID (0xad) as an indicator for Intersil VRMs and
+SVID_VENDOR_PRODUCT_ID (0xbf) for MPS, respectively. Detection based on ipmi i2c is
+also implemented for detecting the connection between VRM and the BMC as mentioned
+in Section 4.3. An example output of PMBusDetect with Supermicro X11SSL-CF is shown
+in Appendix B, while Table 3 shows a summary of the motherboard tested and the scan
+result for VRMs with PMBusDetect. We are aware that our testing—restricted by (lack
+of) access to server hardware— only gives a very limited picture of the use of PMBus and
+VRMs on server hardware. We hence decided to open-source PMBusDetect and build on
+community efforts in the future to obtain a better view of the PMBus landscape.
+Table 3: Tested motherboards and their VRM detection result.
+Name
+BMC
+Chipset
+VRM Address
+PMBus Connects to
+Supermicro X11SSL-CF
+AST2400
+C232
+0x20
+BMC & CPU
+Supermicro X12DPi-NT6
+AST2600
+C621A
+0x30 & 0x34
+—
+ASRock E3C246D4I-2T
+AST2500
+C246
+0x60
+BMC & CPU
+6.1
+ASRock Power-Down Attack
+The ASRock E3C246D4I-2T motherboard uses an Intel Xeon E-2124 CPU with an
+Intel C246 Chipset and ASPEED AST2500 BMC with login credentials defaulting to
+ADMIN:ADMIN. We used the PMBusDetect tool together with manual probing and found
+that the VRM of this motherboard is connected to both the BMC and I2C bus of the
+CPU. In the following attack, we assume that the attacker is a user on a baremetal server
+with root access in the OS.
+The VRM used on this motherboard is an ISL69138. Because it is made by a different
+
+RIGOL
+WAIT
+H
+1.00ms
+250MSa/s
+3.00M pts
+4.00000000ms
+[1
+2.68V
+Horizonta
+Coupling
+DC
+Period
+BW Limit
+20M
+Freg
+Probe
+10X
+Rise Time
+Invert
+OFF
+Fall Tirme
+Volts/Div
+4
+Coarse
++width
+Unit
+[V] 
+width
+DV#1→2=*****
+tmax=-1.210ms
+Max=2.84 #
+Vupper=2.58 y
+AW=1.25 *
+2.00 v
+50.0 V
+.
+1.00 V
+:500mv日Zitai Chen and David Oswald
+17
+manufacture compared to the MP2955, the voltage changing PMBus command sequence
+used for the MP2955 does not work with this VRM. Due to lack of documentation of this
+procedure, we at the moment could not precisely overvolt or undervolt the CPU via the
+PMBus. Yet, we discovered a new attack to disable the VRM and force power-down the
+CPU, leaving the system in a (temporary) inoperable state.
+PMBusDetect shows that the VRM is at address 0x60 on I2C bus 2 of the host CPU.
+Different to the findings for the Supermicro X11SSL-CF, this VRM uses PMBus registers
+on page 0x1 instead of the default 0x0. We then issue the ON_OFF_CONFIG (0x02) and
+OPERATION (0x01) commands: We configure the OPERATION to “Immediate Off” and set
+the “source of enable” only to ON_OFF_CONFIG. This results in a immediate power-off of
+the VRM and crashes the system.
+During testing, we found the PMBus is only writable from the CPU with IPMI over
+KCS interface, but not from the BMC with ipmi i2c commands. As the result, it is not
+possible for the administrator of the system to remotely configure the VRM back to a
+normal state. Simply issuing the ipmi powercycle command with IPMI over LAN will
+leave the system in a infinite boot loop. To recover from this attack, the administrator
+has to physically power-cycle the system, which might increase downtime in a Denial-of-
+Service (DoS) scenario.
+This shows that PMBus as an attack vector does not only affect Supermicro X11SSL-
+CF, but also can have impact on servers from other manufacturers. Besides we believe that
+it might also be possible to conduct CPU bricking attacks if the PMBus voltage changing
+sequence of Intersil VRM is known. We leave this for future work.
+6.2
+Other Supermicro X11 Motherboards
+We also ran the PMBusDetect tool on X11SPG-TF and X11SSE-F Supermicro server
+motherboards—in both cases, the VRM was reachable in the default configuration. To
+test if they are vulnerable to PMFault, we sent PMBus commands through ipmi i2c
+commands and successfully undervolted them to crash the system. This shows that the
+attack chain through the IPMI interface is valid on these systems. As the systems were
+provided by a third party for remote testing, we were not able to attempt overvolting and
+similar, destructive experiments, but believe these motherboards to be equally affected.
+6.3
+Supermicro X12 Motherboards
+We disclosed the vulnerability to Supermicro in May 2022. They confirmed the issue
+and also provided a X12 generation Server for further testing. This system, Supermicro
+X12DPi-NT6, features a dual Intel Xeon Gold 6330 CPU, Intel C621A Chipset, and
+AST2600 BMC. Our investigation shows that mitigations has already been implemented
+on this motherboard to break the attack chain of PMFault before we reported the attack
+to Supermicro. Firstly, the firmware upgrade package is properly signed with RSA and
+verified during the firmware upgrade process, which prevents malicious firmware uploads to
+the BMC via IPMI. This breaks the attack chain though firmware upgrade. Secondly, I2C
+packet filtering has been implemented in the BMC, which prevents IPMI commands to
+directly send packets to the PMBus. Moreover, our PMBusDetect tool shows that the VR
+is not connected to the CPU, which prevents an attack directly from the operating system.
+In conclusion, to the best of our knowledge, we believe that Supermicro X12DPi-NT6
+is not directly vulnerable to the attacks described in this paper. However, we note that
+as-of-yet unknown vulnerabilities might remain in the firmware update process and the
+complex software stack running on the BMC, which warrants further investigation.
+
+18
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+7
+Conclusions and Countermeasures
+In this paper, we demonstrated two remote attacks that use the PMBus interface to control
+the CPU voltage. An undervolting attack can be used to inject fault to the SGX enclave of
+the CPU and e.g., recover a secret key used in cryptography algorithms. The overvolting
+attack causes permanent damage to the CPU.
+The attack affects, to our knowledge, all 11th generation Supermicro systems. It also
+impacts ASRock (tested with ASRock E3C246D4I-2T), though as described the VRM
+behaves differently to Supermicro. We suspect that the attack might also affect other
+vendors (given that BMCs are often similar), but could not further investigate this and
+thus leave it for future work.
+7.1
+Server Platform Security and Embedded System Security
+We first discuss the security considerations for server platforms. Previous security research
+on computer platforms were mainly focused on the security of the software (either running
+on the CPU or the management controller). However, each subsystem on a server platform
+does not act in isolation. Instead, they may interact with each other via the physical
+connections on the motherboard. In our attacks, we show that the hardware design of
+the system with a correctly implemented ipmitool can lead to severe security issues and
+damage to the system.
+Apart from the components on the motherboard, one should also take “plugin” devices
+into consideration when analysing the security of server platforms. During our investigation
+of the system, we found that when a Peripheral Component Interconnect Express (PCI-E)
+device is plugged onto the motherboard, it is also connected to the I2C bus of the
+motherboard. However, if the firmware of a PCI-E device is compromised, it can gain
+access to the PMBus to perform the same attacks described in this paper. On E3-1220V6-
+X11SSL-CF, this connection can be configured with a jumper named JI2C. Although this
+jumper is disconnected by default, the user may not be aware of the security implications
+of connecting this jumper.
+In summary, the server platform is a system that has multiple components and mi-
+crocontrollers. The security of the platforms is not only down to ensuring the security of
+the software running on it, but the overall design of the hardware and embedded systems
+on the motherboard should also go through a thorough security review. Securing such a
+system needs collaborative effort of both software developers and hardware engineers.
+7.2
+SGX Security
+Our attack on SGX enclaves shows that a privileged local attacker can inject a fault to the
+enclave and recover secret information with the server management interface, effectively
+reviving Plundervolt-like software undervolting attacks on Supermicro X11 motherboards.
+We also demonstrate that a malicious service provider (e.g., cloud hoster) can use the
+attack chains described in the paper to break the security guarantee provided by SGX.
+Moreover, the vulnerability currently cannot be detected/mitigated by SGX attestation,
+because the BMC and its firmware are not within the scope of SGX attestation.
