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+arXiv:2301.00064v1  [astro-ph.SR]  30 Dec 2022
+MNRAS 000, 1–11 (2022)
+Preprint 3 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The orbital kinematics of 휂 Carinae over three periastra with a possible
+detection of the elusive secondary’s motion
+Emily Strawn1, Noel D. Richardson1,★ Anthony F. J. Moffat2, Nour Ibrahim1,3,
+Alexis Lane1, Connor Pickett1, André-Nicolas Chené4, Michael F. Corcoran5,6,
+Augusto Damineli7, Theodore R. Gull8,9, D. John Hillier10, Patrick Morris11,
+Herbert Pablo12, Joshua D. Thomas13 Ian R. Stevens14, Mairan Teodoro9, Gerd Weigelt15
+1 Embry Riddle Aeronautical University, Department of Physics and Astronomy, 3700 Willow Creek Road, Prescott, AZ 86301, United States
+2 Département de physique, Université de Montréal, Complexe des Sciences, 1375 Avenue Thérèse-Lavoie-Roux, Montréal (Qc), H2V 0B3, Canada
+3 Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA
+4 NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawai’i, 96720, USA
+5 CRESST & X-ray Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
+6 The Catholic University of America, 620 Michigan Ave., N.E. Washington, DC 20064, USA
+7 Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária, São Paulo, Brasil
+8 Exoplanets & Stellar Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
+9 Space Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD 21218, USA
+10 Department of Physics & Astronomy & Pittsburgh Particle Physics, Astrophysics, & Cosmology Center (PITT PACC),
+University of Pittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260, USA
+11 California Institute of Technology, IPAC, M/C 100-22, Pasadena, CA 91125, USA
+12 American Association of Variable Star Observers, 49 Bay State Road, Cambridge, MA 02138, USA
+13 Department of Physics, Clarkson University, 8 Clarkson Ave, Potsdam, NY 13699, USA
+14 School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
+15 Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The binary 휂 Carinae is the closest example of a very massive star, which may have formed through a merger during its Great
+Eruption in the mid-nineteenth century. We aimed to confirm and improve the kinematics using a spectroscopic data set taken
+with the CTIO 1.5 m telescope over the time period of 2008–2020, covering three periastron passages of the highly eccentric
+orbit. We measure line variability of H훼 and H훽, where the radial velocity and orbital kinematics of the primary star were
+measured from the H훽 emission line using a bisector method. At phases away from periastron, we observed the He ii 4686
+emission moving opposite the primary star, consistent with a possible Wolf-Rayet companion, although with a seemingly narrow
+emission line. This could represent the first detection of emission from the companion.
+Key words: techniques: spectroscopic — stars: massive — stars: variables: S Doradus — stars: winds, outflows — binaries:
+spectroscopic — stars: individual: 휂 Carinae
+1 INTRODUCTION
+The binary star system 휂 Carinae is known for being one of
+the most massive and luminous binaries in our local galaxy
+(Davidson & Humphreys 2012). The two stars are locked in a highly
+eccentric orbit (Damineli 1996a; Damineli et al. 1997). Envelop-
+ing these stars is the Homunculus nebula which was formed by
+a large eruption in the mid-nineteenth century (e.g., Currie et al.
+1996). The Great Eruption that formed the Homunculus nebula was
+recently modeled to be the product of a binary merger in a triple sys-
+tem leading to the current orbit (Portegies Zwart & van den Heuvel
+2016; Hirai et al. 2021), supported by light echo observations (e.g.,
+Smith et al. 2018) and an extended central high-mass torus-like struc-
+★ E-mail: [email protected]
+ture surrounding the central binary (Morris et al. 2017). In this sce-
+nario, the luminous blue variable primary star is currently orbited
+by a secondary star that is a classical Wolf-Rayet star, as discussed
+by Smith et al. (2018). The system began as a hierarchical triple,
+and mass transfer led to the initial primary becoming a hydrogen-
+deficient Wolf-Rayet star. Mass transfer causes the orbits to become
+unstable, which leads to the merger and leaves behind the highly
+eccentric binary system we see today. An alternate model for the
+eruption relies on the fact that 휂 Car is a binary in a highly eccentric
+orbit, and proposes that the periastron events triggered large mass
+transfer events that caused the eruptions (Kashi & Soker 2010). A
+similar model was used to explain the much less massive eruption
+that was seen from the SMC system HD 5980 during its LBV-like
+outburst (e.g., Koenigsberger et al. 2021).
+While the binary nature of the system was inferred by Damineli
+© 2022 The Authors
+
+2
+Strawn et al.
+4830
+4840
+4850
+4860
+4870
+4880
+4890
+4900
+WAVELENGTH (ANGSTROMS)
+0
+5
+10
+15
+NORMALIZED FLUX
+FECH, 12.0096 
+GMOS, 12.0096 
+CHIRON, 13.0081 
+CHIRON, 14.0103 
+-1000
+0
+1000
+2000
+VELOCITY (km s-1)
+Figure 1. A comparison of an example Gemini-GMOS spectrum used by
+Grant et al. (2020) with the CTIO data from the fiber echelle (FECH) in
+2009 and with more recent CHIRON data at the same phase (phases given
+in the legend). Note that the pixel sizes are indicated for the spectra, which
+is most obvious for the GMOS spectrum. The spectra are offset by orbital
+cycle, which highlights the complexities in the echelle spectra compared to
+the GMOS data.
+(1996b) and Damineli et al. (1997), the orbit of the system has mostly
+eluded observers since the discovery of the spectroscopic events by
+Damineli (1996a). Davidson (1997) criticized the first orbit pub-
+lished by Damineli et al. (1997) and published a higher eccentricity
+model using the same data as Damineli et al. (1997). Since these
+first attempts to derive the orbital motion of the system, very few
+observationally derived models have appeared in the literature, with
+most references to the orbit being inferred for modeling purposes.
+Recently, Grant et al. (2020) used archival moderate-resolution Gem-
+ini/GMOS spectra from 2009 to fit the hydrogen lines using multi-
+ple, weighted Gaussians to measure radial velocities corrected to
+account for motion from strong stellar winds. They derived a single-
+lined spectroscopic orbit based on the upper Balmer lines to be
+푇0 = 2454848 (HJD), 푒 = 0.91, 퐾1 = 69 km s−1, and 휔pri = 241◦
+with the period of 2022.7 d that has been widely adopted based
+on multi-wavelength observations (e.g., Teodoro et al. 2016). These
+are broadly consistent with the smoothed-particle hydrodynamical
+(SPH) models used to describe variability across the electromag-
+netic spectrum (e.g., Madura et al. 2013) including the X-ray light
+curves (e.g., Okazaki et al. 2008), optical He i absorption variability
+(Richardson et al. 2016), and the near-UV emission observed with
+the Hubble Space Telescope (Madura & Groh 2012).
+While the results of Grant et al. (2020) establish the orbital pa-
+rameters with greater precision to date, there are potential issues
+with the determination of orbital elements from hydrogen lines in
+휂 Car’s spectrum, as the strong wind of the primary causes the ef-
+fective photospheric radius to be further out from the central star
+for lower energy transitions. Indeed, Grant et al. (2020) found better
+results with higher-order Balmer lines than with the optically thick
+H훼 or H훽. This is a known effect for evolved Wolf-Rayet stars, where
+the observed semi-amplitude can change with the ionization poten-
+tial of the line measured because lower-energy emission lines tend
+to form further out in the wind, where they are more likely to be
+perturbed by the companion star as seen in 훾2 Vel (Richardson et al.
+2017). This effect causes differences from the true orbital motion
+for lower energy transitions, making it difficult to determine accu-
+rate orbits (Grant et al. 2020). Grant & Blundell (2022) confirmed
+that their methods used for emission-line stars worked for the WR
+binaries WR 133 and WR 140 that have combined spectroscopic and
+interferometric orbits (Richardson et al. 2021; Thomas et al. 2021).
+The primary star in the 휂 Car system is a luminous blue variable
+star, with the largest measured value for a mass-loss rate for a mas-
+sive star with �푀 = 8.5 × 10−4푀⊙yr−1 and a terminal wind speed
+of 푣∞ = 420 km s−1 (Davidson & Humphreys 1997; Groh et al.
+2012). Prior to the recent kinematic studies of Grant et al. (2020)
+and Grant & Blundell (2022), the best constraints on the compan-
+ion star parameters, while indirect, came from the X-ray variability
+analyses from RXTE, Swift, and NICER observations of the sys-
+tem (Corcoran et al. 2001, 2017; Espinoza-Galeas et al. 2022). These
+analyses point to a secondary star with a mass-loss rate on the order
+of �푀 ∼ 10−5푀⊙yr−1 and a terminal velocity of 푣∞ ∼ 3000 km s−1
+(Pittard & Corcoran 2002). These values are broadly in agreement
+with the suggestion based on the merger models and mass-loss pa-
+rameters that the remaining secondary would be a Wolf-Rayet star.
+Despite recent work with long-baseline near-infrared interferome-
+try by Weigelt et al. (2021), no direct detection of the companion
+star has been made to date. From the interferometric data, a mini-
+mum primary-secondary flux ratio of ∼50 was derived in the 퐾-band
+(Weigelt et al. 2007). Given the extreme luminosity of the LBV pri-
+mary, this is consistent with any O or WR star in the Galaxy.
+The evolution of the secondary star may well have been signifi-
+cantly modified by interactions and mass exchange during formation
+of the present-day binary, but if the current secondary star is a clas-
+sical H-free Wolf-Rayet star as suggested by Smith et al. (2018) and
+Hirai et al. (2021), or a hydrogen-rich WNh star, possibly the best
+line to detect it in the optical would be the He ii 휆 4686 line, which
+is the dominant line in the optical for the nitrogen-rich WR stars, or
+the hydrogen-rich WNh stars. Most of the observations of He ii were
+made near periastron, where the He ii excess can be explained by
+ionization of He i in the colliding winds in a highly eccentric binary.
+Teodoro et al. (2016) showed that the variability could be explained
+with the smoothed-particle hydrodynamics models of Madura et al.
+(2013). Away from periastron (0.04 < 휙 < 0.96), the He ii line is
+typically not observed with moderate resolving power and a nominal
+S/N of ∼100.
+In this paper, we present our analysis of the spectroscopy collected
+with the CTIO 1.5 m telescope and the CHIRON spectrograph, as
+well as the data collected with the previous spectrograph on that
+telescope with the aim of better constraining the kinematics of the
+system. These observations are described in Section 2. In Section
+3, we review the variability in the two Balmer lines we can easily
+measure (H훼 and H훽). Section 4 describes our techniques of mea-
+suring the radial velocity of the H훽 line, and presents observations of
+He ii away from periastron in the hope of determining the orbit of the
+companion star. We discuss our findings in Section 6, and conclude
+this study in Section 7.
+2 OBSERVATIONS
+We collected high resolution spectra of 휂 Carinae during the peri-
+astron passages of 2009, 2014, and 2020. Many additional spectra
+were taken in the intermediate phases of the binary orbit as well.
+These were collected from the 1.5m telescope at Cerro Tololo Inter-
+American Observatory (CTIO 1.5) and both current CHIRON and
+the former fiber-fed echelle spectrograph (FECH). The data from the
+2009 spectroscopic event spanned from 2008 October 16 to 2010
+March 28, with approximately one spectrum taken every night be-
+tween 2008 December 18 to 2009 February 19, which were previ-
+MNRAS 000, 1–11 (2022)
+
+The spectroscopic orbit of 휂 Car
+3
+ously used by Richardson et al. (2010, 2015) and cover the spectral
+range ∼ 4700 − −7200Å. These spectra with the fiber echelle1 were
+collected in late 2009 and 2010, and often had a signal-to-noise ratio
+around 80–100 per resolution element with 푅 ∼ 40, 000. In total, we
+analyzed 406 spectra of the system.
+The 2014–2020 data were collected with the new CHIRON spec-
+trograph (Tokovinin et al. 2013), and spanned the time between 2012
+March 2 and 2020 March 16, with high-cadence time-series span-
+ning the 2014 and 2020 periastron passages between 2013 Decem-
+ber 29 through 2015 April 21 as well as between 2020 January 3
+to 2020 March 16 when the telescope shut down for the COVID-19
+pandemic. The CHIRON spectra cover the spectral range of ∼4500-
+8000Å, with some spectral gaps between orders in the red portion
+of the spectrum. The data covering the 2014 periastron passage were
+previously used by both Richardson et al. (2016) and Teodoro et al.
+(2016). These data have a spectral resolution of 푅 ∼ 80, 000 and
+typically have a signal-to-noise of 150–200 in the continuum and
+were all reduced with the CHIRON pipeline, which is most recently
+described by Paredes et al. (2021). In addition to the pipeline reduc-
+tions, we perform a blaze correction using fits from an A0V star,
+as done by Richardson et al. (2016), allowing orders to be merged
+if needed. This process resulted in a flat continuum in regions that
+were line-free.
+These observations were all fiber-fed with the fiber spanning
+2.7′′ on the sky, meaning that the data include the nebular emis-
+sion from the Homunculus nebula formed from the eruption of 휂
+Car in the mid-nineteenth century, as well as the Weigelt knots
+(Weigelt & Ebersberger 1986) that are thought to have originated
+from the second eruption in the 1890s. The CHIRON spectra are
+normalized through a comparison with a measured blaze function
+from the star HR 4468 (B9.5V), as was done in the analysis of
+Richardson et al. (2016). Example spectra are shown in Figure. 1,
+with a comparison to a spectrum used by Grant et al. (2020) and
+Grant & Blundell (2022).
+3 MEASURED VARIABILITY IN THE BALMER LINES, H훼
+AND H훽
+Our observations are unique in providing both the spectral resolution
+and signal-to-noise to measure the line strength (equivalent width)
+and profile morphology of the emitting gas for the H훼 and H훽 lines
+of 휂 Carinae. Here, we detail the observations of the variability of
+the hydrogen lines. We estimate errors on equivalent width using the
+methods of Vollmann & Eversberg (2006). We note that the analysis
+of Richardson et al. (2015) includes many optical wind lines near
+the 2009 periastron passage and phases far from periastron. These
+line profiles all show minimum line strength near periastron as the
+secondary’s high ionizing radiation goes behind the primary star’s
+optically thick wind. We use a phase convention in which the low-
+ionization state observed by Gaviola (1953) in 1948 is deemed to be
+cycle 1, so that the low-ionization state starting in Feb. 2020 marks
+the start of cycle 14. We leave the kinematics analysis of the metal
+lines for a future analysis in order to confirm the results of Grant et al.
+(2020) and Grant & Blundell (2022) here, with plans of using higher
+signal-to-noise spectra in a future analysis.
+1 http://www.ctio.noao.edu/noao/content/CHIRON
+3.1 H훼
+Richardson et al. (2010) examined the variability of the H훼 profile
+of 휂 Carinae across the 2009 periastron passage. They found that
+the profile’s strength decreased during the periastron passage and
+reached a minimum a few days following the X-ray minimum. They
+postulated that the changes were caused by the drop in the ionizing
+flux from the secondary when the companion moved to the far side. In
+addition, they observed an appearance of a P Cygni absorption profile
+and an absorption component at −145 km s−1, that also appeared as
+the secondary’s ionizing radiation was blocked by the primary star’s
+optically thick wind. Richardson et al. (2015) expanded upon this
+model to describe the variations of the optical He i profiles while
+documenting the variability of the optical wind lines across the 2009
+periastron passage.
+We measured the equivalent width of H훼 for all of our spectra in
+the range 6500 – 6650Å. These results are shown in Fig. 2, where
+we show the measurements both compared to time and to binary
+phase, assuming a period of 2022.7 d, and the epoch point given by
+Teodoro et al. (2016), which represents the time of the periastron pas-
+sage based on a comparison of the He ii observations (Teodoro et al.
+2016) to SPH models of the colliding winds. Broadly speaking, the
+strength of the line relative to the locally normalized continuum
+shows a fast decrease and recovery near each periastron passage.
+Richardson et al. (2010) found that the variability is smoother when
+considering the photometric flux in the determination of the equiv-
+alent widths. We did not make this correction in these data, but do
+see the similarities of the events in the context of the raw equivalent
+widths.
+There is no strong long-term variability in these observations,
+and the 2014 and 2020 observations were nearly identical in their
+variations. Recently, Damineli et al. (2019, 2021) found that there
+are long-term brightness and spectral changes of the system that has
+been ongoing for decades and accelerated since the mid-1990s, but
+now seems to be stabilizing. The shape of the H훼 variability has
+remained similar over these three well-observed periastron passages,
+and the line strength has stabilized across the past two cycles, which
+could indicate that the system is mostly stable aside from the binary-
+induced variability.
+Richardson et al. (2010) also documented the timing of the ap-
+pearance of the P Cygni absorption component for H훼. In the 2009
+observations we see the absorption occurring at approximately HJD
+2454840.7 (휙 ≈ 12.00) and still persisting through the last observa-
+tion, 2454881.7 (휙 ≈ 12.02). In 2014 a P Cygni absorption occurs at
+2456874.5 (휙 ≈ 13.00) persisting until the object was not observable
+at HJD 2456887.5 (휙 ≈ 13.01). In 2020, the absorption is seen at
+2458886.8 (휙 ≈ 14.01) and still detected through the last observation
+on HJD 2458925.0 (휙 ≈ 14.02).
+A narrow absorption component was observed near −145 km s−1
+in the 2009 observations (Richardson et al. 2010) from 2454836.7
+(휙 ≈ 12.00) through the last day of observation, 2454881.7 (휙 ≈
+12.02). In 2014 an absorption in the same location is observed from
+2456863.5 (휙 ≈ 13.00) – 2456977.8 (휙 ≈ 13.06). There is no ab-
+sorption at this location strong enough to make a definitive detection
+in 2020. Pickett et al. (2022) documented the changes in absorp-
+tion behavior for the Na D complex at these velocities, showing
+that the absorption from these components associated with the Little
+Homunculus, formed during the second eruption in the 1890s, are
+weakening with time and moving to bluer velocities.
+MNRAS 000, 1–11 (2022)
+
+4
+Strawn et al.
+2010
+2012
+2014
+2016
+2018
+2020
+Year
+1000
+2000
+3000
+4000
+5000
+HJD-2454000
+200
+300
+400
+500
+600
+-Wλ (ANGSTROMS)
+CHIRON
+FECH
+0.900
+0.925
+0.950
+0.975
+1.000
+1.025
+1.050
+1.075
+1.100
+ORBITAL PHASE ϕ
+250
+300
+350
+400
+450
+500
+550
+600
+650
+-Wλ (ANGSTROMS)
+11→12
+12→13
+13→14
+Figure 2. Variation in H훼 emission line with respect to time (left) and phase (right); with the data taken between October 2008 and March 2020. Data taken
+from the previous echelle spectrograph is indicated by open squares and data from the new CHIRON spectrograph is indicated by solid dots. In the phase plot,
+we show the different cycles in different colors to clarify the timing of each data set. Furthermore, the errors are typically the size of the points or smaller. The
+phase convention shown in the right panel references the low-ionization spectrum near periastron first observed by Gaviola (1953).