+A supply chain attack is also possible: as the firmware is not securely verified, it is
+possible for a third party to implant malware into the BMC and later launch remote
+attacks on SGX and/or damage the CPU. Such a firmware modification is also conceivable
+while the device is being shipped to the end user. Detecting such attack would be hard, as
+the firmware of the BMC is stored in a separate flash chip. The software running on the
+BMC is thus usually out-of-scope of traditional malware detection methods.
+
+Zitai Chen and David Oswald
+19
+7.3
+Countermeasures
+Overvolting Attack
+According to our experiments, PMBus-based overvolting can lead
+to permanent damage to the CPU and thus permanent DoS of the system.
+The fundamental issue that leads to this attack is the lack of a hardcoded voltage
+limit of the VRM. Simply adding signature verification of the BMC firmware or using
+secure boot to break the attack chain might not be sufficient to prevent overvolting, as
+other, future attacks might also yield PMBus access. Besides, configuring software-based
+PMBus read/write limitations of the VRM through the MFR_PWD_USER command is also
+insufficient to stop the attack. This is because this features only sets a 16-bit passcode,
+which is prone to brute force attack. We suggest the following mitigations be implemented
+for this attack to break the attack chain:
+1. In the short term, the user manual of the relevant system(s) should be updated to
+describe the usage and suggested configuration of the SMBDAT_VRM and SMBCLK_VRM
+jumpers, if they are present on a specific model.
+2. In the long term, an alternative VRM with a hardwired voltage safety limit should be
+used to replace the current VRM.
+3. Another mitigation would be implementing an I2C filter to detect and block malicious
+PMBus packets. MFR_VR_CONFIG, which can be used to set a 10 mV VID table, is one
+of the main commands that need to be blocked. Optionally, other commands that
+involved in the overclocking procedure could be blocked, however, this may affect users
+who actually want to use this feature. Such a filter could be implemented in a small
+microcontroller that listens to the I2C bus and “jams” malicious commands by actively
+pulling the bus low once the command has been detected but before its transmission
+has been completed.
+PMBus-based SGX Undervolting
+To the best of our knowledge, PMFault represents
+the first attack that directly breaches integrity guarantees in the Intel SGX security
+architecture through the PMBus interface. We believe that the fix currently deployed by
+Intel against Plundervolt/V0ltPwn (CVE-2019-11157)—disabling the SVID undervolting
+interface—is insufficient when a remote attacker can get access to the PMBus through
+the BMC or I2C interface of the CPU, as is the case for Supermicro X11 motherboards.
+We note that there might be many other devices connected to the bus, including PCI-E
+devices like graphic cards. It is thus also possible for a compromised PCI-E device to send
+malicious commands to control the CPU voltage.
+Given the potential impact of our findings regarding fault injection into SGX enclaves,
+in the short term, we recommend inserting software-based fault injection countermeasures
+into cryptographic computations in enclaves (e.g., the quoting enclave). However, we note
+that such fixes can only serve as mitigations, but not fully eliminate this attack vector.
+We would like to highlight that in our opinion, this attack surface cannot be easily
+addressed by jumpers to disconnect the VRM from the SMBus or adding signature
+verification of the BMC firmware, as we believe that SGX attestation cannot independently
+verify the relevant system configurations:
+1. The existence of a PMBus/SMBus interface to the VRM and whether it can be controlled
+through the I2C interface of the CPU;
+2. The existence of an external microcontroller on the motherboard and if it has the
+functionality to control the VRM (e.g., BMC or other PCI-E devices);
+3. The firmware security status of the BMC and other devices on the PMBus.
+This will make it impossible to give SGX assurance of the trust status of the system.
+We believe that in the long term, appropriate hardware countermeasures inside the
+CPU package is required: this could on the one hand include continuous monitoring
+of the received supply voltage, as recently presented by Intel for critical parts of their
+systems [NT22], and on the other the use of fully-integrated voltage regulators.
+
+20
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+Acknowledgements
+This research is partially funded by the Engineering and Physical Sciences Research Council
+(EPSRC) under grants EP/R012598/1, EP/R008000/1, and EP/V000454/1. The results
+feed into DsbDtech. We would also like to thank Supermicro for providing a X12DPi-NT6
+server for further investigation of the issue.
+A
+i2cdetect Result for Supermicro X11SSL-CF
+~$ sudo
+i2cdetect 0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+a
+b
+c
+d
+e
+f
+[00 -20]: -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
+30:
+-- -- -- -- -- -- -- 37 -- -- -- -- -- -- -- --
+40:
+-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
+50:
+50 -- -- -- -- -- -- -- 58 -- -- -- -- -- -- --
+60:
+-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
+70:
+-- -- -- -- -- -- -- --
+~$ sudo
+i2cdetect 1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+a
+b
+c
+d
+e
+f
+00:
+-- -- -- -- -- 08 -- -- -- -- -- -- --
+10: 10 -- -- -- -- -- -- -- -- 19 -- -- -- -- -- --
+20: 20 -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
+30: 30 -- -- -- -- 35 36 -- -- -- -- -- -- -- -- --
+40: -- -- -- -- 44 -- -- -- -- -- -- -- -- -- -- --
+50: -- 51 -- -- -- -- -- -- -- -- -- -- -- -- -- --
+60: -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
+70: -- -- -- -- -- -- -- --
+B
+PMBusDetect Result for Supermicro X11SSL-CF
+$ sudo
+modprobe
+i2c_i801
+$ sudo ./ pmbusdetect -d /dev/i2c -1
+Device 0x20
+READ_TEMPERATURE
+success: 0019
+!!!!!!!!!!!
+Detected! Device
+addr: 20 !!!!!!!!!!!
+Device 0x20
+SVID_VENDOR_PRODUCT_ID
+success , data: 2555
+This
+device is likely to be a MPS VRM
+Device 0x20 : 00
+READ_PAGE
+success
+# Save the page
+Page: 00
+Device 0x20 : 00
+WRITE_PAGE
+success
+Device 0x20 : 00
+READ_VOUT
+success: 00D8
+Page: 01
+Device 0x20 : 01
+WRITE_PAGE
+success
+Device 0x20 : 01
+READ_VOUT
+success: 0001
+Device 0x20 : 00
+WRITE_PAGE
+success # Restore
+the page
+
+Zitai Chen and David Oswald
+21
+References
+[asta]
+Aspeed
+24XX/25XX
+I2C
+Controller
+Linux
+Kernel
+5.16
+Driver.
+https://elixir.bootlin.com/linux/latest/source/drivers/i2c/
+busses/i2c-aspeed.c. visited on 2022-09-16.
+[astb]
+Linux device tree file:
+aspeed-g4.dtsi.
+https://github.com/torvalds/
+linux/blob/133d9c53c9dcbb1b8f317e402e79c44d9eb725c9/arch/arm/
+boot/dts/aspeed-g4.dtsi#L438. visited on 2022-09-16.
+[BDL97]
+Dan Boneh, Richard A. Demillo, and Richard J. Lipton. On the Importance
+of Checking Computations. In Proceedings of Eurocrypt’97, pages 37 – 51,
+1997.
+[BECN+06] Hagai Bar-El, Hamid Choukri, David Naccache, Michael Tunstall, and Claire
+Whelan. The sorcerer’s apprentice guide to fault attacks. Proceedings of the
+IEEE, 94(2):370–382, 2006.
+[BJKS21]
+Robert Buhren, Hans-Niklas Jacob, Thilo Krachenfels, and Jean-Pierre Seifert.
+One Glitch to Rule Them All: Fault Injection Attacks Against AMD’s Secure
+Encrypted Virtualization. In Proceedings of the 2021 ACM SIGSAC Confer-
+ence on Computer and Communications Security, CCS ’21, page 2875–2889,
+New York, NY, USA, 2021. Association for Computing Machinery.
+[bmc]
+https://drunkencat.net/misc/SupermicroBIOS.html. visited on 2022-11-
+18.
+[Cor]
+Intel
+Corporation.
+Cryptography
+for
+Intel
+Integrated
+Perfor-
+mance
+Primitives
+Developer
+Reference—RSA
+Primitives.
+https:
+//www.intel.com/content/www/us/en/develop/documentation/ipp-
+crypto-reference/top/public-key-cryptography-functions/rsa-
+algorithm-functions/rsa-primitives.html. visited on 2023-01-05.
+[Cor18]
+Intel Corporation. Intel Xeon Processor E3-1200 v6 Product Family for S
+Platforms, 01 2018. https://www.intel.co.uk/content/dam/www/public/
+us/en/documents/datasheets/xeon-e3-1200v6-vol-1-datasheet.pdf.
+visited on 2022-09-16.
+[CVM+21]
+Zitai Chen, Georgios Vasilakis, Kit Murdock, Edward Dean, David Oswald,
+and Flavio D. Garcia. VoltPillager: Hardware-based fault injection attacks
+against intel SGX enclaves using the SVID voltage scaling interface. In 30th
+USENIX Security Symposium (USENIX Security 21), pages 699–716. USENIX
+Association, August 2021.
+[Ecl18]
+Eclypsium. Insecure firmware updates in server management systems, Sep
+2018. https://eclypsium.com/2018/09/06/insecure-firmware-updates-
+in-server-management-systems/. visited on 2022-09-10.
+[GE17]
+Maxim Goryachy and Mark Ermolov. How to Hack a Turned-Off Computer,
+or Running Unsigned Code in Intel Management Engine, November
+2017.
+Black Hat Europe 2017, https://www.blackhat.com/docs/eu-17/
+materials/eu-17-Goryachy-How-To-Hack-A-Turned-Off-Computer-Or-
+Running-Unsigned-Code-In-Intel-Management-Engine.pdf. Visited on
+2022-01-06.
+
+22
+PMFault: Faulting and Bricking Server CPUs through Management Interfaces
+[KDK+14]
+Yoongu Kim, Ross Daly, Jeremie Kim, Chris Fallin, Ji Hye Lee, Donghyuk
+Lee, Chris Wilkerson, Konrad Lai, and Onur Mutlu. Flipping bits in memory
+without accessing them: An experimental study of DRAM disturbance errors.
+In ISCA, 2014.
+[KFG+20]
+Zijo Kenjar, Tommaso Frassetto, David Gens, Michael Franz, and Ahmad-
+Reza Sadeghi. V0LTpwn: Attacking x86 Processor Integrity from Software.
+In USENIX Security ’20, Boston, August 2020. USENIX Association.
+[MIT17]
+MITRE. CVE-2017-5689, February 2017. https://cve.mitre.org/cgi-bin/
+cvename.cgi?name=CVE-2017-5689. visited on 2022-01-06.
+[MOG+20]
+Kit Murdock, David Oswald, Flavio D. Garcia, Jo Van Bulck, Daniel Gruss,
+and Frank Piessens. Plundervolt: Software-based Fault Injection Attacks
+against Intel SGX. In Proceedings of the 41st IEEE Symposium on Security
+and Privacy (S&P’20), 2020.
+[Mon]
+Monolithic
+Power
+Systems,
+Inc.
+MP2965
+Datasheet.
+https://
+www.monolithicpower.com/en/mp2965.html. visited on 2022-09-10.
+[Nie20]
+Michael Niewöhner. Supermicro BMC firmware image decryptor, 2020. https:
+//github.com/c0d3z3r0/smcbmc. visited on 2022-09-08.
+[NT22]
+Daniel Nemiroff and Carlos Tokunaga. Whitepaper: Fault Injection Counter-
+measures, Deployed at Scale. Technical report, 2022.
+[PGC18]
+Fabien Périgaud, Alexandre Gazet, and Joffrey Czarny. Subverting your
+server through its BMC: the HPE iLO4 case. In Recon Brussels ’18, 2018.
+[pmb]
+PMBus Power System Management Protocol Specification, Part II – Com-
+mand Language. https://470q2hhkn9g15l4bc2btbal1-wpengine.netdna-
+ssl.com/wp-content/uploads/2022/01/PMBus-Specification-Rev-1-3-
+1-Part-II-20150313.pdf. visited on 2022-09-11.
+[QWLQ19]
+P. Qiu, D. Wang, Y. Lyu, and G. Qu. VoltJockey: Breaking SGX by Software-
+Controlled Voltage-Induced Hardware Faults. In AsianHOST ’19, pages 1–6,
+2019.
+[Rak15]
+Brian Rak. Github repo: ipmi_firmware_tools, 2015. https://github.com/
+devicenull/ipmi_firmware_tools. visited on 2022-09-15.
+[RR18]
+Jordan Robertson and Michael Riley. The Big Hack: How China Used a Tiny
+Chip to Infiltrate U.S. Companies, Oct 2018. https://www.bloomberg.com/
+news/features/2018-10-04/the-big-hack-how-china-used-a-tiny-
+chip-to-infiltrate-america-s-top-companies#xj4y7vzkg. visited on
+2022-09-19.
+[Supa]
+Supermicro.
+X11SSL-CF(-nF)
+Quick
+Reference
+Guide.
+https://
+www.supermicro.com/QuickRefs/motherboard/C232/QRG-1782.pdf. visited
+on 2022-09-13.
+[Supb]
+Supermicro. X11SSL-CF X11SSL-nF USER MANUAL Revision 1.1. https:
+//www.supermicro.com/manuals/motherboard/C232/MNL-1782.pdf. visited
+on 2022-09-10.
+[TSS17]
+Adrian Tang, Simha Sethumadhavan, and Salvatore Stolfo. CLKSCREW:
+Exposing the perils of security-oblivious energy management. In USENIX
+Security ’17, pages 1057–1074, Vancouver, BC, August 2017. USENIX Associ-
+ation.
+
+Zitai Chen and David Oswald
+23
+[TW09]
+Alexander Tereshkin and Rafal Wojtczuk.
+Introducing ring -3 rootkits,
+2009. Black Hat USA, https://www.blackhat.com/presentations/bh-usa-
+09/TERESHKIN/BHUSA09-Tereshkin-Ring3Rootkit-SLIDES.pdf. visited on
+2023-01-06.
+[Vaz13]
+Juan Vazquez. Exploiting the Supermicro Onboard IPMI Controller, Nov
+2013. https://www.rapid7.com/blog/post/2013/11/15/exploiting-the-
+supermicro-onboard-ipmi-controller/. visited on 2022-09-12.
+[WS18]
+Nico Waisman and Matias Sebastian Soler. The Unbearable Lightness of
+BMC’s. In BlackHat ’18, 2018.
+

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+ 
+ 
+ 
+ 
+1 
+ 
+Compensation of anisotropy in spin-Hall devices for neuromorphic 
+applications 
+Pankaj Sethi*, Dédalo Sanz-Hernández, Florian Godel, Sachin Krishnia, Fernando Ajejasa), 
+Alice Mizrahi, Vincent Cros, Danijela Marković and Julie Grollier 
+ 
+Unité Mixte de Physique CNRS/Thales, Université Paris-Saclay, 91767 Palaiseau, France 
+ 
+ 
+Spintronic nano-oscillators with reduced non-linearity could offer key benefits for 
+realizing neuromorphic applications such as spike-based neurons and frequency multiplexing 
+in neural networks. Here, we experimentally demonstrate the reduction in non-linearity of a 
+spin-Hall nano-oscillator (SHNO) by compensation of its effective magnetic anisotropy. The 
+study involves optimization of Co/Ni multilayer growth to achieve the compensation, followed 
+by spin diode measurements on patterned microstrips to quantify their anisotropy. The relation 
+between the second (Hk2 = 0.47 mT) and the first order (Hk1eff =  ̶  0.8 mT) anisotropy fields 
+reveals the existence of an easy cone, thereby validating the presence of compensation. 
+Furthermore, we demonstrate a synapse based on the compensated spin diode which has a fixed 
+frequency when the input power is varied. We then study the current-induced auto-oscillation 
+properties of SHNOs on compensated films by patterning nano-constrictions of widths 200 and 
+100 nm. The invariance of the resonance frequency and linewidth of the compensated SHNO 
+with applied dc current indicates the absence of non-linearity. This independence is maintained 
+irrespective of the applied external fields and its orientations. The compensated SHNO obtained 
+has a linewidth of 1.1 MHz and a peak output power of up to 1 pW/MHz emulating a nano-
+neuron with a low linewidth and a fixed frequency.  