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+Year
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+HJD-2454000
+70
+80
+90
+100
+110
+120
+130
+-Wλ (Angstroms)
+0.900
+0.925
+0.950
+0.975
+1.000
+1.025
+1.050
+1.075
+1.100
+ORBITAL PHASE ϕ
+70
+80
+90
+100
+110
+120
+-Wλ (ANGSTROMS)
+12→13
+13→14
+Figure 3. Variation in H훽 emission line with respect to time (left) and phase (right); with the data taken with CHIRON spectrograph as the FECH data were too
+noisy to determine equivalent widths. In the phase plot, we show the two recent cycles in different colors to clarify the timing of each data set. Furthermore, the
+errors are typically the size of the points or smaller. The phase convention shown in the right panel references the low-ionization spectrum near periastron first
+observed by Gaviola (1953).
+3.2 H훽
+While some of the H훽 variability was documented for the 2009
+periastron passage of 휂 Car by Richardson et al. (2015), the full
+variability and timing of the changes is still not well documented
+in the literature. The lack of a more quantitative assessment of the
+variability is in part due to the lower signal-to-noise in the H훽 data
+from the 2009 event. Similar to the H훼 profile, H훽 experiences a P
+Cygni type absorption near −500 km s−1 near periastron. We note the
+absorption appears in 2009 at approximately HJD 2454837.7 (휙 ≈
+12.00) and persist through the last observation taken on 2454879.7
+(휙 ≈ 12.01). In 2014, it appears at approximately 2456863.6 (휙 ≈
+13.00) and ends during a seasonal gap in observations beginning at
+2456887.5 (휙 ≈ 13.01). In 2020, the P Cygni absorption is observed
+between 2458886.8 (휙 ≈ 14.00) and continues through the last day
+of observations on 2458925.0 (휙 ≈ 14.02). This transient absorption
+was determined to be originating from the downstream bowshock by
+Gull et al. (2022).
+A
+narrow
+absorption
+component,
+previously
+observed
+by
+Richardson et al. (2015), is detected near −145 km s−1 in the 2009
+observations from 2454837.7 (휙 ≈ 12.00) and proceeds through the
+end of observations on 2452879.7 (휙 ≈ 12.01). In 2014, this absorp-
+tion is observed between 2456864.5 (휙 ≈ 13.00) and also persists
+through the last day of observations 2456887.5 (휙 ≈ 13.01). As
+with H훼, there is no discernible absorption at −145 km s−1 in 2020
+observations.
+Figure 3 shows the time series variation in the H훽 equivalent width
+over the last two periastron cycles. We note that the 2009 observa-
+tions are not included as they are recorded with the former echelle
+spectrograph and have lower signal-to-noise, though the appearance
+of the P Cygni absorption remains reliable. As with the H훼 equiva-
+lent widths, there is a consistency in the decrease in equivalent width
+for the time period corresponding to times close to periastron.
+4 LINE KINEMATICS
+We measured the bisector velocity of H훽 and the centroid position of
+the He ii 휆4686 line. H훽 measurements were taken during the 2009,
+2014, and 2020 periastron events and the He ii 4686 measurements
+MNRAS 000, 1–11 (2022)
+
+The spectroscopic orbit of 휂 Car
+5
+0
+5
+10
+11.947
+11.998
+12.212
+0
+5
+10
+12.901
+13.007
+13.172
+−750 −500 −250
+0
+250
+500
+750
+0
+5
+10
+13.946
+−750 −500 −250
+0
+250
+500
+750
+14.004
+−750 −500 −250
+0
+250
+500
+750
+14.019
+Radial Velocity (km s−1)
+Normali ed Flux
+Figure 4. Example polynomial fits to H훽 emission lines from 2009, 2014, and 2020 periastron events. The profiles are shown in black with the portion of the
+line wings fit with a polynomial shown in red. The bisector velocity is shown as a vertical line corresponding to the normalized flux at the same level as the
+measurements. Near the edges of these ranges, the bisector often appears to curve due to either profile asymmetries or larger errors in the polynomial fits. The
+bisector velocities between normalized flux levels of 5 and 6, indicated by the dashed lines, were averaged to obtain a final relative velocity for each day. Further
+details are given in Section 4.1.
+were taken for 2014 and 2018 and do not include time within 휙 =
+0.95 – 1.05 to avoid observations affected by periastron caused by
+colliding-wind effects which, to first order, behave with a 퐷−1 trend
+for adiabatic and 퐷−2 or steeper for radiative conditions, where 퐷
+is the orbital separation, which is small and quickly changing at
+periastron. Teodoro et al. (2016) show the behavior of the He ii 4686
+line near periastron in detail. All measurements are tabulated in
+online supplementary data.
+4.1 Bisector velocities of H훽
+The process used to find the bisector velocity of H훽 is demon-
+strated in Fig. 4. Grant et al. (2020) and Grant & Blundell (2022)
+used a method of Gaussian decomposition using many components
+to moderate-resolution spectroscopy taken with Gemini-South and
+GMOS. Their GMOS spectra of 휂 Car are limited in that the highest
+resolving power available is ∼ 4400, whereas our spectroscopy has a
+resolving power of 40, 000 from the fiber echelle, and 80, 000 for the
+CHIRON data. The profiles become more complex at higher spectral
+resolution, making this multiple-Gaussian method more difficult to
+implement, likely requiring more than twice as many components
+compared to the work of Grant et al. (2020).
+In order to create a simpler measurement that has reproducible
+results for any spectroscopic data set, we implemented a bisector
+technique. We began by fitting two fourth degree polynomials, one
+to the red side and another on the blue side of the profile in order to
+smooth over any noise inherent in the data. Through this fit, we were
+then able to establish the bisecting velocity position at each emission
+level with higher precision. Example fits are shown in red in Fig. 4. In
+the regions of heights of 4× the continuum up to 10× the continuum,
+we calculate the bisecting velocity. This area was chosen based on
+the relatively vertical nature of the bisector in this region. We then
+created comparisons of all spectra and found that the bisecting line
+was nearly always vertical in the region of 5 − 6× the normalized
+continuum. We therefore used this region, measuring the velocity at
+every 0.1 increment between these values, and adopting an average
+measurement as the radial velocity for the spectrum. The choice
+of a common emission height with which to measure the bisector
+velocities allows us confidence in the results as it would relate to
+gas emitting from the same region for all spectra, whether the line is
+weak or strong in that particular observation. The resulting velocities
+MNRAS 000, 1–11 (2022)
+
+6
+Strawn et al.
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+Year
+0
+1000
+2000
+3000
+4000
+5000
+HJD-2454000
+−80
+−60
+−40
+−20
+0
+20
+40
+60
+Velocity (km s−1)
+FECH
+CHIRON
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+Phase
+−80
+−60
+−40
+−20
+0
+20
+40
+60
+Velocity (km s−1)
+11→12
+12→13
+13→14
+Figure 5. Radial velocities from H훽 bisector measurements compared to time (left) and orbital phase (right). The orbital fit is described in Section 4.3 and
+typical errors are on the order of the size of the points.
+−400
+−200
+0
+200
+400
+Velocity (km s−1)
+0.98
+0.99
+1.00
+1.01
+1.02
+1.03
+Normalized Flux
+ϕ = 13.08
+Figure 6. Gaussian fit to an example He ii emission line with a vertical line
+plotted at the fitted peak. This particular spectrum had a signal-to-noise ratio
+of 210 per resolution element.
+are shown in Fig. 5. We provide this bisector code via GitHub2 for
+future use on comparable datasets.
+4.2 He ii 휆4686
+The region surrounding, but not blended with, the He ii 휆4686 tran-
+sition is complicated by several features including narrow emission
+lines from the Weigelt knots (Weigelt & Ebersberger 1986) along
+with wind emission from Fe ii and He i lines (for a figure showing
+that region of the spectrum, see Teodoro et al. 2016). While these
+do not directly overlap with the core of the He ii line, they can
+complicate this fitting if not properly avoided. The He ii 휆4686 line
+has usually been observed near periastron passage when the line
+is dominated by the wind-wind collisions, which has been docu-
+mented and modeled by Teodoro et al. (2016). The line was discov-
+ered by Steiner & Damineli (2004). Since then, multiple studies have
+attempted to explain the formation of the stronger line observed near
+periastron (퐿He ii ∼ 300 퐿 ⊙; Martin et al. 2006; Mehner et al. 2011,
+2015; Teodoro et al. 2012; Davidson et al. 2015), but the colliding
+2 https://github.com/EmilysCode/Radial-Velocity-from-a-Polynomial-Fit-
+Bisector.git
+wind model best reproduces the emission near periastron. This emis-
+sion is strongest for times within ±0.05 in phase from periastron, as
+detailed in the recent analysis of Teodoro et al. (2016).
+Outside of the phase intervals near periastron, the He ii 휆4686
+line could only be properly observed with high spectral resolution
+and high signal-to-noise data (Teodoro et al. 2016). Our data taken
+with CHIRON, after the 2014 periastron passage has the necessary
+sensitivity to detect this notably weak emission line. We measure the
+radial velocity of this line outside of 휙 = ±0.05 of periastron, so that
+it minimizes the effects of the colliding winds that peak at periastron.
+As shown in Fig. 6, we fit a Gaussian to the He ii emission line
+and use the centroid position to determine the radial velocity. Un-
+fortunately, the continuum placement for the feature is not reliable
+enough to measure equivalent widths with precision, but the line
+was nearly constant in equivalent width when considering the errors
+of these measurements. Before fitting the 2018 observations near
+apastron, we needed to average up to ten observations to improve
+the signal-to-noise ratio. The resulting velocities are shown in Fig. 7
+with a resulting total of 19 data points. The averaging of the points
+from the 2018 data resulted in a smaller dispersion of the data than
+seen in the earlier points.
+The He ii line is normally absent in the spectra of luminous blue
+variables. The extreme mass-loss rate of 휂 Car does not preclude
+this emission line originating in the primary star’s wind, as there
+are some combinations of parameters used that can create this weak
+emission feature in CMFGEN models. These models and parameters
+are very sensitive and depend on the mass-loss rate and stellar radii
+used. The He ii can be formed through strong wind collisions at
+times close to periastron (e.g., Teodoro et al. 2016). However, this
+line moves in opposition to the primary star’s motion, so we consider
+this feature as originating from the companion during these phases
+far from periastron for the remainder of this analysis.
+4.3 Orbital Kinematics and Observed Elements
+We began our fit of the kinematics of the primary star with the
+BinaryStarSolver software (Milson et al. 2020; Barton & Milson
+2020). The resulting orbit is broadly in agreement with the orbit
+derived with H훽 velocities by Grant et al. (2020), with the orbital
+elements given in Table 1. Our resulting fits are in agreement with
+those of Grant et al. (2020) so we did not perform the same correction
+for the stellar wind effects as in their analysis.
+MNRAS 000, 1–11 (2022)
+
+The spectroscopic orbit of 휂 Car
+7
+Line
+푇0 (HJD-2400000)
+푒
+퐾 (km s−1)
+휔 (degrees)
+훾 (km s−1)
+Source
+Pa훾
+48800 ± 33
+0.63 ± 0.08
+53±6
+286 ±6
+−15 ± 3
+Damineli et al. (1997)
+Pa훾, He i 6678
+48829± 8
+0.802 ± 0.033
+65.4 ± 3.5
+286 ± 8
+-12.1 ± 2.7
+Davidson (1997)
+Pa훾, Pa훿
+50861
+0.75
+50
+275
+-12
+Damineli et al. (2000)
+H훽
+54854.9 +4.5
+−4.1
+0.82 ±0.02
+53.0 +2.1
+−1.9
+254 ±4
+-25.5 ±2.0
+Grant et al. (2020)
+All Balmer lines
+54848.3 ±0.4
+0.91 ±0.00
+69.0 ±0.9
+241 ±1
+. . .
+Grant et al. (2020)
+Upper Balmer lines
+54848.4 ±0.4
+0.89 ±0.00
+69.9 ±0.8
+246 ±1
+. . .
+Grant et al. (2020)
+H훽
+56912.2 ±0.3
+0.8100 ±0.0007
+58.13 ±0.08
+251.43 ±0.19
+6.34 ±0.10
+This work(BinaryStarSolver)
+H훽
+56927.4 ±0.5
+0.8041 ±0.0008
+54.6±0.2
+260.6 ±0.2
+4.83 ±0.09
+This work (PHOEBE)
+He ii
+56973.5 ±0.2
+0.937 ±0.001
+129.5±5.0
+80.6 (fixed)
+63.1 ±0.4
+This work
+Table 1. Orbital elements from previous publications and the results from this work. For the orbits of Grant et al. (2020), Grant & Blundell (2022), and our
+work, the period has been held constant at 2022.7 d, while it was fit in the earlier work of Damineli et al. (1997), Davidson (1997), and Damineli et al. (2000)
+with periods that agree with 2022.7 d within their errors. Note that our errors from the PHOEBE code may be underestimated, especially for the He ii line (see
+text for details).
+6500
+7000
+7500
+8000
+8500
+9000
+HJD - 2,450,000
+-50
+0
+50
+100
+150
+200
+RADIAL VELOCITY (km s-1)
+2014
+2016
+2018
+2020
+YEAR
+Figure 7. Radial velocity as determined using centroid positions in He ii
+emission at phases away from periastron during 2014–2018 with CHIRON.
+We overplotted the He ii orbit from Table 1, along with the H훽 solution from
+our work shifted to the same 훾-velocity as the He ii orbit as a grey dashed
+line.
+In an attempt to fully assess the errors of the parameters, we used
+the PHOEBE code (PHysics Of Eclipsing BinariEs; Prša & Zwitter
+2005; Prša et al. 2016) to verify the orbital elements. The latest
+version of PHOEBE incorporates the Markov Chain Monte Carlo
+package emcee (Foreman-Mackey et al. 2013). Unlike traditional or-
+bit fitting routines, PHOEBE fits using the variable of the projected
+semi-major axis (푎 sin 푖) rather than the semi-amplitude 퐾, but these
+are easily interchangeable using
+푎 sin 푖 = (1 − 푒2)1/2
+2휋
+퐾푃.
+These orbital elements are also similar to the other published
+orbital elements measured with H훽, and the resulting orbit is shown in
+Fig. 5. The distribution of the errors from the Monte Carlo simulation,
+shown in Fig. 8, is tightly constrained but shows that various orbital
+elements have errors that are interdependent with other parameters.
+While this represents the best solution to the entire data set, we
+explored how the parameters change if we kept only the densest of
+the three periastra observed (the 2014 event). Running the PHOEBE
+code with the MCMC package on just those data resulted in the
+eccentricity being slightly larger (푒 = 0.824), the time of periastron
+being later (HJD 2,456,935.31), and the value of 푎 sin 푖 (hence 퐾1)
+being slightly larger at 2620.4 푅⊙. These values are outside the
+limits given with our MCMC fit of all of the data, so we caution that
+the errors in Table 1 are likely underestimated. We include the fit
+parameters in the same style as Fig. 8 in the online Fig. A1.
+Once the orbital elements for H훽 were fit, we proceeded to run a
+simpler model for the He ii emission. For this PHOEBE model, we
+keep 휔 constant to that representing the primary star from the upper
+Balmer line results from Grant et al. (2020). However, we do allow
+the semi-major axis, 훾-velocity, 푒, and time of periastron passage to
+vary. The resulting orbit is more eccentric than that of the primary star
+when derived using H훽 (and a bit more eccentric than the Grant et al.
+(2020) solution) and is shown in Fig. 7. With future observations of
+the He ii line at times away from periastron, a combined double-lined
+orbit of the system with 휔 being consistent for the two stars will be
+possible.
+5 DISCUSSION
+The optical spectrum of 휂 Car is dominated by emission lines from
+the wind of the primary and its ejecta. The dominant emission lines
+are the hydrogen Balmer lines, but there are strong lines from He i
+and Fe ii in the spectrum as well. The He i lines, when considered
+in non-LTE stellar wind models, are a strong function of the adopted
+value of the stellar radius. However, if most of the He i emission
+comes from the colliding wind interaction region, it forces a larger
+stellar core radius value for the primary star, ∼ 120푅⊙ in the pre-
+ferred models (see Hillier et al. 2001, for many further details). The
+model of Groh et al. (2012) improved previous spherically symmet-
+ric models of Hillier et al. (2001) in that the spectrum was modeled
+with a cavity carved from the wind of the secondary, which was in-
+cluded along with a central occulter or “coronagraph" that extended
+∼ 0.033′′ to allow for stronger He i emission, and better agreement
+for the P Cygni absorption lines. Given the spectral modeling agree-
+ment for the spectroscopically similar star HDE 316285 (Hillier et al.
+1998), the strong disagreements for the absorption components and
+He i lines led to an interpretation that the He i lines are formed in
+the wind-wind collision region of the system (Nielsen et al. 2007).
+Indeed, the P Cygni absorption component variability of the optical
+He i lines seems to represent the outflowing shocked gas from the
+wind-wind collision region (Richardson et al. 2016). These results
+all indicate that the best lines in the optical for determination of the
+orbit may indeed be the upper hydrogen Balmer lines, even if they
+are likely modified by the wind collisions.
+All of the measured orbits, including ours, rely on measurements
+taken when the line profiles are most variable near periastron. This
+MNRAS 000, 1–11 (2022)
+
+8
+Strawn et al.
+2595.0+4.0
+−4.0 R⊙
+4.50
+4.65
+4.80
+4.95
+5.10
+vγ (km
+s )
+4.83+0.09
+−0.09
+km
+s
+0.8020
+0.8035
+0.8050
+0.8065
+ebinary
+0.8041+0.0008
+−0.0008
+6
+7
+8
+9
+t0, perpass, binary (d)
++2.45692e6
+2456927.4+0.5
+−0.5 d
+2584
+2592
+2600
+2608
+abinarysinibinary (R )
+259.9
+260.2
+260.5
+260.8
+261.1
+ω0, binary (∘)
+4.50
+4.65
+4.80
+4.95
+5.10
+vγ (km
+s )
+0.8020
+0.8035
+0.8050
+0.8065
+ebinary
+6
+7
+8
+9
+t0, perpass, binary (d)
++2.45692e6
+259.9
+260.2
+260.5
+260.8
+261.1
+ω0, binary (∘)
+260.6+0.2
+−0.2
+∘
+Figure 8. Results of the Markov chain Monte Carlo fit for the H훽 velocities. Note that 휔0 refers to the value of 휔 for the primary star.