+ 
+a) Present address: Department of Physics and Center for Advanced Nanoscience, University of 
+California, San Diego, La Jolla, CA, 92093, USA 
+*Corresponding author: [email protected], [email protected] 
+ 
+
+ 
+ 
+ 
+ 
+2 
+ 
+I. INTRODUCTION 
+Spintronic nano-oscillators with their low device footprint, rich dynamics and 
+multifunctionality can provide an energy efficient solution to realize neuromorphic 
+applications [1–4]. Non-linearity is prevalent in the magnetization dynamics of such nano-
+oscillators. In a non-linear auto-oscillator, the frequency, f, has a component which depends on 
+the precession amplitude or the effective magnetization given by,  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+f = fFMR + Np,    
+(1) 
+ 
+where fFMR is the frequency at ferromagnetic resonance, N is the non-linear frequency shift 
+coefficient and p is the term related to the amplitude of precession  [5]. This non-linearity 
+emerges from an effective anisotropy in the system, which results in non-circular trajectory of 
+the precessing magnetization. It leads to a large frequency tunability with current which 
+provides multifunctionality to these nano-oscillators such as the possibility to be modulated or 
+synchronized. This has been exploited in realizing numerous applications relevant to data 
+communication [6–10] and neuromorphic computing [3,4,11,12]. However, there are certain 
+systems where it is possible to reduce the effective anisotropy and, as a result, the non-linearity. 
+In such compensated systems, the anisotropy field is counterbalanced by the demagnetization 
+field, resulting in circular trajectories of the precessing magnetization. The absence of 
+nonlinearity, which results in a constant frequency with respect to the injected input power or 
+current, also offers key benefits for realizing neuromorphic applications. For instance, multiply-
+and-accumulate (MAC) operations using spintronic resonators employ frequency multiplexing 
+to uniquely address input radio frequency (RF) signals from neurons to the corresponding 
+resonators  [13,14]. This requires the neurons and the corresponding spin diode-based synapses 
+to resonate at a relatively fixed frequency independent of the injected RF power, which can be 
+accomplished by compensating anisotropy in spintronic nano-oscillators and spin diodes, 
+respectively. Secondly, an absence of non-linearity can reduce the phase noise of the nano-
+oscillator by removing the effect of amplitude noise on it. The phase noise, Δf, of an auto-
+oscillator is given by, 
+ 
+ 
+Δf = Δfthermal (1+ N2/Γeff2), 
+(2)
+ 
+ 
+
+ 
+ 
+ 
+ 
+3 
+ 
+where Δfthermal is the contribution from the thermal generation linewidth and Γeff is the effective 
+damping [5]. The second term, which is the contribution from the amplitude noise, can be 
+neglected if N is very small. Thus, a neuron with low linewidth and a relatively fixed frequency 
+can be realized using anisotropy compensation. A third application is the realization of spike-
+based neurons which was recently demonstrated via macro-spin approach and micromagnetic 
+simulations [15]. It was shown that anisotropy compensation in a spin Hall geometry results in 
+circular trajectories of the precessing magnetization and the resulting output is a chain of spikes 
+emulating the biological neurons. Thus, it is important to study systems with compensated 
+anisotropy.  
+Recently, Jiang et al. have demonstrated a linewidth reduction of spin-valve based spin-
+torque nano-oscillators (STNOs) by controlling the perpendicular magnetic anisotropy (PMA) 
+of their films using He-ion irradiation [16]. An alternate planar geometry based on heavy metal 
+and ferromagnetic layers, which utilizes spin current injected from the heavy metal by spin Hall 
+effect to sustain precession in the ferromagnet, benefits from ease of fabrication [17–19]. 
+Moreover, spin Hall nano-oscillators (SHNOs), in the form of a nano-constriction geometry of 
+these layers, exhibit auto-oscillations by way of mode confinement in a potential well formed 
+by non-uniform magnetic field [20–22]. Divinskiy et al. demonstrated the suppression of 
+nonlinear damping by compensation of in-plane dipolar anisotropy with PMA in Co/Ni based 
+disks patterned on Pt heavy metal [23]. However, the detection of auto-oscillations was 
+performed by optical methods which are less suitable for on chip applications.  
+Here, we experimentally demonstrate, by all-electrical measurements, a reduction of non-
+linearity and linewidth of an SHNO, based on Co/Ni multilayers with compensated anisotropy 
+and a Pt heavy metal layer. Compensation is achieved by tuning the thicknesses of the Co/Ni 
+multilayers. The effective anisotropy is estimated using spin diode measurements performed on 
+microstrip waveguides. The relation between the second and the first order anisotropy terms 
+indicate the presence of an easy cone state [24–26] which validates the existence of 
+compensation. The compensated spin diode thus obtained, does not show variation of its 
+frequency with the injected RF power and can function as a synapse. Nano-constriction based 
+SHNOs with different widths are then patterned on the compensated stacks and the output 
+microwave spectra are analysed. The frequency is found to remain nearly constant as a function 
+of dc current for a wide range of magnetic field strengths and orientations. Moreover, an 
+extremely low linewidth close to 1 MHz (quality factor = 7500) is obtained, which does not 
+increase significantly at large applied   dc   currents.  Control SHNO fabricated with an in-plane 
+anisotropy Ni81Fe19/Pt stack exhibits significant shift of frequency and linewidth with the 
+
+ 
+ 
+ 
+ 
+4 
+ 
+applied dc current. The compensated SHNOs can thus operates as a neuron with a fixed 
+frequency and a low linewidth. 
+ 
+II. COMPENSATION OF ANISOTROPY IN SPIN HALL DEVICES 
+A. Sample preparation  
+The stacks consisting of Ta (5) /Pt (6) /[Co (x) /Ni (y)]5 /Co (x) /Al (2) (thicknesses are in 
+nm) are deposited on high resistivity Silicon (001) substrates (resistivity > 10000 Ω-cm) by dc- 
+magnetron sputtering at room temperature. Ta is used as a seed layer to promote adhesion 
+between silicon and the subsequent layer and Pt serves as the heavy metal layer. Co/Ni 
+multilayers are chosen for their large PMA and spin polarization which can be tuned by varying  
+layer thicknesses [27], as demonstrated previously for domain-wall based devices [28,29]. A 
+Co/Ni multilayer repetition of five was chosen to obtain a sizeable absolute magnetization [30].  
+FIG. 1. Alternating gradient magnetometry measurements for Ta (5) /Pt (6)/ [Co (x) / Ni (y)]5/ 
+Co (x)/ Al (2) (thicknesses are in nm) films with (a) in-plane anisotropy (x = 0.5, y = 0.8), (b) 
+compensated anisotropy (x = 0.4, y = 0.9) and (c) perpendicular anisotropy (x = 0.4, y = 0.8). 
+
+(a)
+Co 0.5/Ni 0.8
+1.0
+0.5
+0.0
+Norm.
+-0.5
+OOP
+-1.0
+IP
+(b)
+LCo 0.4/Ni 0.9
+0.5
+0.0
+-0.5
+OOP
+-1.0
+IP
+(c
+Co 0.4/Ni 0.8
+0.5
+0.0
+Norm.I
+-0.5
+OOP
+-1.0
+IP
+-500
+-250
+0
+250
+500
+Hext (mT) 
+ 
+ 
+ 
+5 
+ 
+Thicknesses of Co (x) and Ni (y) are varied to tune the anisotropy and the corresponding M-H 
+loops are measured for in-plane (IP) and out-of-plane (OOP) field orientations using alternating 
+gradient force magnetometry (AGFM). Starting with in-plane anisotropy (IPA) for Co (0.5 nm)  
+and Ni (0.8 nm) [Fig. 1(a)], the thickness of Co is reduced to 0.4 nm and PMA is obtained [Fig. 
+1(c)] due to interfacial anisotropy overcoming the demagnetization field. Henceforth, in this 
+article, Co (0.5 nm) /Ni (0.8 nm) and Co (0.4 nm) /Ni (0.8 nm) multilayers are referred to as 
+IPA and PMA stacks, respectively. Further, when the thickness of Ni is increased to 0.9 nm, 
+the PMA reduces but the anisotropy is neither fully in-plane nor out-of-plane [Fig. 1(b)]. As 
+will be described in what follows, the intermediate anisotropy obtained with Co (0.4 nm) and 
+Ni (0.9 nm) has been compensated and this film is referred to as the compensated stack. The 
+anisotropy fields were extracted using spin diode measurements [18,31]. To carry out the 
+measurements, the multilayers were patterned into microstrip waveguides of width 10 µm and 
+length 25 µm using optical lithography and Ar ion beam etching techniques. Ti (15 nm)/Au 
+(150 nm) metal stacks are deposited as electrodes and patterned into coplanar waveguides 
+overlaying the microstrips using optical lithography and lift-off techniques. The resulting 
+samples are henceforth referred as IPA, PMA and compensated devices, respectively. 
+ 
+B. Spin-diode measurements and estimation of effective anisotropy 
+Figure 2 shows the spin-diode measurement set-up. A microwave current with a power of 
+8 mW (9 dBm) is injected into the microstrip device to generate microwave frequency spin-
+orbit torque (SOT) on the ferromagnetic layers due to the heavy metal Pt [18]. The mixing 
+between the oscillating magneto-resistance and the microwave current produces a dc rectified 
+voltage, Vdc, at the ferromagnetic resonance, which is detected by using a lock-in amplifier. 
+The external field is swept close to the OOP direction for the PMA device (θ = 5 deg) and is 
+swept in-plane (φ = 45 deg) for the compensated and IPA devices. By keeping the field 
+FIG. 2. Schematic illustration of spin-diode measurement set-up 
+
+sΦ x 
+ 
+ 
+ 
+6 
+ 
+orientation close to the anisotropy of the devices we can eliminate the artefacts due to geometry 
+induced local anisotropy variation and simplify the analysis [32]. All measurements are 
+performed at room temperature. Resonance plots obtained for the PMA, the compensated and 
+the IPA devices are shown in Figures 3 (a), (b) and (c), respectively. The amplitudes observed 
+in the resonance plots are not corrected for the non-flat frequency response of the  wire  bonds
+and the cabling in the set-up. However, in our analysis we are only interested in the estimation 
+of the resonance fields which are independent of amplitude losses. The plots can be well fit by  
+FIG. 3. Spin diode resonance plots at different injected microwave frequencies for (a) PMA, 
+(b) compensated and (c) IPA stacks based microstrip waveguides. Resonance frequency as a 
+function of the resonance field for (d) PMA, (e) compensated and (f) IPA stacks based 
+microstrip waveguides. Solid red lines are Kittel fits and dotted blue lines, plotted for 
+guidance, corresponds to Meff = 0.  
+
+a)
+60
+(d)
+ExtractedPeaks
+4 GHz
+50
+(GHz)
+Kittel Fit
+5 GHz
+ - Meft = 0
+6 GHz
+40
+7 GHz
+Meff=-35mT
+30
+8 GHz
+6
+Meff
+&
+20
+5
+10
+0
+100
+200
+300
+400
+120
+160
+200
+240
+280
+Hext (mT)
+Hext (mT)
+(b)
+50
+(e) 8
+ExtractedPeaks
+(GHz)
+Kittel Fit
++ -Mer= 0
+(Λr)
+-50
+Frequency
+6
+3 GHz
+4 GHz
+5
+Meff
+>-100
+5 GHz
+Meff=0.5mT
+6GHz
+-150
+7 GHz
+4
+8GHz
+3
+-200
+100
+200
+300
+400
+80
+120
+160200240280
+Hext (mT)
+Hext (mT)
+(c) 100
+(f)
+..
+Extractedpeaks
+(GHz)
+Fit
+0
+- Mef = 0
+3 GHz
+Frequency
+6
+Mof=86.6mT
+4 GHz
+5 GHz
+5
+-200
+6 GHz
+7 GHz
+4
+-300
+8 GHz
+3
+100
+200
+300
+400
+80
+120
+160
+200
+240
+Hext (mT)
+Hext (mT) 
+ 
+ 
+ 
+7 
+ 
+the sum of symmetric and antisymmetric Lorentzian curves [18]. The resonance field, Hr is 
+extracted for each of the injected microwave frequency (fres) and the Kittel functions (fres vs Hr) 
+are plotted for each of the three configurations. The linear relation obtained in Figure 3 (d) for 
+the PMA device is well explained by the Kittel formula, fres = γ/2π(Hr  ̶  µ0Meff ) [33], where 
+µ0Meff = µ0Ms – Hk, is the effective anisotropy field. The fit of the equation yields an Meff   =  ̶ 
+35 mT. The negative sign of Meff confirms the existence of PMA. Figures 3 (e) and (f) depict 
+the fres vs Hr plots for the compensated and the IPA devices, respectively which are well fit with 
+the equation, fres= γ/2π[Hr(Hr + µ0Meff)]1/2 [18]. The extracted values of Meff are 0.5 mT and 
++86.6 mT for the compensated and the IPA devices, respectively. As a comparison, the Kittel 
+function corresponding to Meff = 0 is also plotted together with the as obtained fits for each of 
+the three devices. Clearly, the compensated stack-based device is closest to the near zero 
+effective anisotropy.  
+Given that the first order anisotropy is close to zero in the compensated device, the possible 
+influence of the second order anisotropy needs to be taken into consideration. The following 
+equations are the more generalized forms which take the second order anisotropy into 
+consideration,  
+ 
+ 
+ 
+ 
+f = γ/2π(H1H2)1/2 
+(3) 
+with 
+H1 = Hr cos(θH  ̶  θM) + Hk1eff cos2θM  ̶  Hk2cos4θM, 
+ 
+H2 = Hr cos(θH  ̶  θM) + Hk1effcos 2θM  ̶  Hk2/2(cos 2θM + cos 4θM), 
+(4) 
+ 
+where θH, θM correspond to the angle of the external magnetic field and the magnetization angle 
+measured from the sample normal, respectively. Hk1eff and Hk2 correspond to the first and the 
+second order effective anisotropy fields, respectively [34]. By adopting Hk1eff, Hk2 and γ as 
+adjustable parameters, the θH dependence of Hr yields the first and the second order anisotropy 
+fields. The energy minimum conditions ∂F/∂θM = 0 and ∂2F/∂θM2 > 0 are used to extract the 
+value for θM, where F is the magnetic energy density [34].  
+ 
+ 
+
+ 
+ 
+ 
+ 
+8 
+ 
+ 
+ 
+Spin-diode measurements are performed by sweeping the magnetic field at different out-
+of-plane angles, θH, in the y-z plane as shown in the schematic of Figure 4. In this geometry, 
+the signal strength of the output voltage is larger due to the spin pumping contributions [35]. 
+The resonance fields, Hr, are extracted from the sum of symmetric and antisymmetric 
+Lorentzians for each of the angles. The measurements are first performed for the IPA and the 
+PMA devices. The extracted Hr as a function of θH are shown in Figures 5 (a) and (b), with 
+input microwave frequencies fixed at 3 GHz and 4 GHz for the IPA and the PMA devices, 
+respectively. The curves display a monotonic behaviour, where the Hr is minimum close to the 
+in-plane angle (θH = ±90 deg) for the IPA device and close to the out-of-plane angle (θH = 0 
+deg) for the PMA device. The nature of the curves is independent of the input microwave 
+frequency, different values are selected for the two devices based on the signal quality. The 
+measurements have been performed for the compensated device at a frequency of 5 GHz and  
+the corresponding Hr vs θH plots are shown in Figure 5 (c). The curves display a non-monotonic  
+FIG. 4. Schematic illustration of spin-diode measurement set-up when external field is rotated 
+out-of-plane. 
+FIG. 5. Resonance field vs field angle for the microstrip waveguide with (a) IPA stack, 
+microwave frequency fixed at 3 GHz (b) PMA stack, microwave frequency fixed at 4 GHz and 
+(c) compensated stack, microwave frequency fixed at 5 GHz. 
+
+Bias-tee
+Input
+Lod:
+am:(a)
+(b)
+(c)
+In-Plane
+PMA
+Compensated
+240
+240
+Exp.
+240
+Fit
+200
+E
+200
+220
+160
+H
+160
+ 200
+120
+80
+3 GHz
+120
+4 GHz
+180
+5 GHz
+-90
+-60-30
+0
+30
+60
+90
+-90
+-60
+-30
+0
+30
+60
+90
+-90-60-30
+0
+30
+60
+90
+Angle (deg)
+Angle (deg)
+Angle Qμ (deg) 
+ 
+ 
+ 
+9 
+ 
+behaviour, where the Hr is minimum at an intermediate angle close to 50 deg. This is referred 
+to as the cone angle and its existence is an indication of compensation of the anisotropy [24,36]. 