+likely causes additional errors in the parameters derived, but we
+tried to always sample emission from the same line formation re-
+gion by taking bisector velocities at the same height. Furthermore,
+our technique produces nearly the same orbital elements as those
+from Grant et al. (2020) in the case of H훽. Grant et al. (2020) pro-
+ceeded to correct the orbital elements by considering the effects of
+the outflowing wind.
+These results all show that the system is a long-period and highly
+eccentric binary where the primary star is in front of the secondary
+at periastron, causing the ionization in our line of sight to drop
+during the “spectroscopic events" due to a wind occultation of the
+secondary at these times. The results of Grant et al. (2020) show that
+the higher-order Balmer lines give different results than that of the
+lower-level lines such as H훼 or H훽, which is expected as the higher
+level lines form deeper in the wind (e.g. Hillier et al. 2001). As such,
+the results of Grant et al. (2020) and Grant & Blundell (2022) should
+be considered the best for the primary star at the current time. Similar
+differences in the orbital kinematics is sometimes inferred for Wolf-
+Rayet stars (e.g., 훾2 Vel; Richardson et al. 2017).
+Despite the detection of the He ii 휆4686 emission at times near
+apastron by Teodoro et al. (2016), the exact formation channel for this
+line remains unclear. The emission lines in colliding wind binaries
+often vary as a function of the orbit due to the colliding wind line
+excess (e.g., Hill et al. 2000), and the modeling of these variations
+has been done in the context of the so-called Lührs model (Lührs
+1997). Recently, the excess emission was observed to be a strong
+cooling contributor when X-ray cooling becomes less efficient in the
+colliding wind binary WR 140 (Pollock et al. 2021). In WR 140, the
+MNRAS 000, 1–11 (2022)
+
+The spectroscopic orbit of 휂 Car
+9
+Lührs model was used by Fahed et al. (2011) to explain the variations
+in the C III 휆5696 line near periastron.
+The L��hrs model can explain changes in the radial velocity and
+the width of the excess emission. As can be seen in Fig. 6, we
+detect the He ii line with our spectra, but the actual characterization
+of this line will have large errors in line width due to the limited
+signal-to-noise for the detection in the spectroscopy. We used the
+models for WR 140 (Fahed et al. 2011) as a starting point, changing
+stellar and binary parameters as appropriate to the 휂 Carinae system
+to investigate if the He ii velocities in Fig. 7 were from colliding
+wind excess emission. For the velocity of the outflow, we can see
+that during the periastron passage of 2014, 휂 Car’s outflow reached
+velocities faster than the primary star’s wind speed based on the
+optical He i lines (Richardson et al. 2016), which are slower than the
+excess absorption seen to reach nearly 2000 km s−1 in the meta-stable
+He i 휆10830 line (Groh et al. 2010). With these velocities, we expect
+to see the observed amplitude of the excess increase between the
+times of 2015 and 2018 like we see in Fig. 7, but with amplitudes of
+at least 1000 km s−1, much greater than the ∼ 100 km s−1 observed.
+Therefore, the analysis of the He ii 휆4686 emission line at times
+away from periastron from the CHIRON spectra is an important
+observation towards understanding the nature of the companion. We
+note that the data indicate a narrower emission line profile then
+expected from the parameters inferred for the secondary. However,
+the primary star dominates the spectrum, and the motion of this peak
+opposite the primary indicate that the He ii excess could be from the
+secondary’s wind. In particular, the Lührs models of the kinematics
+of the He ii line seem to exclude the possibility that the line is formed
+in the colliding winds at times away from periastron.
+The models of Smith et al. (2018) suggest that the companion
+should be a classical Wolf-Rayet star. The classical hydrogen-free
+Wolf-Rayet stars can be split into the WN and WC subtypes. The
+WN stars show strong He and N lines, with the He ii 휆4686 typically
+being the strongest optical line, whereas the WC subtype exhibits
+strong He, C, and O lines with the C IV 휆휆5802,5812 doublet often
+being the strongest optical line. There is also the rare WO subtype,
+which is similar to the WC subtype but shows more dominant O
+lines. The WO stars were recently shown to have higher carbon
+and lower helium content than the WC stars, likely representing the
+final stages of the WR evolution (Tramper et al. 2015; Aadland et al.
+2022). Given the generalized characteristics of WR stars, a WN star
+would seem the most likely companion star if the He ii 휆4686 line is
+from the companion at times further from periastron.
+For contrast, the Carina nebula is also the home to several
+hydrogen-rich Wolf-Rayet stars: WR 22, WR 24, and WR 25
+(Rosslowe & Crowther 2015)3. This type of WR star tends to be
+considered the higher mass and luminosity extension of the main
+sequence. As such, these stars have masses in excess of ∼ 60푀⊙,
+with the R145 system in the LMC having masses of the two WNh
+stars being 105 and 95 푀⊙ (Shenar et al. 2017). Like the classical
+WN stars, these stars have similar nitrogen and helium spectra, along
+with stronger emission blended on the Balmer lines which overlap
+with Pickering He ii lines. The region surrounding the He ii 휆5411
+line in our 휂 Carinae spectra does not exhibit emission lines at the
+same epochs as our observations of He ii 4686, making it difficult to
+quantify the companion’s properties without the higher order He ii
+lines which would also be notably weaker than He ii 휆4686.
+With the assumption that the He ii orbit shown in Table 1 is from
+the companion star, and that the semi-amplitude from the higher-
+3 http://pacrowther.staff.shef.ac.uk/WRcat/
+order Balmer lines for the primary star (Grant et al. 2020), then the
+semi-amplitude ratio shows that the primary star is 2–3 times more
+massive than the secondary star. This is also an indicator that the
+companion is not likely a WNh star, as that would imply the primary
+star could have a mass of in excess of 100 푀⊙. Models of the system,
+such as those by Okazaki et al. (2008) and Madura et al. (2013),
+typically have the masses of the primary and secondary as 90 and
+30 푀⊙ respectively, broadly in agreement with the kinematics of
+the orbits presented here. On the other hand, if 휂 Carinae A has a
+mass of > 100푀⊙, the secondary would have a mass on the order
+of 50–60 푀⊙. This is similar to the nearby WNh star in the Carina
+nebula: WR22. The mass of this WNh star in an eclipsing system is
+56–58 푀⊙ (Lenoir-Craig et al. 2022). The tidally-induced pulsations
+observed by Richardson et al. (2018) were modeled with stars of
+masses 100 and 30 푀⊙, and therefore may also support the higher
+masses suggested here.
+Most models for 휂 Car have a preferred orbital inclination of
+130–145◦ (Madura et al. 2012), which agrees with forbidden [Fe iii]
+emission observed with Hubble Space Telescope’s Space Telescope
+Imaging Spectrograph. This inclination can be used with the mass
+function derived from the primary star’s orbit
+푓 (푀) =
+푚3
+2 sin3 푖
+(푚1 + 푚2)2 = (1.0361 × 10−7)(1 − 푒2)3/2퐾3
+1푃[M⊙]
+to constrain the system’s masses with the mass function using the
+standard units measured and our measured H훽 orbit using PHOEBE
+(Table 1). The mass function is 푓 (푀) = 8.30 ± 0.05 M⊙, and would
+indicate a companion star with a mass of at least 60 푀⊙ if we assume
+a primary mass of ∼ 90 푀⊙. Given the actual mass functions for the
+measured upper Balmer lines and He ii orbits, the minimum masses
+required for these measured orbits are 푀 sin3 푖 = 102푀⊙ for the
+LBV primary and 푀 sin3 푖 = 55푀⊙ for the secondary, making the
+companion star’s identification as a WNh star more likely. These
+results are still preliminary and require follow-up observations to
+constrain the orbits.
+A WNh star can account for the mass of the secondary star in 휂 Car,
+but could cause some difficulty for the modeling of the Great Eruption
+models of Hirai et al. (2021). In that scenario, the companion star
+would be a hydrogen-stripped star, contrary to the hydrogen content
+of the WNh stars. Recently modeled WNh systems such as R144
+(Shenar et al. 2021) show that the surface fraction of hydrogen is
+about 0.4. This does show some amount of lost hydrogen on the
+surface, so the scenario could still be relevant even if the final star
+is not a fully stripped classical Wolf-Rayet star, assuming that the
+evolution of the secondary star has not been significantly influenced
+by mass exchange prior to or during the merger event hypothesized
+by both Portegies Zwart & van den Heuvel (2016) and Hirai et al.
+(2021).
+6 CONCLUSIONS
+In this paper, we provide an orbital ephemeris for 휂 Carinae mea-
+sured with a bisector method and high resolution ground-based spec-
+troscopy of the H훽 emission line, along with an ephemeris for the
+He ii 휆4686 emission line at times far from periastron. Our findings
+can be be summarized as follows:
+• The H훽 emission profile tracks the primary star, and our bisec-
+tor method provides similar results as the multiple-Gaussian fitting
+method used by Grant et al. (2020). The results show a high ec-
+centricity orbit of the system with the primary star in front of the
+secondary at periastron.
+MNRAS 000, 1–11 (2022)
+
+10
+Strawn et al.
+• The weak He ii 휆4686 emission tracks opposite the kinematics
+of the primary star, suggesting it is formed in the secondary star’s
+windat timesawayfromperiastron. Thiscouldsupport thehypothesis
+of the scenarios presented by Hirai et al. (2021) for a stellar merger
+being the cause of the Great Eruption as the secondary could be a
+Wolf-Rayet star that has leftover hydrogen on its surface.
+• With the assumed inclination of 130–145◦, the masses of the
+stars could be around ∼100 푀⊙ for the primary and at least 60 푀⊙
+for the secondary. However, the mass ratio derived by comparing the
+two semi-amplitudes is about 1.9. New observations will be needed
+to better determine precise masses.
+Future studies will be able to better measure the He ii 4686 orbit
+and refine its parameters. As shown in Grant et al. (2020), the upper
+Balmer lines are more likely to reflect the orbital motion of the
+stars, and the upper Paschen lines will also be useful. However, our
+work shows that a simpler bisector measurement of higher resolution
+spectroscopy results in the same derived orbital elements as that of
+Grant et al. (2020). Furthermore, with better signal-to-noise spectra,
+we can better determine if the He ii emission near periastron can
+be reproduced with a Lührs model or if it is a signature of the
+companion. With this information, we will be able to more precisely
+measure the kinematics of the two stars and the mass function, and
+then we can begin to better understand the current evolutionary status
+of the system.
+ACKNOWLEDGEMENTS
+We thank our referee, Tomer Shenar for many suggestions that
+improved this paper. These results are the result of many alloca-
+tions of telescope time for the CTIO 1.5-m telescope and echelle
+spectrographs. We thank internal SMARTS allocations at Geor-
+gia State University, as well as NOIR Lab (formerly NOAO) al-
+locations of NOAO-09B-153, NOAO-12A-216, NOAO-12B-194,
+NOAO-13B-328, NOAO-15A-0109, NOAO-18A-0295, NOAO-19B-
+204, NOIRLab-20A-0054, and NOIRLab-21B-0334. This research
+has used data from the CTIO/SMARTS 1.5m telescope, which
+is operated as part of the SMARTS Consortium by RECONS
+(www.recons.org) members Todd Henry, Hodari James, Wei-Chun
+Jao, and Leonardo Paredes. At the telescope, observations were car-
+ried out by Roberto Aviles and Rodrigo Hinojosa. C.S.P. and A.L.
+were partially supported by the Embry-Riddle Aeronautical Univer-
+sity Undergraduate Research Institute. E.S. acknowledges support
+from the Arizona Space Grant program. N.D.R., C.S.P., A.L., E.S.,
+and T.R.G. acknowledge support from the HST GO Programs #15611
+and #15992. AD thanks to FAPESP (2011/51680-6 and 2019/02029-
+2) for support. AFJM is grateful for financial aid from NSERC
+(Canada). The material is based upon work supported by NASA un-
+der award number 80GSFC21M0002. The work of ANC is supported
+by NOIRLab, which is managed by the Association of Universities
+for Research in Astronomy (AURA) under a cooperative agreement
+with the National Science Foundation.
+DATA AVAILABILITY
+All measurements can be found in Appendix A. Reasonable requests
+to use the reduced spectra will be granted by the corresponding
+author.
+REFERENCES
+Aadland E., Massey P., Hillier D. J., Morrell N. I., Neugent K. F., Eldridge
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+APPENDIX A: APPENDIX
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–11 (2022)
+
+12
+Strawn et al.
+2620.4
++4.6
+−4.7
+R
+⊙
+3.60
+3.75
+3.90
+4.05
+v
+γ
+ (
+km
+s
+)
+3.811
++0.08
+−0.08
+km
+s
+0.8200
+0.8225
+0.8250
+0.8275
+e
+binary
+0.824
++0.0013
+−0.0013
+4.5
+5.0
+5.5
+6.0
+6.5
+t
+0,
+perpass,
+binary
+ (d)
++2.45693e6
+2456935.31
++0.28
+−0.29
+d
+2608
+2616
+2624
+2632
+a
+binary
+sin
+i
+binary
+ (R
+⊙
+)
+260.8
+261.2
+261.6
+262.0
+ω
+0,
+binary
+ (
+⊙
+)
+3.60
+3.75
+3.90
+4.05
+v
+γ
+ (
+km
+s
+)
+0.8200
+0.8225
+0.8250
+0.8275
+e
+binary
+4.5
+5.0
+5.5
+6.0
+6.5
+t
+0,
+perpass,
+binary
+ (d)
++2.45693e6
+260.8
+261.2
+261.6
+262.0
+ω
+0,
+binary
+ (
+⊙
+)
+261.43
++0.22
+−0.22
+⊙
+Figure A1. Results of the Markov chain Monte Carlo fit for the H훽 velocities for only the data surrounding the 2014 periastron passage. Note that 휔0 refers to
+the value of 휔 for the primary star.
+MNRAS 000, 1–11 (2022)
+

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@@ -0,0 +1,1072 @@
+1 
+ 
+Low-Temperature Chemical Vapor Deposition of Copper(II) 
+Sulfide Crystals and its Nonlinear Optical Response 
+Abdulsalam Aji Suleimana*, Reza Rahighia, Amir Parsia, and Talip Serkan Kasirgaa,b* 
+aInstitute of Materials Science and Nanotechnology, Bilkent University UNAM, Ankara 
+06800, Turkey 
+bDepartment of Physics, Bilkent University, Ankara 06800, Turkey 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+*Corresponding authors; 
+Email: [email protected]; [email protected]  
+ 
+
+2 
+ 
+ABSTRACT 
+The need for novel multifunctional nanomaterials capable of meeting new demands in the realm 
+of nanotechnology coupled with versatility of chemical vapor deposition (CVD) technique (in 
+large-area growth of crystals), encourages innovative methods for synthesis of untried two-
+dimensional (2D) crystals. While there exist reports on both top-down and bottom-up synthesis 
+methodologies of different Cu2S-based nanostructures, CVD-based synthesis of 2D crystals of 
+copper(II) sulfide (CuS) has not been investigated. This work represents details of CVD method 
+in systematic growth of highly crystalline 2D CuS sheets as thin as ~ 6 nm with lateral sizes 
+exceeding 60 μm, at a relatively low temperature of 560 °C in ambient pressure. Samples were 
+characterized via X-ray diffraction, Raman, atomic force microscopy, and high-resolution 
+transmission electron microscopy. SAED revealed a 6-fold symmetric structure and identical 
+atomic ratio of copper:sulphur was corroborated from the energy-dispersive X-ray spectroscopy. 
+The as-prepared 2D CuS sheets were successfully utilized in second harmonic generation (SHG) 
+and their strong response was found to be highly polarization angle-sensitive as well. The CVD-
+synthesized 2D CuS crystals in this study are considered to be of great significance in a diverse 
+range of future applications, as in energy storage, next-generation solar cells, nonlinear 
+optoelectronic-related devices, and even bioelectronics pursuits. 
+KEYWORDS: 2D materials, CVD method, Covellite, Nonlinear optics, Second harmonic 
+generation.
+
+3 
+ 
+ToC 
+
+560C
+2D lattice of
+Arflow~
+Copper (l) Sulfide
+S
+CuCl
+2W
+w
+SecondHarmonic
+Generation yia CuS4 
+ 
+Two-dimensional (2D) materials could make the way for myriad of unprecedented functional 
+devices such as high on/off ratio field-effect transistors (at room temperature),1 electric nose,2 
+mode-locked laser,3 and broad-band photodetectors.4-5 However, top-down methods of 
+chemical exfoliation,6 solvothermal,7 or supercritical8 lead to defective structures that deviate 
+from the required parameters regarding fabrication of highly efficient nanodevices sought in 
+different areas of optoelectronics, bioelectronics,9 and spintronics.10-13 Among bottom-up 
+approaches for synthesis of highly-ordered crystalline 2D structure14-16, chemical vapor 
+deposition (CVD) provides high-quality and high-yield products.17-18 Engineering its 
+corresponding parameters such as temperature, substrate, gas flow, dwell time, fast/natural 
+cooling, can herald novel 2D structures, pursued in realization of innovative devices.19-21. 
+Copper(II) sulfide (covellite, CuS), being highly conductive, chemically stable, and 
+having ultralow thermal conductivity, is widely used in solar cells, batteries, and photothermal 
+treatments.22-27 CuS nanoparticles have been reported to be prepared via wet-chemistry, as a 
+promising candidate for a lithium-ion battery.28 In another report, their photocatalysis property 
+was investigated and attributed to the large specific surface area.29 In addition, light harvesting 
+and charge separation activities can be significantly enhanced by nanosheets of ZnIn2S4/CuS,30 
+without necessity of co-catalyst thanks to the strong interactions between assembled p-n 
+heterostructures. Some nanoflakes of CuCrS2 showing switchable ferroelectric polarization 
+have been also reported to be synthesized recently.31 While important phenomenon of 
+superconductivity has been theoretically predicted from 2D lattice of CuS,32 the 
+physicochemical properties of 2D copper-based chalcogenides have scarcely been studied and 
+there exists no report on CVD growth of 2D CuS yet. 
+Highly crystalline 2D CuS synthesized in this work, are produced using a single-step 
+CVD technique at a relatively low temperature of 560 °C. The as-grown 2D CuS sheets having 
+nanometer thickness, were characterized by atomic force microscopy (AFM), X-ray diffraction 
+
+5 
+ 
+(XRD), Raman, high-resolution transmission electron microscopy (HRTEM), and energy-
+dispersive X-ray spectroscopy (EDX). A 6-fold symmetric structure was revealed via selected 
+area diffraction (SAED). As an application of 2D CuS, a single sheet of it was utilized in second 
+harmonic generation (SHG) with the nonlinear susceptibility of up to 1.4 × 10−11 m/V. In 
+addition, the nonlinear optical characteristic of 2D CuS crystals was utilized in broad-spectrum 
+wavelength and polarization-resolved SHG.  