+The curves are well fit with (4) and are used to extract Hk1eff =  ̶  0.8 mT and Hk2 = 0.47 mT. 
+The obtained parameters also satisfy the following conditions for the existence of an easy cone: 
+Hk1eff < 0; Hk2 >0 and Hk2 >  ̶ Hk1eff/2  [24].These measurements thus demonstrate that a device 
+with compensated anisotropy has been fabricated that can be employed to realize a synapse 
+with a fixed frequency. 
+ 
+ 
+ 
+FIG. 6. (a) Comparison of shift in resonance field as a function of input rf power for a spin 
+diode in IPA, compensated and PMA configuration. (b) Resonance curves as a function of input 
+rf power for (b) IPA and (c) compensated (synapse) spin diodes 
+
+(a)1.5
+In-Piane
+Comp.
+1.0
+PMA
+0.5
+res
+0.0
+-0.5
+-1.0
+-1.5
+2345678910
+RFpower(mW)
+(b)
+40
+UncompensatedDevice(IPA)
+0
+0
+(μV)
+-40
+-4
+-80
+>
+-8
+RFpower
+-120
+-12
+1mW
+-160
+10mW
+-16
+45
+60
+75
+90
+(c) 80
+Hext (mT)
+8
+40
+CompensatedDevice
+0
+0
+-40
+-4
+-80
+-8
+%-120
+-12
+-160
+FRFpower
+-16
+-200
+1mW
+-20
+-240
+10mW
+-24
+75
+90
+105
+120
+Hext (mT) 
+ 
+ 
+ 
+10 
+ 
+C. Input independent spin-Hall synapse with fixed frequency 
+A synapse can be realized using spin-diodes. Leroux et al. demonstrated a MAC operation 
+using magnetic tunnel junctions as spin diodes [14]. In a MAC operation, the output voltage Uj 
+can be represented by a weighted sum of the input power, Uj = ΣPiWji. The above equation can 
+be mapped to a spin-diode equation in the linear zone close to resonance, where the weights are 
+represented by the resonator frequencies. During the frequency multiplexing in a MAC 
+operation, each injected input power Pi, should be able to uniquely address the corresponding 
+synapse by its frequency. This imposes a constraint on the frequency of the synapse which 
+should not change with the injected rf power. In a spintronic resonator, this criterion is usually 
+not satisfied on account of the inherent non-linearity. However, the compensated spin diode can 
+be operated as an input independent synapse with a fixed frequency. Figure 6 (a) shows the 
+shift in Hr as a function of the injected input rf power for the IPA, the compensated and the 
+PMA spin diodes. Starting at the minimum input power ( = 1 mW), the shift is normalized to 0 
+for all the three devices. As the input power is increased, the IPA and the PMA devices exhibit 
+an increase in the shift of Hr, whereas, the compensated device shows a negligible shift in Hr. 
+Figure 6 (b) and (c) show the comparison of the resonance plots for the IPA and the 
+compensated devices, respectively, as a function of the input power. Clearly, there is no visible 
+shift in the resonance field and the equivalent frequency with the injected rf power for the 
+compensated device as compared to the IPA device. Thus, the compensated spin diode can 
+function as an input independent spin-Hall synapse.    
+ 
+III. AUTO-OSCILLATIONS IN COMPENSATED SPIN HALL DEVICES – 
+NEURON OPERATION 
+A. Device fabrication and measurement set-up 
+Nano-constrictions with widths of 100 nm and 200 nm are fabricated on the compensated 
+Co/Ni stacks using electron-beam lithography and Ar ion beam etching. Ti (15 nm)/Au (150 
+nm) metal stacks are deposited as electrodes and patterned into coplanar waveguides overlaying 
+the nano-constrictions using optical lithography and lift-off.  The device geometry is similar to 
+the one used in previous reports for realizing an SHNO [21,22]. As a comparison, in-plane 
+SHNO based on Py/Pt stacks are also patterned into nano-constrictions (Py = Permalloy = 
+Ni81Fe19).  
+The scanning electron microscopy image of a 200 nm nano-constriction along with the 
+measurement set-up to detect the auto-oscillations is shown in Figure 7 (a). A dc current, Idc, is  
+ 
+
+ 
+ 
+ 
+ 
+11 
+ 
+ 
+ 
+ 
+FIG. 7. (a) SEM image of 200 nm nano-constriction and a schematic to study the microwave 
+emission from the SHNO. (b) Auto-oscillation spectra for the compensated Co/Ni SHNO obtained 
+at Idc = + 2.8 mA, Hext = 300 mT (θH = 15 deg, ϕH = 50 deg). (c) Auto-oscillation spectra for the in-
+plane Py/Pt SHNO obtained at Idc =  ̶  3.5 mA, Hext = 50 mT (θH = 85 deg, ϕH = 42 deg). Linewidth 
+as a function of Idc sweep for (d) compensated Co/Ni SHNO and (e) in-plane Py/Pt SHNO. Power 
+spectral density plots showing frequency vs Idc sweep for (f) compensated Co/Ni SHNO and (g) in-
+plane Py/Pt SHNO.  
+ 
+
+300 nm
+t,xy
+8.0
+20
+1.0
+ee
+200
+9150
+Frequency (GHz)
+7.8
+(zHW/Md)
+0.8
+Af=1.1MHz
+6
+10
+0.6
+Compensated
+(zHW)
+5
+7.6
+-3.0 -3.5
+Pt/(Co/Ni)5
+0
+alove
+0.4
+7.4
+PSD
+-10
+0.2
+7.2
+Ise
+0.0
+20
+7.00
+7.25
+7.50
+7.75
+8.00
+-3.0
+-3.5
+-4.0
+-4.5
+-5.0
+2.5-3.0-3.5-4.0-4.5-5.0
+Frequency (GHz)
+Idc (mA)
+(c)
+(e)
+'dc (mA)
+0.5
+300
+Emitted
+20
+0.4
+250
+(GHz)
+6.0
+(zHW/Md) 
+Af=7.85MHz
+10
+In-plane
+200
+0.3
+(zHW)
+Frequency
+5.8
+150
+Py 5/Pt 5
+5.6
+0
+0.2
+PSD
+100
+-10
+0.1
+5.4
+noise
+50
+0.0
+0
+5.2
+20°
+B
+5.00
+5.25
+5.50
+5.75
+6.00
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+2.5
+3.0
+3.54.0
+4.5
+5.0
+Frequency (GHz)
+Idc (mA)
+Idc (mA) 
+ 
+ 
+ 
+12 
+ 
+injected into the nano-constriction via the dc port of a bias-tee. An external magnetic field is 
+applied at an in-plane angle, ϕH and an out-of-plane angle, θH. The SHNO emits microwave 
+power which is extracted from the rf port of the bias-tee and amplified by 38 dB using a low 
+noise wide-band amplifier. The output spectra are sampled using a spectrum analyzer. All 
+measurements are performed at room temperature.     
+ 
+B. Electrical microwave measurements for compensated and in-plane devices 
+Figure 7 (b) shows the emission spectra for the 200 nm SHNO realized using the 
+compensated Co/Ni stack at Idc =  ̶ 2.8 mA (+ x-direction) and Hext = 300 mT (θH = 15 deg, ϕH 
+= 50 deg). The linewidth (Δf) obtained from the Lorentz fit is 1.1 MHz with the peak power 
+spectral density (PSD), after subtracting the amplifier gain, as high as 1 pW/MHz. To the best 
+of our knowledge, the quality factor (Q ≈ 7500) obtained is more than the highest reported     
+using a single  constriction based SHNO  [3,37]. As a comparison, the above measurements are 
+also performed on Py/Pt based SHNO devices. Figure 7 (c) shows the corresponding spectra 
+obtained at Idc = +3.5 mA and Hext = 50 mT (θH = 85 deg, ϕH = 42 deg). It is worth noting that 
+the field orientation is maintained close to the in-plane direction for this device to excite the in-
+plane modes and the sign of Idc is positive as the SOT is from the top interface. The minimum 
+linewidth obtained from the Lorentz fit is 7.85 MHz and is much larger than that achieved using 
+the compensated Co/Ni SHNO. The above observations can be explained from (2), which 
+indicate a reduction of Δf if N reduces. To further validate this claim, we sweep the injected Idc 
+and record the variation of the frequency and Δf for the two SHNOs at the above-mentioned 
+external fields and orientations, respectively. Figures 7 (d) and (e) show Δf as a function of Idc 
+for the compensated Co/Ni and the in-plane Py/Pt SHNOs, respectively. Figure 7 (d) is plotted 
+for Idc larger than the critical current of auto-oscillations (Ic =  ̶  2.7 mA), which is the region of 
+interest, and the inset shows the data for I < Ic as well. When Idc < Ic, Δf increases with the 
+reduction in current for both the devices, as expected.  At large Idc, the Py/Pt SHNO shows an 
+increase in Δf due to the inherent non-linearity, which is not the case with the compensated 
+Co/Ni SHNO which shows a near constant Δf. The evidence for the absence of non-linearity in 
+the compensated Co/Ni SHNO becomes stronger when we compare its frequency vs Idc shown 
+in the power spectral density plots in Figure 7 (f) to that obtained for Py/Pt SHNO in Figure 7 
+(g). Clearly, the rate of change of frequency with the current (df/dI) is minimal for the 
+compensated Co/Ni SHNO (= 10 MHz/ mA) and significant for the in-plane Py/Pt SHNO (= 
+500 MHz/mA). However, for Idc >  ̶ 4.5 mA, some non-linearity can be observed in Figure 7  
+ 
+
+ 
+ 
+ 
+ 
+13 
+ 
+ 
+(f), which could be ascribed to the device heating or frequency shift due to the Oersted field or 
+the field-like torque [30]. The above observations are a direct validation of a reduction in the 
+non-linearity as indicated in (1). The measurements are repeated at different applied external 
+magnetic fields to the compensated Co/Ni SHNO and are shown in Figure 8. As is the case, the 
+FIG. 8. Auto-oscillation frequency as a function of Idc sweep for compensated Co/Ni SHNO 
+performed at external fields of 165 mT, 300 mT and 500 mT. 
+FIG. 9. (a) Linewidth as a function of Idc sweep at Hext = 180 mT (θH = 15 deg, ϕH = 50 deg) for 
+the compensated Co/Ni SHNO with 100 nm width. (b) Comparison of frequency vs Idc sweep 
+when Hext = 180 mT is applied along out-of-plane angles of 22, 30 and 46 deg to the 100 nm 
+compensated Co/Ni SHNO  
+ 
+
+11
+10
+9
+8
+7
+6
+165mT
+5
+300mT
+500 mT
+4
+3
+-2.5
+-3.0
+-3.5
+-4.0
+-4.5
+Idc (mA)a)
+35
+250
+30
+150
+25
+15
+10
+5
+0
+-2.0
+-2.4
+-2.8
+-3.2
+-3.6
+Idc (mA)
+(b)
+6.4
+6.2
+6.0
+Out-of-plane angle
+22deg
+5.8
+30deg
+46deg
+5.6
+5.4
+-1.5
+-2.0
+-2.5
+-3.0
+-3.5
+Idc (mA) 
+ 
+ 
+ 
+14 
+ 
+external fields only change the frequency of the ferromagnetic resonance and not the slope 
+which are nearly zero for the compensated Co/Ni SHNO.  
+To further validate the existence of compensation across different devices, the 
+measurements are repeated on a 100 nm constriction. Figure 9 (a) shows the variation of ∆f vs 
+Idc for this device, performed at Hext = 180 mT (θH = 15 deg, ϕH = 50 deg). The plot indicates a 
+high ∆f for Idc < Ic (=  ̶ 1.8 mA), as shown in the inset, upon which it does not increase 
+significantly at higher currents. A larger ∆f in excess of 5 MHz as opposed to 1.1 MHz is 
+obtained when the width of the constriction is reduced from 200 to 100 nm, which is expected 
+due to a smaller mode volume. We also performed frequency vs Idc for this device at different 
+orientations of the external magnetic field (Hext = 180 mT). The measurements are performed 
+for three different angles, θH = 22, 30 and 46 degrees, respectively keeping ϕH fixed at 90 deg. 
+Figure 9 (b) shows the results of frequency vs Idc at different out-of-plane angles of the external 
+field. At each angle, the frequency is different as expected, and is minimum at 46 deg which is 
+close to the cone angle of precession. However, the frequency remains nearly constant with 
+respect to Idc, even at different angles, thus providing a strong evidence for the absence of non-
+linearity in the compensated SHNO device.  
+ 
+ 
+IV. CONCLUSION 
+In summary, we experimentally demonstrate a strong reduction of non-linearity in the 
+magnetization dynamics of an SHNO by compensation of its effective magnetic anisotropy. 
+Co/Ni multilayers with a Pt heavy metal form the system for the study. The thicknesses of Co 
+and Ni are tuned to change the magnetization anisotropy, which is estimated using spin-diode 
+measurements. An easy cone anisotropy is obtained for the compensated stack when the PMA 
+is counterbalanced by the demagnetization field. The relation between the second and the first 
+order anisotropy fields thus obtained, satisfies the condition for the existence of an easy cone. 
+The spin-diode signal is shown to be independent of the input power as required to operate as 
+a synapse in neuromorphic computing applications. Auto-oscillations in the SHNO are 
+examined using nano-constrictions fabricated from the compensated stacks and are compared 
+with the emission spectra of Py/Pt based SHNO with an in-plane anisotropy. The frequency and 
+the linewidth are found to be independent of the applied dc current for the compensated SHNO 
+even at different external fields and orientations. The linewidth obtained is as low as 1.1 MHz 
+and the peak emission power is as high as 1 pW/MHz. Thus, the compensated SHNO can 
+operate as an artificial neuron with a fixed frequency and a low linewidth. This study opens up 
+
+ 
+ 
+ 
+ 
+15 
+ 
+a possibility of realizing neuromorphic applications such as frequency multiplexing in a 
+multiply-and-accumulate (MAC) operation, and spike-based neurons exploiting easy-plane 
+oscillations in a compensated SHNO.   
+ 
+ACKNOWLEDGMENTS 
+This work is supported by the Agence Nationale de la Recherche Project ANR-20-CE24-0002 
+(SpinSpike). J.G. and D. H. S. acknowledge support from Q-MEEN-C, an Energy Frontier 
+Research Center funded by the U. S. Department of Energy, Office of Science, Basic Energy 
+Science, under Grant No. DE-SC0019273, for work on neuromorphic computing with SHNO. 
+ 
+[1] 
+J. Torrejon et al., Neuromorphic Computing with Nanoscale Spintronic Oscillators, 
+Nature 547, 428 (2017). 
+[2] 
+M. Zahedinejad, H. Fulara, R. Khymyn, A. Houshang, M. Dvornik, S. Fukami, S. Kanai, 
+H. Ohno, and J. Åkerman, Memristive Control of Mutual Spin Hall Nano-Oscillator 
+Synchronization for Neuromorphic Computing, Nat Mater 21, 81 (2022). 
+[3] 
+M. Zahedinejad, A. A. Awad, S. Muralidhar, R. Khymyn, H. Fulara, H. Mazraati, M. 
+Dvornik, and J. Åkerman, Two-Dimensional Mutually Synchronized Spin Hall Nano-
+Oscillator Arrays for Neuromorphic Computing, Nat Nanotechnol 15, 47 (2020). 
+[4] 
+M. Romera et al., Vowel Recognition with Four Coupled Spin-Torque Nano-Oscillators, 
+Nature 563, 230 (2018). 
+[5] 
+A. Slavin and V. Tiberkevich, Nonlinear Auto-Oscillator Theory of Microwave 
+Generation by Spin-Polarized Current, IEEE Trans Magn 45, 1875 (2009). 
+[6] 
+A. S. Jenkins, L. S. E. Alvarez, P. P. Freitas, and R. Ferreira, Digital and Analogue 
+Modulation and Demodulation Scheme Using Vortex-Based Spin Torque Nano-
+Oscillators, Sci Rep 10, (2020). 
+[7] 
+H. S. Lee, S. H. Kim, T. H. Jang, H. G. Park, B. C. Min, S. Y. Park, and C. S. Park, 
+Power-Efficient Spin-Torque Nano-Oscillator-Based Wireless Communication with 
+CMOS High-Gain Low-Noise Transmitter and Receiver, IEEE Trans Magn 55, (2019). 