+ 
+Copper(I) chloride (CuCl) powder was opted as the Cu source for growth of 2D CuS 
+lattices, due to its suitable chemical property and the relatively low melting point temperature. 
+An asymmetric tiny crucible was chosen to this end, filled with scant amount of CuCl powder, 
+and put in middle of tubular CVD furnace as can be seen in the schematic setup in the supporting 
+information (Figure S1). During the synthesis process, the optimum growth temperature was 
+found to be about 560 °C, way lower than the other CVD synthesis33 of copper-based 
+chalcogenides. Additional information regarding the CVD growth process is provided in the 
+experimental section. Figure 1a shows a typical optical image of CuS crystals grown on a mica. 
+The lateral length of the grown 2D CuS crystals can be up to 70 µm (Figure 1b). Mica is used 
+as a substrate because of its atomic-level smooth and inert surface, which has been widely 
+reported as a favorable substrate for 2D material synthesis.34 AFM height trace map given in 
+Figure 1c confirms that the surface of 2D CuS is very smooth and the thickness was found to 
+be about 14 nm according to AFM measurements. 
+
+6 
+ 
+ 
+Figure 1: Typical OM image of the as-grown 2D CuS crystals (a and b), the scale bars are 10 
+and 20 μm, respectively. AFM image of the CuS crystal (c), and its height profile of the in the 
+inset. XRD pattern of 2D CuS crystals on SiO₂/Si substrate (d). EBSD inverse pole figure 
+(IPF) map along the c-axis of 2D CuS crystal on SiO₂/Si substrate (e), the length of the scale 
+bar corresponds to 5 μm. Color coded map type of IPF (f). 
+ 
+XRD pattern of as-grown (Figure S2) and transferred CuS crystals on the SiO2/Si 
+substrate depicted in Figure 1d clearly identifies the hexagonal phase of CuS (PDF No. 06-
+0464). The strong characteristic peaks of (002), (006), and (008) show that 2D CuS crystals 
+preferentially grow in the basal plane (00l). The plane interspacing can be estimated using 
+Bragg’s relation, 𝑛𝜆 =  2𝑑(ℎ𝑘𝑙)𝑠𝑖𝑛(𝜃) where n is an integer corresponds to the other of 
+diffraction peak and λ is wavelength of X-ray. The mean dimensions of the crystallite 
+perpendicular to the (00l) plane (L002) can be determined by using Scherrer equation, 
+𝐿(ℎ𝑘𝑙)  =  𝐾𝜆/𝛽𝑐𝑜𝑠(𝜃) 
+
+a)
+b)
+c)
+30nm
+Position(μm)
+d)
+(006)
+f)
+Intensity (a.u.)
+1010
+Cus crystal
+PDF06-0464
+(002)
+IS
+(008)
+0001
+2110
+10
+20
+30
+40
+50
+60
+20 (deg.)7 
+ 
+where K is a constant (0.89) and β is the integral full widths at half maximum (in radians, in 
+our case 0.07 regarding 2θ peak at 11°). The average number of layers can be estimated by 
+simply dividing L(002) over d(002). Therefore, the average number of layers was found ~ 16, 
+implying the presence of multi-layer sheets in the structure, consistent with AFM investigations. 
+In addition, we utilized electron backscatter diffraction (EBSD), a powerful method for 
+identifying the microstructural characterization of materials, to determine the crystallographic 
+orientation of the 2D CuS crystals.35-36 Figure 1e-f displays a uniform color contrast of the 
+EBSD inverse pole figure (IPF) map within the hexagonal domains along the basal plane of 
+CuS ([00l] direction), implying a single-crystalline nature and ordered in-plane orientation 
+throughout the hexagonal CuS crystal, which is consistent with the XRD results. As illustrated 
+in Figure S3, the CuS structure belongs to the space group P63/mmc (hexagonal symmetry) 
+with Z = 6 per unit cell. Cu atoms exist in two types of environments: CuS3 (triangular planes) 
+and CuS4 (rectangular planes) (tetrahedra). The unit cell can be assumed as plates connected by 
+S-S bonds, and through triangular planes, vortices merge the tetrahedral units. According to 
+previous research, the Cu(1)-S(1) bonds (∼2.19 Å ) which occur in triangle units, have a length 
+much shorter than the Cu-S bonds seen in most other copper sulfides (∼2.33 Å).37 Because of 
+this, it is rather conceivable that the Cu(1)-S(1) bond will have a stronger bond. It has also been 
+reported that Cu(1) ions in the [Cu(1)-S(1)3] triangles exhibit significantly high thermal motion 
+along the c-axis.38 
+Raman spectroscopy with a 532 nm excitation laser was used to investigate the intrinsic 
+properties and identify the fingerprint of the 2D CuS crystal structure. As shown in Figure 2a, 
+the Raman spectrum of the 2D CuS crystal shows four distinct Raman peaks at 90.1, 130, 279, 
+and 471 cm-1, representing E2g, A1, Eg
+1, and A1 modes, respectively. Among these peaks, the 
+strong characteristic peak at 471.0 cm-1 can be attributed to the stretching mode of the S-S bond, 
+corresponding to the S2 groups of the recognized crystal structure of 2D CuS lattice.39-40 
+
+8 
+ 
+ 
+Figure 2: Raman spectrum of the 2D CuS lattice on mica substrate (a). Spatially resolved 
+Raman mapping images (b) of the 2D CuS characteristic peaks E2g
+3  (P1), A1g
+2  (P2), E2g
+2  (P3), 
+and A1g
+1  (P4), scale bars: 5 μm. Temperature-dependent spectra (c) of 2D CuS crystal (80-300 
+K, step: 20 K). Raman peak positions of 2D CuS (P1-4) as a function of the measured 
+temperature (d). 
+ 
+Temperature-dependent Raman spectroscopy is a classical method to study the atomic 
+bonding and thermal expansion of 2D materials.41-42 The spatially resolved Raman mapping 
+images (Figure 2b) of the four characteristic peaks (60, 138, 267, and 471 cm–1) exhibit 
+uniformity throughout the 2D crystalline sheet of CuS. Figure 2c shows the typical 
+temperature-dependent Raman spectra for the grown 2D CuS crystal at temperatures ranging 
+
+a)
+b)
+P2
+Intensity (a.u.)
+P4
+A2
+2c
+100
+200
+300
+400
+500
+600
+Raman shift (cm-1)
+c)
+d
+P1
+P4
+476.1
+P4
+P2
+P3
+Fit
+300 K
+473.8
+471.5
+Slope=-0.02535
+Raman shift (cm'
+Intensity (a.u.)
+267.5
+P3
+Fit
+265.0
+262.5
+Slope=-0.02563
+140.0
+P2
+Fit
+137.2
+134.4
+Slope=-0.02677
+P1
+60.48
+Fit
+59.85
+80 K
+59.22
+Slope=-0.00662
+60
+120
+100
+200300400500
+180
+240
+300
+600700
+800
+Raman shift (cm-1)
+Temperature (K)9 
+ 
+from 80 to 300 K. It can be clearly seen that the positions of the peaks exhibit a slight "redshift" 
+with increasing temperature, which is mainly due to anharmonic vibrations of the lattice 
+induced by the thermal expansion of the lattice at elevated temperatures.43 The correlation 
+between them can be described by a linear equation: 𝜔(𝑇) = 𝜔0 + 𝜒𝑇, where 𝜔0, T, and χ 
+are the Raman peak position at 0 K, the Kelvin temperature, and the first-order temperature 
+coefficient, respectively. As shown in previous reports,44-45 the first-order temperature 
+coefficient of 2D materials is related to the van der Waals interaction between the neighboring 
+layers and is usually used to explain the temperature dependence of the Raman peak shift. 
+Notably, the derived χ-values for P1, P2, P3, and P4 of CuS crystals are - 0.00662, -0.02677,  -
+0.02563, and - 0.02535 cm-1K-1, respectively (Figure 2d), which is larger than that of ordinary 
+layered materials.46-47 
+Further analyses, such as high-resolution transmission electron microscopy (HR-TEM), 
+SAED, and EDX, were carried out to investigate the crystal structure and atomic composition 
+of the 2D CuS crystal. Figure 3a shows Fast Fourier Transform-filtered HR-TEM image and it 
+can be clearly seen that the atoms are arranged hexagonally. The interplanar spacing of the two 
+planes crossing at an angle of 120° is 0.35 nm, corresponding to planes (100) and (010), 
+respectively. The corresponding SAED image shows a 6-fold symmetric structure with an [001] 
+axis presented in Figure 3b, and the EDX spectrum of the 2D CuS crystal is shown in Figure 
+3c. The detected peaks suggest that the crystal is made entirely of Cu and S components, which 
+is supported by X-ray photoelectron spectroscopy (XPS) results which is shown in Figure 3d-
+e. 
+
+10 
+ 
+ 
+Figure 3: Structural and chemical compositional characterization of 2D CuS crystals. 
+High-resolution TEM image. Scale bar: 5 nm (a). The SAED patterns, scale bar: 5 nm-1 (b), 
+EDX spectrum, inset shows atomic ratios of the chemical composition (c), XPS spectra 
+deconvoluted peaks of Cu2p (d), and S2p (e) core levels. 
+ 
+New materials that have nonlinear optical response, can be of beneficial application in 
+different areas ranging from photon generation, imaging, and photon manipulation in ultrafast 
+
+0.35 nm
+0.35 nm
+(010)
+(100)
+b)
+Intensity (a.u.)
+Cu
+s
+cu
+52 %
+48 %
+（100
+S
+Cu
+（010)
+cu
+0
+2
+4
+6
+8
+10
+Energy (keV)
+d)
+Cu,2p3/2
+e
+Intensity (a.u.)
+S 2p3/2
+Intensity (a.u.),
+S 2p1/2
+Cu 2p1/2
+096
+950
+940
+930
+920
+170
+168
+166
+164
+162
+Binding energy (eV)
+Binding energy (eV)11 
+ 
+lasers, optical modulators, and pulse characterization.48-53 Stacking faults existing in the 
+synthesized CuS crystals (in this study), such as an interlayer slip, dislocation, and undulation 
+of the atomic layers, can induce multi-oriented domains in the crystal and deemed responsible 
+for the observed nonlinear optical behavior.54 The SHG is a very useful technique, where the 
+incident laser (ω) generates an (2ω) response, as shown in Figure 4a. The SHG response of a 
+CuS crystal (14.5 nm) under various incident laser wavelengths from the edge of visible light 
+to near-infrared (760 to 1020 nm) is presented in Figure 4b, which shows a wide spectrum 
+response with distinct wavelength selectivity. Moreover, the SHG mappings display a uniform 
+response throughout the entire 2D CuS lattice (Figure 4b inset). 
+Evolution of SHG intensity with changing incident laser power was also further 
+systematically investigated. With increasing the incident laser power from 0.7 to 1.6 mW under 
+800 nm laser excitation, the intensity of the SHG signal at 400 nm exhibits significant 
+enhancement (Figure 4c). The relationship between SHG intensity and laser power was fitted 
+linearly in the log-log coordinate, as displayed in Figure 4d. Interestingly, the slope of 2.05 is 
+close to the theoretical value of 2 calculated from the electric dipole theory.41 
+ 
+ 
+ 
+ 
+
+12 
+ 
+ 
+Figure 4: SHG characterization of 2D CuS crystal. Basic mechanism of nonlinear optical 
+effects (a), The SHG spectra of 2D CuS crystal under various excitation wavelengths (760 - 
+1020 nm). Inset is the SHG mapping of 2D CuS crystal under 800 nm laser excitation, scale 
+bar corresponds to 3 μm (b), The SHG spectra of the 2D CuS crystal with different incident 
+powers (c), The SHG intensities as a function of incident power (d), Polarization angle-
+dependent SHG intensity under parallel (e), and perpendicular (f) polarization configurations 
+(The excitation laser is 800 nm with a power of 1.2 mW). 
+ 
+
+a)
+b)
+(a.u.)
+2=760-1020 nm
+5k
+3
+2w
+m
+2w
+4k
+SHG intensity
+3k
+3
+2k
+SHG
+1k
+Mica substrate
+380400420440460480500
+520
+Wavelength (nm)
+6k
+0.7 mW
+1.3 mW
+Data
+(a.u.
+5k
+0.8 mW
+1.4 mW
+Linear fit
+0.9 mW
+1.5 mW
+a
+4k
+Intensity
+1.0 mW
+1.6 mW
+3k
+1.2 mw
+Intensi
+2k
+Slope = 2.05
+1k
+385
+390
+395
+400
+405
+410
+415
+0.6
+0.9
+1.2
+1.5
+1.8
+Wavelength (nm)
+Laser power (mw)
+e)
+f)
+06
+xX
+06
+120
+60
+120
+60
+XY
+Fit
+Fit
+150
+30
+150
+30
+180
+0
+180
+0
+210
+330
+210
+330
+240
+300
+240
+300
+270
+27013 
+ 
+The nonlinear susceptibility of our newly synthesized CuS crystal was estimated to be 
+𝜒𝐶𝑢𝑆
+(2) = 1.4 × 10−11 m/V. Before assessing polarization, we rotated the sample to a position 
+where the highest SHG response could be generated by setting the initial azimuthal angle to 0°. 
+In parallel (XX) and perpendicular (XY) directions, the typical 6-fold symmetry pattern fitted 
+proportionally with sin2 3𝜃 and cos2 3𝜃 can be detected, as presented in Figure 4e-f. It implied 
+broken inversion symmetry property that is characteristic of hexagonal-symmetric structures 
+similar to other SHG sensitive materials (Table 1). This feature imparts 2D CuS crystals with 
+promising properties of interest in the field of applied nonlinear optics. In addition, utilizing 
+Piezoresponse force microscopy (PFM), we explored the unexpected SHG response in CuS, 
+and interestingly, we detected switchable hysteretic behavior in the dual-pass remnant 
+hysteresis measurement on numerous CuS crystals (Figure S4). The findings of the PFM 
+support the SHG response. 
+Table 1: Properties of the synthesized 2D CuS sheets and comparison with other 
+nanomaterials used in SHG (C: centrosymmetric, and N: noncentrosymmetric). 
+2D 
+Material 
+Synthesis 
+method 
+Sample 
+Thickness 
+(nm) 
+C 
+N 
+𝝌(𝟐) 
+Refs. 
+CuS 
+CVD 
+14.5 
+* 
+ 
+1.4 × 10–11 
+This 
+work 
+MoS2 
+ 
+Exfoliation 
+0.8 
+ 
+* 
+1 × 10–7 
+[55] 
+GaSe 
+ 
+CVD 
+0.83 
+ 
+* 
+0.7 × 10-9 
+[56] 
+SnP2S6 
+ 
+Exfoliation 
+8 
+* 
+ 
+4 × 10-9  
+[57] 
+WS2 
+ 
+CVD 
+0.65 
+ 
+* 
+4.5 × 10-9 
+[58] 
+RhI3 
+ 
+Exfoliation 
+12 
+* 
+ 
+- 
+[59] 
+ 
+
+14 
+ 
+To summarize, in a single-step CVD technique (at a growth temperature of less than 
+600 °C), highly crystalline 2D lattice CuS was synthesized for the first time. The as-grown 2D 
+CuS sheets with nanoscale thickness were thoroughly characterized (phase and orientation of 
+its lattice were verified). A sheet of 2D CuS crystal was utilized in SHG, with a nonlinear 
+susceptibility of up to 1.4 × 10-11 m/V and the underlying mechanism was discussed. The 
+nanoscale 2D sheets of CuS are therefore expected to have a wide range of optoelectronic 
+applications.  
+ 
+Materials and Characterization 
+CVD growth: 2D CuS crystals were grown in a tubular furnace with a single temperature zone 
+and atmospheric pressure CVD conditions. A quartz boat containing a CuCl powder (97%, 
+Sigma Aldrich) was placed in the middle of the temperature zone. S powder (99.5%, Sigma 
+Aldrich) was inserted at the upstream end of the tube, and the temperature was maintained at 
+200. Substrates, e.g., cleaved fluorphlogopite mica, were positioned 8 cm apart from the 
+furnace's center in the downstream position. The tube was pumped and cleaned with 500 sccm 
+Ar flow to drain air prior to heating. Then, the furnace was heated to 560 °C at a rate of 30 
+⁰C/min using steady 50 sccm Ar as the carrier gas, and it was held at that temperature for 30 
+minutes. After the procedure was concluded, the furnace was allowed to cool naturally. 
+Characterizations: 2D CuS crystal morphologies were examined using an OM (BX51, 
+OLYMPUS) and an AFM (Bruker Dimension Icon). The crystalline structure, orientation, and 
+composition were investigated using XRD (λ: 1.54 Å, D2 phaser, Bruker), XPS (AXIS-ULTRA 
+DLD-600W, Kratos), EBSD (FEI Quanta650), and TEM (Tecnai G30 F30, FEI). Raman 
+spectra were acquired using a confocal Raman system (Alpha 300R, WITec) equipped with a 
+532 nm laser. 
+
+15 
+ 
+SHG measurements: SHG measurements were performed in an (alpha300RS+, WITec) 
+Raman system with a reflection mode under normal incidence excitation using a femtosecond 
+laser as the excitation source. A mode-locked Ti: sapphire laser with a pulse duration of 140 fs 
+and repetition rate of 80 MHZ generated the output laser with a continually varying wavelength 
+ranging from 340 nm to 1600 nm, which was then filtered into an optical parametric oscillator 
+(Chameleon Compact OPO-Vis). A dichroic beam splitter was used to reflect the laser beam 
+into the 100x objective lens with a spot size of roughly 1.8 μm and communicate the reflected 
+SHG signal. The reflected SHG signal was then filtered with a short pass (SP) filter before 
+being sent to the spectrometer and CCD. The collected polarized SHG signal was sent through 
+a linear polarized analyzer for SHG polarization measurement by rotating the sample with a 
+step of 10° relative to fixed light polarization (Figure S5). All experiments were carried out in 
+a natural setting. 
+ASSOCIATED CONTENT 
+AUTHOR CONTRIBUTION 
+A.A.S: Synthesis, characterizations, conceptualization, data curation, writing-original draft, co- 
+corresponding. R.R: Writing, data curation and editing. A.P: Characterization and editing. 
+T.S.K: Supervision, conceptualization, funding, editing, corresponding. All authors have 
+agreed on the final version of the manuscript. 
+CONFLICT OF INTEREST 
+No competing financial interest are declared.  
+ACKNOWLEDGMENTS 
+The authors acknowledge funding from the Scientific and Technological Research Council of 
+Turkey (TUBITAK) under grant number 120N885.  
+
+16 
+ 
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+Honeycomb RhI3 Flakes with High Environmental Stability for Optoelectronics. Adv Mater 
+2020, e2001979. 