+[8] 
+R. Sharma, R. Mishra, T. Ngo, Y. X. Guo, S. Fukami, H. Sato, H. Ohno, and H. Yang, 
+Electrically Connected Spin-Torque Oscillators Array for 2.4 GHz WiFi Band 
+Transmission and Energy Harvesting, Nat Commun 12, (2021). 
+[9] 
+A. Litvinenko et al., Ultrafast Sweep-Tuned Spectrum Analyzer with Temporal 
+Resolution Based on a Spin-Torque Nano-Oscillator, Nano Lett 20, 6104 (2020). 
+[10] A. Litvinenko, P. Sethi, C. Murapaka, A. Jenkins, V. Cros, P. Bortolotti, R. Ferreira, B. 
+Dieny, and U. Ebels, Analog and Digital Phase Modulation and Signal Transmission 
+with Spin-Torque Nano-Oscillators, Phys Rev Appl 16, (2021). 
+
+ 
+ 
+ 
+ 
+16 
+ 
+[11] S. Tsunegi, T. Taniguchi, K. Nakajima, S. Miwa, K. Yakushiji, A. Fukushima, S. Yuasa, 
+and H. Kubota, Physical Reservoir Computing Based on Spin Torque Oscillator with 
+Forced Synchronization, Appl Phys Lett 114, (2019). 
+[12] M. Romera et al., Binding Events through the Mutual Synchronization of Spintronic 
+Nano-Neurons, Nat Commun 13, (2022). 
+[13] A. Ross et al., Multilayer Spintronic Neural Networks with Radio-Frequency 
+Connections, ArXiv Preprint ArXiv:2211.03659 (2022). 
+[14] N. Leroux, D. Marković, E. Martin, T. Petrisor, D. Querlioz, A. Mizrahi, and J. Grollier, 
+Radio-Frequency Multiply-and-Accumulate Operations with Spintronic Synapses, Phys 
+Rev Appl 15, (2021). 
+[15] D. Marković, M. W. Daniels, P. Sethi, A. D. Kent, M. D. Stiles, and J. Grollier, Easy-
+Plane Spin Hall Nano-Oscillators as Spiking Neurons for Neuromorphic Computing, 
+Phys Rev B 105, (2022). 
+[16] S. Jiang, R. Khymyn, S. Chung, T. Q. Le, L. H. Diez, A. Houshang, M. Zahedinejad, D. 
+Ravelosona, and J. Åkerman, Reduced Spin Torque Nano-Oscillator Linewidth Using 
+He + Irradiation, Appl Phys Lett 116, (2020). 
+[17] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Conversion of Spin Current into Charge 
+Current at Room Temperature: Inverse Spin-Hall Effect, Appl Phys Lett 88, (2006). 
+[18] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Spin-Torque Ferromagnetic 
+Resonance Induced by the Spin Hall Effect, Phys Rev Lett 106, 36601 (2011). 
+[19] J. E. Hirsch, Spin Hall Effect, Phys Rev Lett 83, 1834 (1999). 
+[20] H. Mazraati, S. Chung, A. Houshang, M. Dvornik, L. Piazza, F. Qejvanaj, S. Jiang, T. 
+Q. Le, J. Weissenrieder, and J. Åkerman, Low Operational Current Spin Hall Nano-
+Oscillators Based on NiFe/W Bilayers, Appl Phys Lett 109, 242402 (2016). 
+[21] V. E. Demidov, S. Urazhdin, A. Zholud, A. v. Sadovnikov, and S. O. Demokritov, 
+Nanoconstriction-Based Spin-Hall Nano-Oscillator, Appl Phys Lett 105, (2014). 
+[22] B. Divinskiy, V. E. Demidov, A. Kozhanov, A. B. Rinkevich, S. O. Demokritov, and S. 
+Urazhdin, Nanoconstriction Spin-Hall Oscillator with Perpendicular Magnetic 
+Anisotropy, Appl Phys Lett 111, (2017). 
+[23] B. Divinskiy, S. Urazhdin, S. O. Demokritov, and V. E. Demidov, Controlled Nonlinear 
+Magnetic Damping in Spin-Hall Nano-Devices, Nat Commun 10, (2019). 
+[24] A. Okada, S. Kanai, S. Fukami, H. Sato, and H. Ohno, Electric-Field Effect on the Easy 
+Cone Angle of the Easy-Cone State in CoFeB/MgO Investigated by Ferromagnetic 
+Resonance, Appl Phys Lett 112, (2018). 
+[25] B. M. S. Teixeira, A. A. Timopheev, N. Caçoilo, S. Auffret, R. C. Sousa, B. Dieny, and 
+N. A. Sobolev, Stabilization of the Easy-Cone Magnetic State in Free Layers of Magnetic 
+Tunnel Junctions, Phys Rev B 100, (2019). 
+
+ 
+ 
+ 
+ 
+17 
+ 
+[26] Y. Fu, I. Barsukov, J. Li, A. M. Gonçalves, C. C. Kuo, M. Farle, and I. N. Krivorotov, 
+Temperature Dependence of Perpendicular Magnetic Anisotropy in CoFeB Thin Films, 
+Appl Phys Lett 108, (2016). 
+[27] G. H. O. Daalderop, P. J. Kelly, and F. J. A. den Broeder, Prediction and Confirmation 
+of Perpendicular Magnetic Anisotropy in Co/Ni Multilayers, Phys Rev Lett 68, 682 
+(1992). 
+[28] P. Sethi, C. Murapaka, G. J. Lim, and W. S. Lew, In-Plane Current Induced Domain 
+Wall Nucleation and Its Stochasticity in Perpendicular Magnetic Anisotropy Hall Cross 
+Structures, Appl Phys Lett 107, 192401 (2015). 
+[29] S. Fukami, Y. Nakatani, T. Suzuki, K. Nagahara, N. Ohshima, and N. Ishiwata, Relation 
+between Critical Current of Domain Wall Motion and Wire Dimension in 
+Perpendicularly Magnetized Co/Ni Nanowires, Appl Phys Lett 95, 232504 (2009). 
+[30] N. Figueiredo-Prestes et al., Magnetization Switching and Deterministic Nucleation in 
+Co/Ni Multilayered Disks Induced by Spin–Orbit Torques, Appl Phys Lett 119, 032410 
+(2021). 
+[31] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. 
+D. Djayaprawira, N. Watanabe, and S. Yuasa, Spin-Torque Diode Effect in Magnetic 
+Tunnel Junctions, Nature 438, 339 (2005). 
+[32] J. G. Choi, J. Park, M. G. Kang, D. Kim, J. S. Rieh, K. J. Lee, K. J. Kim, and B. G. Park, 
+Voltage-Driven Gigahertz Frequency Tuning of Spin Hall Nano-Oscillators, Nat 
+Commun 13, (2022). 
+[33] J.-M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent, 
+Ferromagnetic Resonance Linewidth in Ultrathin Films with Perpendicular Magnetic 
+Anisotropy, Phys Rev B 80, 180415 (2009). 
+[34] A. Okada, S. Kanai, M. Yamanouchi, S. Ikeda, F. Matsukura, and H. Ohno, Electric-
+Field Effects on Magnetic Anisotropy and Damping Constant in Ta/CoFeB/MgO 
+Investigated by Ferromagnetic Resonance, Appl Phys Lett 105, (2014). 
+[35] A. Okada, Y. Takeuchi, K. Furuya, C. Zhang, H. Sato, S. Fukami, and H. Ohno, Spin-
+Pumping-Free Determination of Spin-Orbit Torque Efficiency from Spin-Torque 
+Ferromagnetic Resonance, Phys Rev Appl 12, (2019). 
+[36] Y. Fu, I. Barsukov, J. Li, A. M. Gonçalves, C. C. Kuo, M. Farle, and I. N. Krivorotov, 
+Temperature Dependence of Perpendicular Magnetic Anisotropy in CoFeB Thin Films, 
+Appl Phys Lett 108, (2016). 
+[37] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli, A. Houshang, A. A. Awad, P. 
+Dürrenfeld, B. G. Malm, A. Rusu, and J. Åkerman, Spin-Torque and Spin-Hall Nano-
+Oscillators, Proceedings of the IEEE 104, 1919 (2016). 
+ 
+ 
+