+ 
+
+S1 
+ 
+SUPPORTING INFORMATION 
+Low-Temperature Chemical Vapor Deposition of Copper(II) 
+Sulfide Crystals and its Nonlinear Optical Response 
+Abdulsalam Aji Suleimana*, Reza Rahighia, Amir Parsia, and Talip Serkan Kasirgaa,b* 
+aInstitute of Materials Science and Nanotechnology, Bilkent University UNAM, Ankara 
+06800, Turkey 
+bDepartment of Physics, Bilkent University, Ankara 06800, Turkey 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+*Corresponding authors; 
+Email: [email protected]; [email protected]  
+
+S2 
+ 
+ 
+ 
+Figure S1. Schematic image of the CVD setup (a). CVD growth temperature curve (b).
+
+a)
+560°C
+Arflow
+2D CuS sheets
+S
+Cucl
+d)
+T/°C
+Growth time
+T1
+rate
+°C/min
+30
+-
+1
+0
+t1
+t2
+t3
+t/minS3 
+ 
+ 
+Figure S2. OM mages of CuS crystals grown on various substrates of mica, Si/SiO2, and Si 
+(a-c), scale bars are 10, 20, and 20 μm, respectively. Corresponding AFM images of CuS 
+crystals with their height profiles 40, 28, and 6 nm respectively (d-f). Raman spectra of CuS 
+crystals grown on mica, SiO2, and Si, respectively (g-i).
+
+a)
+b)
+c)
+d)
+e
+f)
+16.7 nm
+48.1nm
+12.5nm
+-8.3 nm
+-12.8nm
+-3.7nm
+HeightSensor
+2.0um
+Height Sensor
+3.0um
+HeightSensor
+4.0um
+g)
+h)
+D
+Intensity (a.u.)
+Al1g
+3
+(n
+Intensity (a.
+(a.
+Intensity
+E2
+E2g
+A1g
+E2
+100200300400500
+600
+100200300400500
+600
+100200300400500
+600
+Raman shift (cm-1)
+Raman shift (cm-1)
+Raman shift (cm-1)S4 
+ 
+ 
+Figure S3: Side view and top view of CuS crystal structure. 
+ 
+
+Topview
+CuS4
+Cu
+CuS3
+S
+CuS2
+S-S
+CuS4
+CuS3
+CuS4S5 
+ 
+ 
+Figure S4: On-field hysteresis loops for (a) PFM amplitude and (b) PFM phase on CuS 
+crystal. 
+ 
+
+a)
+b)
+100
+100
+Amplitude (pm)
+80
+50
+Phase (°)
+60
+40
+-50
+20
+100
+0
+-150
+8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+-8
+9-
+-2
+0
+2
+4
+6
+8
+Bias (V)
+Bias (V)S6 
+ 
+Calculation of Second-order nonlinear susceptibility 
+Using similar estimation of second order susceptibility χ(2) as reported by several studies 
+in the literature, the χ(2) value of thin CuS crystal could be calculated: 
+𝐼2𝜔 = [𝜒(2)]
+2𝐼𝜔
+28𝜖0𝑐3.
+1
+𝑛2𝜔𝑛𝜔
+2  .
+𝜔2𝑑2
+8𝜖0𝑐3, 
+where 𝐼𝜔 and 𝐼2𝜔 are the intensity of excitation laser and SHG signal, respectively; 𝜒(2) is 
+the second-order susceptibility; 𝜖0 is the vacuum dielectric constant; 𝑐 is the speed of light 
+in vacuum; 𝑛𝜔 ≈ 2.6 and 𝑛2𝜔 ≈ 2.6 are respectively the refractive index of CuS at 
+frequency ω of excitation laser and at frequency 2ω of SHG field1; 𝑑 = 14.3 nm is the 
+thickness. However, it is challenging to obtain 𝐼2𝜔  as it involves many experimental 
+parameters including the optical absorptions of the optical setup, the detector efficiencies 
+and laser frequency and duration. For this reason, typically the susceptibility is referenced 
+with respect to the monolayer MoS2 (𝜒𝑀𝑜𝑆2
+(2)
+= 4.05 × 10−10 m/V) using the following 
+relation2: 
+𝜒𝐶𝑢𝑆
+(2) = √
+𝐼2𝜔−𝐶𝑢𝑆
+𝐼2𝜔−𝑀𝑜𝑆2
+.
+𝑑𝑀𝑜𝑆2
+𝑑𝐶𝑢𝑆
+. √ 𝑛2𝜔−𝐶𝑢𝑆
+𝑛2𝜔−𝑀𝑜𝑆2
+𝑛𝜔−𝐶𝑢𝑆
+2
+𝑛𝜔−𝑀𝑜𝑆2
+2
+. 𝜒𝑀𝑜𝑆2
+(2)   
+Our result was attained after giving due consideration to the efficacy of signal collection 
+and detection as 𝜒(2) = 1.4 × 10−11  m/V for 800 nm. This calculation was only an 
+approximation of the order of magnitude because the value of χ(2) depends on many 
+accurate experimental parameters. 
+
+S7 
+ 
+ 
+Figure S5: Setup for measuring second harmonic generation. 
+
+Pinhole
+Mirror
+Spectrometer
+Polarizer
+SP Filter
+Camera
+Mirror
+Dichroic beamsplitter
+Beam
+expander
+Ti-Sapphire
+OPO
+Laser
+Mirror
+N
+Len
+100x
+SampleS8 
+ 
+ 
+References 
+(1) Aziz, S. B.; Abdulwahid, R. T.; Rsaul, H. A.; Ahmed, H. M., In situ synthesis of 
+CuS nanoparticle with a distinguishable SPR peak in NIR region. J. Mater. Sci. Mater. 
+2016, 27 (5), 4163-4171. 
+(2) Shi, J.; Yu, P.; Liu, F.; He, P.; Wang, R.; Qin, L.; Zhou, J.; Li, X.; Zhou, J.; Sui, 
+X.; et al., 3R MoS2 with broken inversion symmetry: a promising ultrathin nonlinear 
+optical device. Adv. Mater. 2017, 29 (30). 
+ 
+ 
+

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+J. Astrophys. Astr. (0000) 000:
+DOI
+Probing Cosmology beyond ΛCDM using the SKA
+Shamik Ghosh1,2, Pankaj Jain3, Rahul Kothari4,*, Mohit Panwar3, Gurmeet Singh3, Prabhakar
+Tiwari5
+1CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of
+Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China
+2School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, China
+3Department of Physics, Indian Institute of Technology, Kanpur-208016, India.
+4Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa.
+5National Astronomical Observatories, Chinese Academy of Science, Beijing 100101, P.R.China.
+*Corresponding author. E-mail: [email protected]
+MS received 1 January 2022; accepted 1 January 2022
+Abstract. The cosmological principle states that the Universe is statistically homogeneous and isotropic at large
+distance scales. There currently exist many observations which indicate a departure from this principle. It has been
+shown that many of these observations can be explained by invoking superhorizon cosmological perturbations and
+may be consistent with the Big Bang paradigm. Remarkably, these modes simultaneously explain the observed
+Hubble tension, i.e., the discrepancy between the direct and indirect measurements of the Hubble parameter. We
+propose several tests of the cosmological principle using SKA. In particular, we can reliably extract the signal of
+dipole anisotropy in the distribution of radio galaxies. The superhorizon perturbations also predict a significant
+redshift dependence of the dipole signal which can be nicely tested by the study of signals of reionization and
+the dark ages using SKA. We also propose to study the alignment of radio galaxy axes as well as their integrated
+polarization vectors over distance scales ranging from a few Mpc to Gpc. We discuss data analysis techniques that
+can reliably extract these signals from data.
+Keywords.
+cosmological principle—superhorizon perturbations—square kilometre array.
+1. Introduction
+Current observations support an expanding universe. If
+we extrapolate this back in time, we can infer that the
+Universe started from a very hot and dense state. This
+event, known as Big Bang, marked the origin of the
+Universe in a very high temperature state.
+In order to make the problem of expansion dynam-
+ics tractable, we assume that the Universe is spatially
+isotropic and homogeneous. This assumption is also
+known as Cosmological Principle (hereafter CP) (Kolb
+& Turner, 1994; Einstein, 1917; Aluri et al., 2022). It
+turns out that Hubble’s law is a direct consequence of
+CP (Coles & Lucchin, 2003). Furthermore, it can be
+shown that the most general spacetime metric that de-
+scribes a universe following CP is the FLRW metric
+(Weinberg, 1972; Coles & Lucchin, 2003). It is also im-
+portant to mention that CP is an independent assump-
+tion and does not follow from symmetries of the Ein-
+stein’s Equations.
+The FLRW metric describes a Universe with a
+smooth background having an exact isotropic and ho-
+mogeneous matter distribution.
+But observationally,
+the Universe also possesses structure in the form of
+stars, galaxies, etc. These structures arise due to cur-
+vature perturbations which are seeded during the epoch
+of exponential expansion called inflation. The resulting
+cosmological model, including dark matter and dark
+energy is called ΛCDM.
+Although, these perturbations aren’t isotropic and
+homogeneous per se, they satisfy these properties in a
+statistical sense. For example, in the cosmic frame of
+rest, the matter density is expected to be the same at
+all points provided we average over a sufficiently large
+distance scale. The precise value of this distance scale
+is still not clear but is expected to be of order 100 Mpc
+(see, for example Kim et al. (2021)).
+It has been speculated that during an epoch, be-
+fore inflation ensued, the Universe may be described
+by a complicated metric whose nature is currently
+poorly understood. However, it quickly evolves to the
+isotropic and homogeneous FLRW metric during infla-
+© Indian Academy of Sciences
+1
+arXiv:2301.03065v1  [astro-ph.CO]  8 Jan 2023
+
+Page 2 of
+J. Astrophys. Astr. (0000) 000:
+tion, perhaps within the first e-fold. Wald (1983) for the
+first time, gave an explicit demonstration for Bianchi
+Universes (except type IX). Some other results also
+exist for inhomogeneous metric (Stein-Schabes, 1987;
+Jensen & Stein-Schabes, 1986). We may speculate that
+the idea generalizes to a larger class of metrics1. The
+Big Bang paradigm is therefore consistent with an early
+anisotropic and/or inhomogeneous phase of the Uni-
+verse. Given the existence of such a phase, it is clearly
+important to ask whether it has any observational con-
+sequences.
+Observationally, the Universe is found to be consis-
+tent with CP to a good approximation. But currently
+there exist many observations in CMB and large scale
+structures (LSS henceforth) which appear to violate CP
+(Ghosh et al., 2016). We review these anomalies later
+in §2. For an expansive review, see Aluri et al. (2022).
+There exist many theoretical attempts to explain these
+observations. It has been suggested that superhorizon
+modes, i.e., perturbations of wavelengths larger than
+the horizon size (Grishchuk & Zeldovich, 1978a,b),
+may explain some of these observations (Gordon et al.,
+2005; Erickcek et al., 2008a,b; Ghosh, 2014; Das et al.,
+2021; Tiwari et al., 2022). Additionally, these can ac-
+count for low-ℓ alignments (Gao, 2011), though these
+can’t extenuate the present accelerated expansion of
+the Universe (Hirata & Seljak, 2005; Flanagan, 2005).
+It is assumed that such large wavelength modes are
+aligned with one another and hence do not obey CP.
+An intriguing possibility is that such modes might orig-
+inate during an anisotropic and/or inhomogeneous pre-
+inflationary phase of the Universe (Aluri & Jain, 2012;
+Rath et al., 2013). Hence, despite being in violation
+with CP, they would be consistent with the Big Bang
+paradigm.
+1.1 Mathematical Formulation and Ramifications
+In order to relate theory with observations, we seek en-
+semble averages of the fields under consideration. Er-
+godicity hypothesis (Ellis et al., 2012) allows us to re-
+late this ensemble averaging to the space averaging. It
+is known that for the gaussian random fields, all the
+statistical information is contained in the 2 point cor-
+relation functions (2PCF). However, in the presence of
+non-gaussianities, we need higher order correlators like
+bispectrum (3PCF) or trispectrum (4PCF), etc., in or-
+der to extract optimal cosmological information. CP
+dictates that the nPCF only be a function of distances
+between the points xi ≡ (zi, ni). Thus
+�ρ(x1)ρ(x2) . . . ρ(xn)� = f(x12, x13, . . . , xi j, . . .),
+(1)
+1There are exceptions to these results as well (Sato, 1988).
+A
+O
+C
+B
+Figure 1. Illustration of statistical isotropy. In this Figure,
+A, B and C are given points on the spherical surface such
+that ∠AOB = ∠AOC.
+where xij = |xi − xj| = xji and i � j. Clearly, this
+makes this nPCF invariant under arbitrary translations
+and rotations. The condition (1) for 2PCF in case of a
+2D field, e.g., CMB temperature, takes the usual form
+�T(x1)T(x2)� ≡ �T( ˆm)T(ˆn)� = f( ˆm · ˆn),
+(2)
+with x1 ≡ (z∗, ˆn) and x2 ≡ (z∗, ˆm), z∗ being the red-
+shift to decoupling. Eq. (2) is the familiar result for
+the 2PCF, which dictates that the temperature correla-
+tion depends only upon the angle between the locations.
+This is illustrated in Figure 1, where three points A, B
+and C are chosen in a manner such that ∠AOC = ∠AOB.
+Thus we must have �T(A)T(C)� = �T(A)T(B)�, since
+A · C = A · B.
+2. Observations at tension with ΛCDM
+Our observations in the past two decades have firmly
+planted the inflationary ΛCDM cosmology as the stan-
+dard paradigm. A vast set of observables from CMB
+to LSS broadly agree with ΛCDM predictions. Despite
+the successes of ΛCDM, we have a growing set of ob-
+servations that are at tension with our expectations from
+ΛCDM. We will summarise some of the observed ten-
+sions, in the context of the model discussed in this pa-
+per. See Perivolaropoulos & Skara (2022) for a review.
+2.1 Observed violations of Statistical Isotropy
+As we discussed before, CP implies statistical isotropy
+and homogeneity. Due to our fixed vantage point, it
+is not possible to directly test statistical homogene-
+ity. However, we can test statistical isotropy. Various
+
+J. Astrophys. Astr. (0000)000:
+Page 3 of
+observational tests, performed on different cosmologi-
+cal datasets, amply attest statistical isotropy violations.
+Some of these are reviewed in Ghosh et al. (2016).
+2.1.1 The kinematic dipole:
+As explained in the §1.,
+CP is valid only in the cosmic frame of rest. We as ob-
+servers are not stationary with respect to this frame on
+account of the motion of Earth, the Sun, and the Milky
+Way. This gives rise to an effective peculiar velocity to
+our observation frame. This peculiar velocity results in
+a Doppler boost of the CMB temperature fluctuation,
+further culminating in a kinematic dipole in the CMB
+temperature fluctuations. Interpreting the CMB dipole
+to be of kinematic origin (Planck Collaboration et al.,
+2014) leads to the peculiar velocity of our local frame
+to be 384 ± 78 km s−1.
+The peculiar velocity v of our observation is ex-
+pected to give rise to a dipole in the observed num-
+ber count of sources. The local motion would cause
+a Doppler and aberration effects, both of which con-
+tribute to a dipole in the observed number counts (Ellis
+& Baldwin, 1984). For sources with flux following a
+power law relation in frequency: S ∝ ν−α, and with dif-
+ferential number count N(S, ˆn) = S −1−x, the expected
+dipole is given by:
+D = [2 + x(1 + α)] v/c,
+(3)
+where c is the speed of light, α is the frequency scal-
+ing spectral index, and (1 + x) is the slope of ln N v/s
+− ln S plot. We can use the estimates of our peculiar
+velocity from the CMB and use it to predict the esti-
+mated dipole in the large-scale structure data. Assum-
+ing α ≈ 0.75 and x ≈ 1, we find the expected dipole
+Dth ∼ 0.005. Measurements of the dipole in LSS sur-
+veys at z ∼ 1 have all yielded results that are consis-
+tent with CMB direction but the magnitude is found
+to be double or more of the predicted value. In Table
+1, we list the measured value of dipole in the NVSS,
+NVSS+WENSS, NVSS+SUMSS and CatWISE cata-
+logs. The dipole measured in the LSS has a much larger
+magnitude than expected from CMB measurements but
+is consistent with the CMB dipole direction. The devia-
+tion is found to be at 4.9σ in the CatWISE data (Secrest
+et al., 2021).
+We point out that the assumed power law depen-
+dence of number counts on S is not strictly valid (Ti-
+wari et al., 2015).
+This leads to a difference in the
+dipole in number counts and in sky brightness. It also
+introduces a dipole in the mean flux per source. Hence
+this provides a nontrivial test of whether the dipole is
+indeed of kinematic origin. This idea has been gener-
+alised in Nadolny et al. (2021) who develop a method
+to extract kinematic dipole independently from an in-
+trinsic dipole.
+Authors
+|D| (×10−2)
+(l, b)
+Singal (2011)
+1.8 ± 0.3
+(239◦, 44◦)
+Rubart & Schwarz (2013)
+1.6 ± 0.6
+(241◦, 39◦)
+Tiwari et al. (2015)
+1.25 ± 0.40
+(261◦, 37◦)
+Tiwari & Nusser (2016)
+0.9 ± 0.4
+(246◦, 38◦)
+Colin et al. (2017)
+1.6 ± 0.2
+(241◦, 28◦)
+Secrest et al. (2021)
+1.5
+(238◦, 29◦)
+Table 1. Results for the dipole in LSS exceed the expected
+value of 5 × 10−3.
+2.1.2 Alignment of quadrupole (ℓ = 2) and octupole
+(ℓ = 3):
+Both ℓ = 2, 3 CMB multipoles are aligned
+with preferred direction pointing roughly along the
+CMB dipole (de Oliveira-Costa et al., 2004). Physi-
+cally, both of these multipoles form a planar structure,
+such that the perpendicular to this plane is aligned with
+the CMB dipole.
+2.1.3 Alignment of galaxy axes and polarizations:
+There have been many observations, both in optical
+(Hutsem´ekers, 1998) and radio (Tiwari & Jain, 2013;
+Taylor & Jagannathan, 2016) data sets that suggest
+alignment of galaxy axes and integrated linear polar-
+izations. These observations can be nicely explained in
+terms of the correlated magnetic field which may be of
+primordial origin (Tiwari & Jain, 2016). Intriguingly,
+the optical alignment is seen to be very prominent in the
+direction of the CMB dipole (Ralston & Jain, 2004).
+2.1.4 Dipole in radio polarization offset angles:
+The
+integrated polarizations of radio galaxies are known to
+be aligned approximately perpendicular to the galaxy
+position axes. Remarkably, the angle between these
+two axes shows a dipole pattern in the sky with pre-
+ferred axis again pointing roughly along the CMB
+dipole (Jain & Ralston, 1999). Hence, we see that sev-
+eral diverse observations appear to indicate the same
+preferred direction. Taken together, they are strongly
+suggestive of a violation of the CP (Ralston & Jain,
+2004).
+2.1.5 Dipole modulation and the Hemispherical Asym-
+metry:
+We find that the CMB temperature fluctua-
+tions have slightly higher power in the southern ecliptic
+hemisphere than the northern one. This is called the
+hemispherical power asymmetry and was first observed
+in the WMAP data (Hoftuft et al., 2009) and continues
+to persist in the Planck measurements (Planck Collab-
+oration et al., 2020). It is also observed that the CMB
+temperature fluctuations appear to be modulated by a
+
+Page 4 of
+J. Astrophys. Astr. (0000) 000:
+dipole that points close to the south ecliptic pole. This
+implies that the CMB temperature fluctuation along
+line-of-sight direction ˆn is given by:
+∆T(ˆn) = ∆Tiso [1 + Aλ · ˆn] ,
+(4)
+where ∆Tiso satisfies CP, A is the amplitude of the
+dipole and λ is the preferred direction. Current Planck
+measurements (Planck Collaboration et al., 2020) give
+A = 0.070+0.032
+−0.015 and λ = (221◦, −21◦) ± 31◦. Such a
+dipole modulation would lead to difference in powers
+in the two hemispheres along ˆλ.
+2.1.6 Other CMB observations:
+Other observations
+of SI violations in the CMB are low in significance,
+albeit they are present in both WMAP and Planck data.
+For low-ℓ values, the even multipoles are anomalously
+smaller than the odd multipole modes in power. This
+is called the parity asymmetry. The largest asymme-
+try are evidenced in the lowest multipoles, viz., ℓ ∈
+[2, 7].
+These low multipoles show an anomalously
+small power, which is called the low power on large
+scales in the CMB temperature fluctuations.
+2.2 Hubble Tension
+The Hubble tension is the disagreement in measured
+value of the Hubble parameter H0 from different meth-
+ods.
+The local universe measurements of H0 using
+the ‘distance ladder’ method with Cepheids and super-
+novae type Ia (SNIa) or strong lensing systems dif-
+fer from the measurements from the CMB assuming
+ΛCDM. Other methods like tip of the red giant branch
+(TRGB) (Reid et al., 2019) or gravitational wave events
+(Gayathri et al., 2020; Mukherjee et al., 2020) have
+measured value somewhere between the two. Broadly
+speaking, H0 measurements from the local universe
+is larger than the measurements from the CMB at
+nearly 5σ significance (Anchordoqui & Perez Bergli-
+affa, 2019).
+The Cepheid-SNIa measurements use Cepheid
+variables in host galaxies of SNIa, to calibrate the dis-
+tance.
+These calibrated type Ia supernovae are then
+used to calibrate magnitude and redshift of a large
+sample of SNIa. The full sample of SNIa probes the
+Hubble flow and is used to directly infer the Hub-
+ble parameter. Riess et al. (2019) estimate the value
+H0 = 74.03 ± 1.42 kms−1Mpc−1. This agrees with the
+Freedman et al. (2012) estimate of H0 = 74.3 ± 2.1
+kms−1Mpc−1.
+The H0LiCOW team’s (Wong et al.,
+2020) recent measurement, using the time delay for
+a system of six gravitationally lensed quasars, yields
+H0 = 73.3+1.7
+−1.8 kms−1Mpc−1 that agrees very well with
+Cepheid (Freedman et al., 2001) measurements.
+In addition to the aforementioned ‘direct’ measure-
+ments, CMB can also be used to infer the value of
+the Hubble parameter ‘indirectly’. The CMB T and E
+mode measurements are used to fit the ΛCDM model.
+In its basic form, ΛCDM has only six parameters. The
+Hubble parameter can be estimated indirectly from the
+best fit. This indirect estimation of the H0 gives a value
+lower than the direct measurements. Planck Collabora-
+tion et al. (2018) gives H0 = 67.27 ± 0.60 kms−1Mpc−1
+using only T and E mode data. Estimates of H0 us-
+ing other CMB experiments like ACT (Dunkley et al.,
+2011) and SPTpol (for ℓ < 1000) (Henning et al., 2018)
+give consistent results with Planck.
+3. Superhorizon perturbation model
+It has been suggested that the superhorizon perturba-
+tions can explain the observed violations of statisti-
+cal isotropy. These are perturbations with wavelengths
+larger than the particle horizon (Erickcek et al., 2008a).
+Such modes necessarily exist in a cosmological model.
+However, in order to explain the the observed viola-
+tions of isotropy (Gordon et al., 2005) we also need
+them to be aligned with one another. In Gordon et al.
+(2005), such an alignment is attributed to a stochas-
+tic phenomenon known as spontaneous breakdown of
+isotropy. Alternatively, the alignment may be attributed
+to an intrinsic violation of the cosmological principle.
+A very interesting possibility is presented in Aluri &
+Jain (2012) and Rath et al. (2013). It is argued that
+during its very early phase, the Universe may not be
+isotropic and homogeneous. As explained in §1., it ac-
+quires this property during inflation (Wald, 1983). The
+modes which originate during the early phase of in-
+flation when the Universe deviates from isotropy and
+homogeneity may not obey the cosmological principle
+(Rath et al., 2013).
+We postulate that these are the
+aligned superhorizon modes.
+3.1 Resolution of various anomalies
+Cosmological implications of this phenomenon have
+been obtained by assuming the existence of a single
+adiabatic mode (Erickcek et al., 2008a,b; Ghosh, 2014;
+Das et al., 2021; Tiwari et al., 2022). Working in the
+conformal Newtonian Gauge, such a mode can be ex-
+pressed as,
+Ψp = ϱ sin(κx3 + ω)
+(5)
+Thus a superhorizon mode is characterised by its am-
+plitude ϱ, wavenumber κ and phase factor ω � 0. In
+Eq. (5), we have taken the mode to be aligned along
+the x3 (or z) axis which we also assume to be the direc-
+tion of CMB dipole. For a superhorizon mode, we have
+
+J. Astrophys. Astr. (0000)000:
+Page 5 of
+κ/H0 ≪ 1.
+It has been shown that such a superhorizon mode is
+consistent with all existing cosmological observations
+(like CMB, NVSS constraints etc.) for a range of pa-
+rameters (Ghosh, 2014; Das et al., 2021; Tiwari et al.,
+2022). Some parameter values are given in Table 2.
+It can affect the large scale distribution of matter and
+can potentially explain the enigmatic excess dipole sig-
+nal observed in the radio galaxy distribution (Singal,
+2011; Gibelyou & Huterer, 2012; Rubart & Schwarz,
+2013; Tiwari et al., 2015; Tiwari & Jain, 2015; Tiwari
+& Nusser, 2016; Colin et al., 2017).
+The observed matter dipole, Dobs is expressed as:
+Dobs = �Dkin + Dgrav + Dint
+�ˆx3,
+(6)
+where Dkin, Dgrav and Dint respectively denote the am-
+plitudes of the kinematic, gravitational and intrinsic
+dipoles.
+These components are redshift dependent.
+Thus we can write the magnitude of the observed dipole
+between the redshifts z1 and z2, due to the superhorizon
+mode (5) as
+Dobs(z1, z2) =
+�
+A1(z1, z2) + A2(z1, z2)
++ C(z1, z2)
+�ϱκ cos ω
+H0
++ B
+(7)
+where the term
+B = [2 + x(1 + α)]v
+c
+(8)
+is the redshift independent kinematic dipole compo-
+nent. The explicit expressions for other redshift depen-
+dent factors A1(z1, z2), A2(z1, z2), C(z1, z2) are given in
+(Das et al., 2021). Notice that in the absence of a su-
+perhorizon mode, i.e., ϱ → 0, the dipole magnitude in
+Eq. (7), as expected, becomes redshift independent and
+equal to (3).
+3.1.1 The Matter Dipole:
+Due to the presence of an
+aligned superhorizon mode, an additional contribution
+to our velocity arises with respect to LSS in the CMB
+dipole direction. This is given in Eq. (2.12) of (Das
+et al., 2021). Hence it leads to a change in Dkin in com-
+parison to its prediction based on CMB dipole (Das
+et al., 2021).
+Furthermore, the superhorizon mode
+contributes through the Sachs-Wolfe (SW) and the in-
+tegrated Sachs-Wolfe (ISW) effects (Erickcek et al.,
+2008a), thereby leading to Dgrav in Eq. (6). Finally,
+the superhorizon mode leads to an intrinsic anisotropy
+in the matter distribution and hence contributes to Dint.
+Eq. (7) is the explicit expression considering all these
+effects. From the equation, it is clear that for a given
+value of ϱ > 0, the dipole contribution is maximum if
+ω = π. All the contributions due to the mode depend on
+redshift since the mode has a systematic dependence on
+distance and hence the predicted dipole is redshift de-
+pendent.
+It is interesting to note that the contributions of the
+superhorizon mode to the CMB dipole cancel out at
+the leading order (Erickcek et al., 2008a). Such a can-
+cellation does not happen in the case of matter dipole
+(Das et al., 2021). We may understand this as follows.
+As per Eq. (6), there are three different contributions
+to the matter dipole – (a) the kinematic dipole which
+arises due to our velocity relative to the source, (b) the
+gravitational dipole (SW and ISW) and (c) the intrin-
+sic dipole. In the case of CMB, these three add up to
+zero. In the case of matter dipole, the kinematic dipole
+explicitly depends on the parameter α which arises in
+the spectral dependence of the flux from a source, as
+well as the parameter x (see Eq. (8)) which arises in
+the number count distribution. Furthermore, the gravi-
+tational effect also depends on α. The intrinsic dipole,
+however, does not depend on either of these parame-
+ters. We point out that both of these parameters arise
+at non-linear order in the theory of structure formation
+and furthermore the assumed power law distribution is
+only an approximation (Tiwari et al., 2015). These pa-
+rameters are best extracted from observations and can-
+not be reliably deduced theoretically. Hence, the sit-
+uation is very different in the case of matter dipole in
+comparison to CMB dipole and we do not expect that
+the two would behave in the same manner. We clar-
+ify that in the case of matter dipole, the superhorizon
+perturbation is treated at first order in perturbation the-
+ory. However, the small wavelength modes which are
+responsible for structure formation have to be treated at
+nonlinear order. In Das et al. (2021), the existence of
+structures is assumed as given with their properties de-
+duced observationally and the calculation focuses only
+on the additional contribution due to the superhorizon
+mode. However, a complete first principles calculation
+would have to treat small wavelength modes at nonlin-
+ear order.
+There are some further issues, associated with
+gauge invariance (Challinor & Lewis, 2011; Bonvin &
+Durrer, 2011), which are not addressed in Das et al.
+(2021). These issues are very important, but to the best
+of our understanding, they are expected to lead to small
+corrections to the calculational framework used in Das
+et al. (2021) and not expected to qualitatively change
+their results. It will be very interesting to repeat these
+calculations using the gauge invariant framework, but
+this is beyond the scope of the present paper. Such a
+calculation must also take into account the fact that the
+aligned superhorizon modes we are considering do not
+arise within the ΛCDM model but perhaps due to an
+
+Page 6 of
+J. Astrophys. Astr. (0000) 000:
+anisotropic/inhomogeneous early phase of cosmic ex-
+pansion (Aluri & Jain, 2012; Rath et al., 2013).
+3.1.2 Alignment of Quadrupole and Octupole:
+Fur-
+ther, the superhorizon mode can also explain the align-
+ment of CMB quadrupole and octupole (Gordon et al.,
+2005). With x3 axis along the CMB dipole, it leads
+to non-zero spherical harmonic coefficients T10, T20,
+T30 in the temperature anisotropy field (Erickcek et al.,
+2008a).
+We obtain constraints on the mode param-
+eters in Eq.
+(5) by requiring that T20 and T30 are
+less than three times the measured rms values of the
+quadrupole and octupole powers respectively (Erickcek
+et al., 2008a). It turns out that the dipole contribution
+does not lead to a significant constraint. These con-
+tributions can explain the alignment of quadrupole and
+octupole if we assume the presence of an intrinsic con-
+tribution to T10 and T20 which is partially cancelled by
+the contribution due to the superhorizon mode. Note
+that this intrinsic contribution is statistical in nature and
+hence its exact value cannot be predicted.
+3.1.3 Hubble Tension:
+It has been shown by Das
+et al. (2021) that a superhorizon mode leads to a per-
+turbation in the gravitational potential between distant
+galaxies and us. This culminates in a correction in ob-
+served redshift of galaxies.
+1 + zobs = (1 + z)(1 + zDoppler)(1 + zgrav)
+(9)
+Thus we see that in the presence of superhorizon
+modes, the galaxy at redshift z is observed instead at a
+redshift zobs. In the above equation, the redshifts zDoppler
+and zgrav are respectively due to our velocity relative to
+LSS and perturbation in potential introduced by the su-
+perhorizon mode. We can express zobs as (Das et al.,
+2021; Tiwari et al., 2022),
+zobs = ¯z + γ cos θ + . . .
+(10)
+where the first and the second terms on the RHS are the
+monopole and dipole terms. Here θ is the polar angle
+of the source with x3 axis along the CMB dipole and γ
+the dipole amplitude. Interestingly, the monopole term
+in Eq. (10) resolves the Hubble tension (Tiwari et al.,
+2022). For that we need to choose the phase ω � π.
+The range of parameters which explain both the matter
+dipole and the Hubble tension is given in (Tiwari et al.,
+2022). In Table 2, we quote some of those values.
+The superhorizon modes are also likely to leave
+their signatures in other cosmological observables like
+Baryon Acoustic Oscillations, epoch of reionziation
+etc.
+ω
+ϱ
+κ/H0
+1
+0.81π
+0.97
+2.58 × 10−3
+2
+0.81π
+0.48
+6.4 × 10−3
+Table 2. Some parameter values for the superhorizon mode
+(Eq. 5) explaining NVSS excess dipole and also resolving
+the Hubble tension.
+These values also satisfying CMB
+constraints are taken from Tiwari et al. (2022).
+4. Constraints using SKA
+4.1 Superhorizon perturbation observation
+The superhorizon model predicts several observations
+which can be tested with SKA and other future surveys.
+An interesting feature is the significant dependence of
+dipole on the redshift in the presence of superhorizon
+modes.
+Hence, the dipole measurements in redshift
+bins with SKA1 and SKA2 continuum survey can work
+as a potential test of the model. For a radio contin-
+uum survey, it is unlikely that we would have spec-
+troscopic redshift information.
+In the past, redshifts
+of radio galaxies have been estimated by taking cross
+correlation with well known redshift surveys (Blake &
+Wall, 2002; Tiwari et al., 2015). Such strategies are
+still viable by using data from the GAMA survey fields
+(Baldry et al., 2018). However, new techniques like
+template fitting (Duncan et al., 2018) or machine learn-
+ing based photometric redshift computations (Brescia
+et al., 2021) make it possible for the SKA radio con-
+tinuum survey galaxies to contain redshift information.
+This added information provides a unique possibility to
+test superhorizon mode physics with the SKA.
+4.2 Predictions
+Here, we demonstrate how precisely evident the dipole
+predictions with a superhorizon model would be. Fur-
+ther, we demonstrate how much they are constrained
+using SKA observations.
+Assuming superhorizon
+modes that (a) satisfy the present NVSS and Hubble pa-
+rameter measurements and (b) are consistent with CMB
+and other cosmological measurements (see Table 2);
+we obtain the dipole magnitude Dobs in redshift bins
+using the formalism described in (Tiwari et al., 2022).
+The dependence of dipole signal on the redshift z is
+shown in Figure 2 where we have shown the redshift
+dependence of Dobs for two cases
+1. Cumulative Redshift Bins: For this case, we fix
+z1 = 0 in Eq. (7) and vary z2 = z. In other words,
+we calculate Dobs(0, z).
+2. Non-overlapping Redshift Bins: In this case, we
+obtain the non overlapping z dependence by eval-
+
+J. Astrophys. Astr. (0000)000:
+Page 7 of
+1
+2
+3
+4
+5
+6
+z
+0.010
+0.011
+0.012
+0.013
+0.014
+0.015
+0.016
+Dobs
+cumulative redshift bins
+non-overlapping redshift bins
+Figure 2. The dipole signal observation in the presence of
+superhorizon modes with SKA1 (z ≤ 5) and SKA2 (z ≤ 6)
+continuum surveys.
+The inner and outer shaded regions
+respectively represent the optimistic and realistic uncer-
+tainties for both SKA1 & SKA2. Here we have assumed a
+superhorizon mode (Tiwari et al., 2022) satisfying present
+NVSS dipole observation (Tiwari et al., 2015).
+Flux Density
+SKA1
+SKA2
+Optimistic
+> 10
+> 1
+Realistic
+> 20
+> 5
+Table 3. Optimistic and realistic flux densities (in µJy) for
+SKA1 and SKA2 surveys.
+uating Dobs(z − ∆z, z + ∆z) with ∆z = 0.25. For a
+fix ∆z, this thus gives Dobs at z.
+4.3 Estimating Uncertainties
+We employ Alonso et al. (2015) ‘Ultra-large scales’
+codes2 (for continuum surveys) to determine the num-
+ber densities for SKA surveys. We further assume that
+SKA1 and SKA2 will observe the sky up to respective
+declinations of 15◦ and 30◦. The optimistic and realis-
+tic flux densities’ limits for SKA1 and SKA2 (Square
+Kilometre Array Cosmology Science Working Group
+et al., 2020; Bengaly et al., 2018) are given in Table 3.
+Additionally, we note that SKA1 is expected to
+probe up to 0 ≤ z ≤ 5, whereas the SKA2 will reach
+up to redshift 6. We mock SKA1 and SKA2 contin-
+uum sky to determine the observational implications
+of the superhorizon model.
+We produce 1000 num-
+ber density simulation of SKA1 and SKA2 continuum
+survey for each (optimistic and realistic) flux thresh-
+old using HEALPix software (Go´rski et al., 2005),
+with Nside = 64.
+The mean number of galaxies in
+2http://intensitymapping.physics.ox.ac.uk/codes.html
+a pixel is determined using number density obtained
+from Alonso et al. (2015) code and by modelling a
+dipole with magnitude and direction expected in pres-
+ence of a superhorizon mode. Given the mean number
+density in a pixel, we call random Poisson distribution
+to emulate the galaxy count in the pixel. The galaxy
+mock thus neglects the cosmological galaxy clustering.
+This is justified since the clustering dipole in LSS is
+≈ 2.7×10−3 (Nusser & Tiwari, 2015; Tiwari & Nusser,
+2016), which is roughly five times less than the appar-
+ent dipole in LSS3 and inconsequential for our simula-
+tions. This is roughly equal to the uncertainties in the
+measured dipole is LSS using NVSS galaxies (see Ta-
+ble 1). We consider SKA1, SKA2 sky coverage, i.e.,
+mask declination above 15◦ and 30◦, respectively. Ad-
+ditionally, we mask the galactic latitudes (|b| < 10◦) in
+order to remove Milky Way contamination. The galac-
+tic plane cut is often chosen to be anywhere between
+|b| < 5◦ and |b| < 15◦, and in most studies one tests
+the robustness of the results with varying redshift cuts.
+For tests on mock data, here we choose a typical cut
+of |b| < 10◦ that should balance the exclusion of galac-
+tic plane contamination and loss of sky fraction. Next,
+we use Python Healpy4 (Go´rski et al., 2005; Zonca
+et al., 2019) fit dipole function and obtain dipole
+for each 1000 mock maps. From these 1000 dipole val-
+ues, we calculate the standard deviation to determine
+the uncertainty in measurements. The shaded regions
+in Figure 2 show the results obtained for SKA1 and
+SKA2 optimistic and realistic number densities in non-
+overlapping and cumulative redshift bins.
+4.4 Other Anisotropy tests with SKA
+If the universe does not follow CP at large distance
+scales then every observable should have directional
+dependence characteristics. Out of all, three observ-
+ables are of particular interest as these are independent
+of the number density over the sky. So these are more
+robust under unequal coverage and systematics of the
+sky. These observables are
+• Mean spectral index (¯α) – As we said in §2.1.1,
+the spectral index for a radio source is defined
+between flux density and frequency through S ∝
+ν−α. In the Healpy pixelation scheme, the sky is
+divided into equal area pixels. For a given pixel p
+with Np sources having spectral indices αi,p, we
+3These estimates correspond to NVSS galaxies.
+Assuming the
+NVSS measured dipole in LSS is true, we expect similar numbers
+from SKA continuum surveys.
+4https://healpy.readthedocs.io/en/latest/index.html
+
+Page 8 of
+J. Astrophys. Astr. (0000) 000:
+define mean spectral index
+¯αp = 1
+Np
+�
+i
+αi,p
+(11)
+here i runs over all the sources in pixel p
+• Exponent (x) of differential number count – De-
+fined using N(S, ˆn) ∝ S −1−x
+• Average Flux Density ( ¯S ) – We define this quan-
+tity for a pixel p
+¯S p = 1
+Np
+�
+i
+S i,p
+(12)
+where again i runs over all the sources in p and
+Np is the number of sources in the pixel
+The spectral index characterises the morphology of
+an astronomical source. Angular dependence of ¯α has
+not been much looked at in the literature. Analysing
+the dipole anisotropy in ¯α has been a challenge since it
+requires reliable multi-frequency continuum radio sky
+survey. Such an SKA survey can be used to estimate
+this anisotropy if the flux density of the sources at dif-
+ferent frequencies is measured with sufficient accuracy.
+For x, the angular dependency was analysed by (Ghosh
+& Jain, 2017) in NVSS data using likelihood maximisa-
+tion and the results were found to be consistent with CP.
+However, with a larger expected source count of SKA
+that is almost twice in comparison to NVSS, angular
+dependence analysis of x may provide a more stringent
+test of CP.
+SKA will also be able to test the phenomenon of
+alignment of radio galaxy axes and integrated polariza-
+tions, as claimed in earlier radio observations (Tiwari
+& Jain, 2013, 2016; Taylor & Jagannathan, 2016).
+5. Conclusion and Outlook
+In this paper, we have reviewed several cosmological
+signals which appear to show a violation of CP. We
+have also reviewed a model, based on aligned super-
+horizon modes which can explain some of these obser-
+vations along with the Hubble tension (Tiwari et al.,
+2022). The model can be theoretically justified by pos-
+tulating a pre-inflationary phase during which the Uni-
+verse may not be homogeneous and isotropic (Rath
+et al., 2013). The model leads to several cosmolog-
+ical predictions which can be tested at SKA. By us-
+ing the best fit parameters with current observations,
+we have determined the redshift dependence of the pre-
+dicted dipole in radio galaxy number counts and associ-
+ated uncertainties. As can be seen from Figure 2, SKA
+can test this very reliably. If this prediction is confirmed
+by SKA, it may provide us with a first glimpse into the
+Physics of the pre-inflationary phase of the Universe.
+We have also suggested other isotropy tests with
+SKA using other variables which are independent of
+number density and thus are more robust under unequal
+coverage and systematics of the sky. These variables
+are (a) mean spectral index ¯α, (b) exponent of the dif-
+ferential number count x & (c) average flux density ¯S .
+Acknowledgements
+RK is supported by the South African Radio Astron-
+omy Observatory and the National Research Founda-
+tion (Grant No. 75415). PT acknowledges the support
+of the RFIS grant (No. 12150410322) by the National
+Natural Science Foundation of China (NSFC) and the
+support by the National Key Basic Research and De-
+velopment Program of China (No. 2018YFA0404503)
+and NSFC Grants 11925303 and 11720101004.
+SG
+is supported in part by the National Key R & D Pro-
+gram of China (2021YFC2203100), by the Fundamen-
+tal Research Funds for the Central Universities under
+grant no: WK2030000036, and the NSFC grant no:
+11903030. We are also very thankful to the anonymous
+referee whose comments were really helpful in improv-
+ing the presentation of the paper.
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+Bootstrap Embedding on a Quantum Computer
+Yuan Liu,∗,† Oinam R. Meitei,‡ Zachary E. Chin,† Arkopal Dutt,¶ Max Tao,†
+Troy Van Voorhis,∗,‡ and Isaac L. Chuang†,§
+†Department of Physics, Co-Design Center for Quantum Advantage, Massachusetts
+Institute of Technology, Cambridge, Massachusetts 02139, USA
+‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge,
+Massachusetts 02139, USA
+¶Department of Mechanical Engineering, Massachusetts Institute of Technology,
+Cambridge, Massachusetts 02139, USA
+§Department of Electrical Engineering and Computer Science, Massachusetts Institute of
+Technology, Cambridge, Massachusetts 02139, USA
+E-mail: [email protected]; [email protected]
+Abstract
+We extend molecular bootstrap embedding to make it appropriate for implementa-
+tion on a quantum computer. This enables solution of the electronic structure problem
+of a large molecule as an optimization problem for a composite Lagrangian governing
+fragments of the total system, in such a way that fragment solutions can harness the
+capabilities of quantum computers. By employing state-of-art quantum subroutines
+including the quantum SWAP test and quantum amplitude amplification, we show how
+a quadratic speedup can be obtained over the classical algorithm, in principle. Utiliza-
+tion of quantum computation also allows the algorithm to match – at little additional
+computational cost – full density matrices at fragment boundaries, instead of being
+limited to 1-RDMs. Current quantum computers are small, but quantum bootstrap
+1
+arXiv:2301.01457v1  [quant-ph]  4 Jan 2023
+
+embedding provides a potentially generalizable strategy for harnessing such small ma-
+chines through quantum fragment matching.
+1
+Introduction
+Determining the ground state of large-scale interacting fermionic systems is an important
+challenge in quantum chemistry, materials science, and condensed matter physics. Just as
+electronic properties of molecules underpin their chemical reactivity,1–3 phase diagrams of
+solid state materials are also determined to a large degree by their ground state electronic
+structure.4–6 However, close to exact solution to the time-independent Schrodinger equation
+of a practical many-electron system remains a daunting task because the dimension of the
+underlying Hilbert space grows exponentially with the number of orbitals, and the computa-
+tional resources required to perform calculations over such a large space can quickly exceed
+the capacity of current classical or quantum hardware.
+One promising approach to fit a large electronic structure problem into a limited amount
+of computational resources is to break the original system into smaller fragments, where
+each fragment can be solved individually from which a solution to the whole is then ob-
+tained.7–9 Efforts along this direction have successfully led to various embedding schemes
+that significantly expand the complexity of the systems solvable using classical computa-
+tional resources, such as density-based embedding theories,10,11 density-matrix embedding
+theories (DMET),12–16 various Green’s function embedding theories6,17–21 and the bootstrap
+embedding theory.22–24 The essence of such embedding-based methods is to add an additional
+external potential to each fragment Hamiltonian and then iteratively update the potential
+until some conditions on certain observables of the system are matched. Nevertheless, due to
+the significant cost in solving the fragment Hamiltonian itself as the fragment size increases,
+the applicability of such methods are limited to relatively small fragments, which may lead to
+incorrect predictions in systems with long-range correlations.25 While approximate fragment
+2
+
+solvers such as the coupled-cluster theory or many-body perturbation theory have greatly
+enhanced the applicability of such embedding methods at a reduced cost,26–28 these approx-
+imations tend to fail for strongly correlated systems due to limited treatment of electron
+correlation. In addition, because of limitations on computing k-electron reduced density
+matrices (k-RDMs for k > 2), embedding and observable calculations beyond 2-RDM are
+difficult in general.
+Quantum computers are believed to be promising in tackling electronic structure prob-
+lems more efficiently,29 despite the possibility of an exponential speedup still being unclear.30
+One natural idea to circumvent the problems of classical eigensolvers is to use a quantum
+computer to treat the fragments. By mapping each orbital to a constant (small) number of
+qubits, the exponentially large (in the number of orbitals) Hilbert space of an interacting
+fermionic system can be encoded in only a polynomial number of qubits and terms. Indeed,
+quantum eigensolvers such as the quantum phase estimation (QPE)31 algorithm has been
+proposed to achieve an exponential advantage given a properly prepared input state32 with
+non-exponentially small overlap with the exact ground state. More recently, various variants
+of the variational quantum eigensolver (VQE)33–37 have been demonstrated experimentally
+on NISQ devices to achieve significant speedup without sacrificing accuracy as compared
+to classical methods. Moreover, k-RDMs (for any k) can be measured through quantum
+eigensolvers38,39 that may circumvent the difficulty encountered on classical computers.
+To take the full advantage of these quantum eigensolvers within the embedding frame-
+work,18,40–44 two open questions immediately arise as a result of the intrinsic nature of
+quantum systems. Firstly, the wave function of a quantum system collapses when measured.
+This means any measurement of the fragment wave function is but a statistical sample (akin
+to Monte Carlo methods), and many measurements are needed to obtain statistical averages
+with sufficiently low uncertainty in order to achieve a good matching condition for the em-
+bedding. Secondly, the best way to perform matching between fragments using results from
+quantum eigensolvers is not clear, and most likely a new approach needs to be formulated
+3
+
+to match fragments. Admittedly, it would be straightforward to first estimate the density
+matrices by collecting a number of quantum samples and then use the estimated density ma-
+trices to minimize the cost function as in classical embedding theories.12,22 But this approach
+would be very costly especially given the increasing number of elements in qubit reduced
+density matrices (RDMs) that need to be estimated.45 Could there be a quantum method
+for matching, as opposed to a statistical sampling-based classical approach?
+We address the two challenges by providing a quantum coherent matching algorithm and
+an adaptive sampling schedule, leading to a quantum bootstrap embedding (QBE) method
+based on classical bootstrap embedding.22 Instead of matching the RDM element-by-element,
+the quantum matching algorithm employs a SWAP test46,47 to match the full RDM between
+overlapping regions of the fragments in parallel. Moreover, the quantum amplitude estima-
+tion algorithm48,49 allows an extra quadratic speedup to reach a target accuracy on estimating
+the fragment overlap. In addition, the adaptive sampling changes the number of samples
+as the optimization proceeds in order to achieve an increasingly better matching conditions.
+The present work invites a viewpoint of treating quantum computers as coherent sampling
+machines which have three major advantages, as compared to their classical counterparts.
+First, the exponentially large Hilbert space provided by a quantum computer allows more
+efficient exact ground state solver (QPE) than their classical counterpart (exact diagonal-
+ization). Second, in the case of truncation for seeking approximate solutions, the abundant
+Hilbert space of quantum computers enable more flexible and expressive variational ansatz
+than classical computers, leading to more accurate solutions. Third, the coherent nature of
+quantum computers allows sampling to be performed at a later stage, e.g. after quantum
+amplitude amplification of matching conditions to extract just the feedback desired, instead
+of having to read out full state of a system.
+The rest of the paper is organized as follows. Sec. 2 overviews bootstrap embedding
+method at a high level and analyzes its scaling on classical computers, in order to motivate
+the need for bootstrap embedding on quantum computers. This section serves to set the
+4
+
+notation and baseline of comparison for the rest of the paper. Sec. 3 presents the theoretical
+framework of quantum bootstrap embedding in detail as constraint optimization problems.
+In Sec.
+4, we give details of the QBE algorithm to solve the optimization problem.
+In
+Sec. 5, we apply our methods to hydrogen chains under minimal basis where both classical
+and quantum simulation results are shown to demonstrate the convergence and sampling
+advantage of our QBE method. We conclude the paper in Sec. 6 with prospects and future
+directions.
+2
+Ideas of Bootstrap Embedding
+The idea of Bootstrap Embedding (BE) for quantum chemistry has recently led to a promis-
+ing path to tackle large-scale electronic structure problems.22,23,50 In this section, we establish
+the terminology and framework that will be used in the rest of the paper. We first briefly
+review BE and outline the main framework of BE for computation on a classical computer in
+Sec. 2.1 and 2.2 for non-chemistry readers, to set up the notation. We then begin presenting
+new material by discussing typical behavior and computational resource requirements for BE
+on classical computers in Sec. 2.3, which leads to the quest for performing BE on a quantum
+computer in Sec. 2.4.
+2.1
+Fragmentation and Embedding Hamiltonians
+To provide a foundation for a more concrete exposition of the bootstrap embedding method,
+we first establish some rigorous notation for discussing molecular Hamiltonians and their
+associated Hilbert spaces. We will work with the molecular Hamiltonian under the second
+quantization formalism.
+Specifically, given a particular molecule of interest, define O =
+{φµ | µ = 1, . . . , N} to be an orthonormal set of single-particle local orbitals (LOs), where
+N is the total number of orbitals; in this work, these LOs are generated through L¨owdin’s
+symmetric orthogonalization method.51 The full Hilbert space H for the entire molecular
+5
+
+system is thus given by H = F(O), where F(O) denotes the Fock space determined by
+the LOs in the set O. Further define the creation (annihilation) operator c†
+µ (cµ) which
+creates (annihilates) an electron in the LO φµ, the molecular Hamiltonian is written in the
+second-quantized notation
+ˆH =
+N
+�
+µν=1
+hµνc†
+µcν + 1
+2
+N
+�
+µνλσ=1
+Vµνλσc†
+µc†
+νcσcλ
+(1)
+where hµν and Vµνλσ are the standard one- and two-electron integrals.
+Note that the number of terms in the full molecular Hamiltonian ˆH scales polynomially
+with the total number of orbitals N, but the dimension of H scales exponentially with N.
+Clearly, for large N, it will become prohibitively expensive to directly compute the exact
+full ground state. To circumvent this issue, we divide the full molecule into multiple smaller
+fragments, each equipped with its own “embedding Hamiltonian” which contains a number of
+terms that only scales polynomially with the number of orbitals in the fragment. Given that
+there are potentially far fewer orbitals in each fragment than in the whole molecular system,
+computing the ground state of each fragment’s embedding Hamiltonian can be significantly
+less expensive than computing the ground state of the full system.
+Furthermore, using
+the bootstrap embedding procedure to be described later, the ground states of individual
+fragments can, to a high degree of accuracy, be algorithmically combined to recover the
+desired electron densities prescribed by the exact ground state of the full system. Thus, this
+combination of fragmentation and bootstrap embedding can be used to reconstruct the full
+molecular ground state more efficiently than by direct computation alone.
+We now briefly review the construction of embedding Hamiltonians for each fragment.
+Consider a single fragment associated with a label A, without loss of generality, define
+O(A) = {φµ | µ = 1, . . . , NA} with NA ≤ N to be the set of LOs contained in fragment A;
+we will refer to O(A) as the set of fragment orbitals. Note that O(A) ⊆ O, the set of LOs
+for the entire molecular system. The construction of the embedding Hamiltonian ˆH(A) for
+6
+
+fragment A begins with any solution of the ground state of the full system ˆH. For simplicity,
+the Hartree-Fock (HF) solution |ΦHF⟩ is often used because it is easy to obtain on a classical
+computer. By invoking a Schmidt decomposition, we can write |ΦHF⟩ with the following
+tensor product structure for ∀ A
+|ΦHF⟩ =
+� NA
+�
+i=1
+λ(A)
+i
+|f (A)
+i
+⟩ ⊗ |b(A)
+i
+⟩
+�
+⊗ |Ψ(A)
+env⟩ .
+(2)
+In the above decomposition, the |f (A)
+i
+⟩ represent single-particle fragment states contained
+in the Fock space F(O(A)) of fragment orbitals. On the other hand, the |b(A)
+i
+⟩ and |Ψ(A)
+env⟩
+represent Slater determinants contained in the “environment” Fock space F(O \ O(A)) of
+the N − NA orbitals not included in the fragment. The key difference between the single
+environment state |Ψ(A)
+env⟩ and the various “bath” states |b(A)
+i
+⟩ is that the bath states |b(A)
+i
+⟩ are
+entangled with the fragment states |f (A)
+i
+⟩ while |Ψ(A)
+env⟩ is not; this entanglement is quantified
+by the Schmidt coefficients λ(A)
+i
+. Crucially, since the HF solution is used, the sum in Eq.
+(2) only has NA terms (as opposed to 2NA for a general many-body wave function). Denote
+the collection of the NA entangled bath orbitals as O(A)
+bath = {βµ |µ = 1, . . . , NA}, where each
+of the LOs βµ are linear combinations of the original LOs not included in the fragment,
+βµ ∈ Span{O \ O(A)}. Furthermore, we denote the Fock space that corresponds to this set
+of entangled bath orbitals as F(O(A)
+bath).
+This tensor product structure of |ΦHF⟩ allows us to naturally decompose the Hilbert space
+H for the full molecular system into the direct product of two smaller Hilbert spaces, namely
+H = H(A) ⊗ H(A)
+env,
+(3)
+where
+H(A) = F(O(A)) ⊗ F(O(A)
+bath)
+(4)
+7
+
+is the active fragment embedding space and H(A)
+env contains the remaining states, including
+|Ψ(A)
+env⟩. Note that since both sets O(A) and O(A)
+bath have size NA, the fragment Hilbert space
+H(A) is a Fock space spanned of just 2NA single-particle orbitals. The core intuition mo-
+tivating this decomposition is that, in the exact ground state of the full system, states in
+H(A)
+env are unlikely to be strongly entangled with the many-body fragment states (consider the
+approximate HF ground state in Eq. (2), where they are perfectly disentangled); therefore,
+in a mean-field approximation, it is reasonable to entirely disregard the states in H(A)
+env when
+calculating the ground state electron densities on fragment A. Following this logic, we can
+define an embedding Hamiltonian ˆH(A) for fragment A only on the 2NA LOs in H(A), which
+will have the form
+ˆH(A) =
+2NA
+�
+pq
+h(A)
+pq a(A)†
+p
+a(A)
+q
++ 1
+2
+2NA
+�
+pqrs
+V (A)
+pqrsa(A)†
+p
+a(A)†
+q
+a(A)
+s a(A)
+r
+,
+(5)
+given some creation and annihilation operators a(A)†
+p
+and a(A)
+p , which respectively create and
+annihilate electrons in orbitals from the combined set O(A) ∪ O(A)
+bath for H(A). The new one-
+and two- electron integrals h(A)
+pq and V (A)
+pqrs can be computed by projecting ˆH into the smaller
+Hilbert space H(A) (consult the Supporting Information (SI) Sec. S1 for details). Note that
+since we can choose 2NA ≪ N, the ground state of this embedding Hamiltonian can be
+solved at a significantly reduced cost when compared to that of the full system Hamiltonian.
+We are hence prepared to generate an embedding Hamiltonian for any arbitrary frag-
+ment of the original molecular system. However, the ground state electron densities of the
+fragment embedding Hamiltonian are unlikely to exactly match those of the full system
+Hamiltonian because, as mentioned above, the embedding process may neglect some small
+(but nonzero) entanglement of the fragment orbitals with the environment. Because we can
+expect interactions in the molecular Hamiltonian to be reasonably local, we anticipate that
+the electron densities on orbitals near the edge of the fragment (those closest to the “envi-
+ronment”) will deviate most significantly from their true values, while electron densities on
+8
+
+orbitals toward the center of the fragment will be most accurate.
+To improve the accuracy of the fragment ground state wave function near the fragment
+edge, we employ the technique of bootstrap embedding. Broadly speaking, we first divide the
+full molecule into overlapping fragments such that the edge of each fragment overlaps with
+the center of another. Fig. 1i illustrates this fragmentation strategy: for example, we see
+that the edge of fragment A (labeled as orbital 3) coincides with the center of fragment B.
+We then apply additional local potentials to the edge sites of each fragment to match their
+electron densities to those on overlapping center sites of adjacent fragments. Because we
+expect the electron densities computed on the center sites to be closer to their true values,
+these added local potentials should improve the accuracy of each fragment wave function
+near the edges. In the next section, we will formalize this edge-to-center matching process
+rigorously and discuss its implementation on a classical computer.
+2.2
+Matching Electron Densities: an Optimization Problem
+As mentioned in the previous section, we intend to correct the electron density error near
+a fragment’s edge by applying a local potential to the edge; this local potential serves to
+match the edge electron density of the fragment to the center electron density of an adjacent
+overlapping fragment, which we expect to be more accurate. In principle, to achieve an
+exact density matching, all k-electron reduced density matrices (k-RDM, for any k) on the
+overlapping region have to be matched. However, in practice, such matching beyond the 2-
+RDM is difficult on a classical computer due to the mathematical challenge that the number
+of terms in k-RDM in general increases exponentially as k. In addition, almost all electronic
+structure codes available on classical computers are programmed to deal with only 1- and
+2-RDMs, despite the importance of k-RDMs (k > 2) for computing observables such as
+entropy and other multi-point correlation functions.52 Due to this reason, the discussion of
+density matching process in classical BE in this section will be based on 1-RDMs. We note
+that the matching process applies similarly if k-RDMs are matched.
+9
+
+Figure 1: Schematic of bootstrap embedding on classical (left, blue arrows) and quantum
+(right, red arrows) computers. The arrows indicate BE iterative loops that are used to
+optimize the corresponding objective functions. Starting from panel (i) (upper center), the
+original system is first broken into overlapping fragments (Fragmentation), where each
+fragment is solved using a classical (iic) (upper left) or quantum eigensolver (iiq) (upper
+right). In classical matching, the 1-electron reduced density matrices (1-RDM) on the
+overlapping sites of adjacent fragments are used to obtain the matching condition (iiic)
+(lower left), while in the quantum case a coherent matching protocol based on SWAP tests of
+overlapping sites combined with a single qubit measurement (iiiq) (lower right). The
+matching results are then used by classical computers to generate the bootstrap embedding
+potential VBE (iv) (lower center) and the updated fragment embedding Hamiltonian
+Hemb + VBE (back to panel (i) in order to minimize a target objective function L in both
+classical and quantum case.
+10
+
+(iic)
+(i)
+Fragmentation
+(ig) Quantum Eigensolver
+Classical
+TTOTOTTOOTOTO
+(QPE, VQE, ...)
+Eigensolver
+000000
+Frag A
+Hemb+VBE
+0-010!
+FragA
+FCI
+CCSD
+000000
+FragB @10i0
+Frag B
+VMC
+FragA
+Frag C
+000000
+010010
+Frag B
+4
+FragD
+000000
+Classical BE
+Quantum BE
+1-RDMs
+VBE
+RDMs
+L=(Hemb)+Qx
+L = <H.
+(ilic)
+Classical Matching
+Generate BE Potential VBE
+(ilig) Coherent Matching
+(iv)
+P)
+Frag A
+VBE
+<ol
+H
+HHAM
+Frag A
+Lp(a)
+(1
+3
+<M>
+Frag A
+QESA
+[29]
+[亚)
+1-RDM
+Subsystem
+P(2)
+P(c)
+[[]
+Frag B
+difference
+overlap
+Frag B
+QESB
+[P()
+ad
+P(P)We begin by introducing some rigorous notation. Recall that a fragment A is defined by
+a set of local orbitals O(A) which constitute the fragment. We partition this set of LOs into
+a subset of edge sites (or orbitals), denoted E(A), and a subset of center sites, denoted C(A),
+such that E(A) ∪ C(A) = O(A) and E(A) ∩ C(A) = ∅. Given the ground state wave function
+|Ψ(A)⟩ of the embedding Hamiltonian, we further define the 1-electron reduced density matrix
+(1-RDM) P(A) according to
+P (A)
+pq
+= ⟨Ψ(A)| a(A)†
+p
+a(A)
+q
+|Ψ(A)⟩
+(6)
+where p, q = 1, . . . , 2NA and the operators a(A)†
+p
+and a(A)
+q
+are defined in the previous section.
+Suppose, for example, that the edge of fragment A overlaps with the center of another
+fragment B so that E(A) ∩ C(B) ̸= ∅. On a high level, the goal of bootstrap embedding is to
+find a ground state wave function |Ψ(A)⟩, perturbed by local potentials on the edge sites of
+A, such that |P (A)
+pq
+− P (B)
+pq | → 0 for indices p and q that correspond to orbitals in the set of
+overlapping sites E(A) ∩ C(B). More generally, and more rigorously, the goal is to find a wave
+function which minimizes the fragment Hamiltonian energy
+|Ψ(A)⟩ = arg min
+Ψ(A)⟨ ˆH(A)⟩A
+(7)
+subject to the constraints
+⟨a(A)†
+p
+a(A)
+q
+⟩A − P (B)
+pq
+= 0
+(8)
+for all other fragments B with E(A) ∩ C(B) ̸= ∅ and for all p, q corresponding to orbitals in
+E(A) ∩C(B). Here, we explicitly write the expectation ⟨·⟩A = ⟨Ψ(A)|·|Ψ(A)⟩ in terms of |Ψ(A)⟩
+to indicate that the optimization is over the wave function of A.
+We can formulate this constrained optimization problem as finding the stationary solution
+to a Lagrangian by associating a scalar Lagrange multiplier (λ(A)
+B )pq to Eq. (8). Since Eq. (8)
+11
+
+has to be satisfied for any p, q and B that overlaps with A, these constraint can be rewritten
+in a more compact vector form λ(A)
+B
+· Q1-RDM(Ψ(A); P(B)) where the dot product conceals
+the implicit sum over p, q, and each component of the vector Q1-RDM(Ψ(A); P(B))pq represents
+the constraint associated with Lagrange multiplier (λ(A)
+B )pq, given by the left hand side of Eq.
+(8). With this notation, we arrive at the following Lagrangian with the constraint added as
+an additional term
+L(A) =⟨ ˆH(A)⟩A + E(A) �
+⟨Ψ(A)| Ψ(A)⟩ − 1
+�
++
+�
+B
+λ(A)
+B
+· Q1-RDM(Ψ(A); P(B)),
+(9)
+where once again the B are fragments adjacent to A with E(A) ∩C(B) ̸= ∅ and p, q are indices
+of orbitals contained in the overlapping set E(A) ∩ C(B). Here, the additional constraint with
+Lagrange multiplier E(A) is also included to ensure normalization of the ground state wave
+function |Ψ(A)⟩. Solving for the stationary solution of the Lagrangian in Eq. (9) will only
+result in a ground state wave function for fragment A whose 1-RDM elements at the edge
+sites match those at the center sites of adjacent overlapping fragments. However, we would
+instead like to solve for such a ground state for all fragments in the molecule simultaneously.
+Toward this regard, we can combine all individual fragment Lagrangians (of the form of Eq.
+(9)) into a single composite Lagrangian for the whole molecule, given by
+L =
+Nfrag
+�
+A=1
+L(A) + µP
+(10)
+where Nfrag is the number of fragments in the molecule. Observe that we have added one
+additional constraint
+P =
+�
+�
+Nfrag
+�
+A=1
+�
+p′∈C(A)
+⟨a(A)†
+p′
+a(A)
+p′ ⟩A
+�
+� − Ne
+(11)
+with Lagrange multiplier µ to restore the desired total number of electrons in the molecule,
+12
+
+Ne. Note in Eq. (11) that p′ is summed over indices corresponding to orbitals only in C(A);
+this is to ensure that there is no double-counting of electrons in the whole molecule. By
+self-consistently finding ground states |Ψ(A)⟩ for A = 1, . . . , Nfrag which make the composite
+Lagrangian in Eq. (10) stationary, we will have completed the density matching procedure
+for all fragments, and the process of bootstrap embedding will be complete.
+We can gain insight into which wave functions |Ψ(A)⟩ will make the composite Lagrangian
+L stationary by differentiating L with respect to |Ψ(A)⟩ for some fixed fragment A and setting
+the resulting expression equal to zero. Upon some algebraic manipulation, we can recover
+the eigenvalue equation
+( ˆH(A) + VBE) |Ψ(A)⟩ = −E(A) |Ψ(A)⟩ ,
+(12)
+where VBE, the local bootstrap embedding potential, is given by
+VBE =
+�
+B
+�
+p,q
+(λ(A)
+B )pqa(A)†
+p
+a(A)
+q
++ µ
+�
+p′
+a(A)†
+p′
+a(A)
+p′
+(13)
+where the p, q are indices of orbitals in the overlapping set E(A) ∩C(B), and the p′ are indices
+of orbitals in the fragment center C(A). We see that, when the composite Lagrangian is
+made stationary with respect to the fragment wave functions, the bare fragment embedding
+Hamiltonians become dressed with a potential VBE that contains a component local to the
+edge sites of each fragment (see the left term of Eq. (13)). This observation confirms our
+intuition that adding a local potential to the edge of one fragment will allow the edge site
+electron density to be matched to that of a center site on an overlapping neighbor. Note
+that VBE also contains an additional potential on the center sites of each fragment (see the
+right term of Eq. (13)); this is simply to conserve the total electron number in the molecule.
+Moreover, VBE as in Eq. (13) only contains one-body terms because only 1-RDM is used for
+density matching. In general, VBE will contain up to k-body terms if k-RDMs are used for
+matching.
+13
+
+On a classical computer, the composite Lagrangian in Eq. (10) is made stationary through
+an iterative optimization algorithm22 until the edge-to-center matching condition for all
+fragments is satisfied by some criterion. One possible criterion is to terminate the algorithm
+when the root-mean-squared 1-RDM mismatch, given by
+ϵ =
+�
+�
+1
+Nsites
+Nfrag
+�
+A
+�
+B
+�
+p,q
+(P (A)
+pq
+− P (B)
+pq )2
+�
+�
+1
+2
+,
+(14)
+drops below some predetermined threshold. Note again that p, q are indices corresponding to
+orbitals in the overlapping set E(A)∩C(B); also, Nsites denotes the total number of overlapping
+sites in the whole molecule, equal to Nsites = �Nfrag
+A
+�
+B
+�
+p,q 1. The final set of density-
+matched fragment wave functions {|Ψ(A)⟩} for A = 1, . . . , Nfrag which solve the composite
+Lagrangian can then be used to reconstruct the electron densities and other observables for
+the full molecular system, as desired.
+2.3
+Resource Requirement and Typical Behavior of BE on Clas-
+sical Computers
+Given the notation established for classical BE, we now begin presenting new material. We
+discuss the computational resource requirement and typical behaviors of performing BE on
+classical computers to set the stage for a quantum BE theory. The details of the classical BE
+algorithms are omitted for succinctness, and we refer the reader to Ref.22–24,50 for details.
+The space and time resource requirement to perform the classical BE can be broken
+down into two parts: a) the number of iteration steps to reach a fixed accuracy for ϵ (Eq.
+(14)); b) the runtime of the fragment eigensolver. For a), numerical evidence suggests an
+exponentially fast convergence on total system energy as the number of bootstrap iteration
+increases (black trace in Fig. 2 for FCI), while a proof of the convergence rate has yet to be
+established.
+14
+
+We focus on resource requirement in b) in the following. Admittedly, an exact classical
+eigensolver such as full configuration interaction (FCI) can be used to solve the embedding
+Hamiltonian in Eq.
+(5).
+However, both the storage space and time requirement scales
+exponentially as the the number of orbitals (see blue symbols and dashed line in Fig. 3 for
+the runtime scaling of FCI). Even with the state-of-the-art classical computational resources,
+exact solutions using FCI are only tractable for systems up to 20 electrons in 20 orbitals.53
+As a result, classical computation of BE resorts to approximate eigensolvers with only
+polynomial cost in practice, by properly truncating or sampling from the fragment Hilbert
+space. One example for truncation is the coupled-cluster singles and doubles (CCSD),54
+which scales with N 6 with N being the number of orbitals. Alternately, different flavors of
+stochastic electronic structure solvers can be employed as fragment solvers in BE. Depending
+on implementation, these stochastic solvers can be biased or unbiased (if unbiased, with a
+cost of introducing the phase problem in general).55–58 Collecting each sample on a classical
+computer usually has similar cost as a mean field theory (roughly O(N 3)), while the overall
+target accuracy ϵ on observable estimation can be achieved with a sampling overhead of
+roughly O( 1
+ϵ2) with a constant prefactor depending on the severity of the sign problem.
+Importantly, the sampling feature of these stochastic electronic structure methods on
+classical computers are strikingly similar to the nature of quantum computers where mea-
+surement necessarily collapses the wave function. As a result, only a classical sample (in
+terms of measurement results) can be obtained from a quantum computer. This similarity
+suggests a general strategy that many sampling techniques in stochastic classical algorithms
+can be deployed to design better quantum algorithms. For example, sophisticated impor-
+tance sampling techniques59,60 can be employed to greatly improve the sampling efficiency
+in both classical58 and quantum cases.61
+Due their shared feature on sampling between classical stochastic algorithm and quantum
+eigensolvers, we shall use one approximate sign-problem-free flavor of stochastic electronic
+structure method, the variational Monte Carlo (VMC), to serve as an additional baseline
+15
+
+scenario for comparison with quantum BE in later sections. In addition to BE convergence
+behavior with a FCI solver, Fig. 2 also shows, for a VMC eigensolver, the density mismatch
+converges exponentially fast initially as iteration number increases with varying number of
+samples. However, due to the statistic noise on estimating the 1-RDM (thus the gradient for
+the optimization), the final density mismatch plateaus to a finite biased value. Comparing
+among the VMC solver with different number of samples, the accuracy improves as the
+number of samples increases (dashed horizontal lines).
+Figure 2: Typical convergence of density mismatch with respect to the number of
+eigensolver calls in classical bootstrap embedding with a deterministic eigensolver (FCI,
+black circle) and a stochastic eigensolver (VMC) with different number of samples (grey,
+blue, and orange solid lines). The horizontal dashed lines shows the final plateaued value of
+the density mismatch for VMC, while the FCI data converges to 10−6 after 700 eigensolver
+calls (not shown on the figure). The discrete jumps around 200 and 300 eigensolver calls
+are due to switching to the next BE iteration. The data is obtained for an H8 linear chain
+under STO-3G basis. See SI Sec. S9 B for computational details.
+The increasing accuracy of density mismatch with respect to BE iteration also suggests
+an increasing number of samples are needed. Thus, an optimal number of samples at each
+BE iteration must be determined to achieve the desired accuracy in the matching conditions.
+A careful design of such a sampling schedule can potentially save a large amount of compu-
+tational resources. We defer a thorough discussion of this point to later sections on quantum
+BE.
+16
+
+4 × 10-3
+3 × 10-3
+Density Mismatch
+M
+2X1
+FCI
+Q
+VMC (40k samples)
+6 × 10-4
+VMC (160k samples)
+VMC (640k samples)
+4 × 10-4
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Eigensolver Calls2.4
+The Quest for BE on Quantum Computers
+By employing the coherent superposition and entanglement of quantum states, the limita-
+tion of an exact classical solver can be overcome by substituting it with an exact quantum
+eigensolver such as the quantum phase estimation (QPE) algorithm.31 Fig. 3 compares the
+runtime (gate depth) of FCI and QPE for finding the ground state of linear hydrogen chain
+Hn for different system size n. Clearly, the QPE runtime scales only polynomially as the
+system size increases as expected,30,32 while its classical counterpart (FCI) has an exponen-
+tially increasing runtime. Note the runtime is normalized to the case of n = 1 for each
+solver separately (see SI Sec. S9 for details). The dramatic advantage in the runtime scaling
+of quantum over classical eigensolvers demonstrated above suggests formulating BE on a
+quantum computer can bring significant benefits.
+Figure 3: Runtime (normalized) as a function of system size n for finding the ground state
+of a linear hydrogen chain Hn at STO-3G basis, comparing an exact classical solver (FCI,
+blue square) and an exact quantum solver (QPE, red circle) on real classical and quantum
+devices. Red (blue) dashed line shows a polynomial (exponential) fit to the QPE (FCI)
+runtime. Note the crossover at large system size.
+One might think that the eigensolver at the heart of the classical BE algorithm could
+simply be replaced with a quantum one.
+However, as mentioned before, there are two
+outstanding challenges for such a quantum bootstrap embedding (QBE) method. First, just
+17
+
+108
+QPE
+FCI
+Time (normalized)
+106
+104
+/
+102
+/
+Q
+口
+口
+口
+口
+100
+口
+3
+1
+5
+7
+9
+1113
+15
+17
+19
+System Sizeas in classical stochastic methods, the results of a quantum eigensolver need to be measured
+for later use, but quantum wave functions collapse after measurement. Therefore, sampling
+from the quantum eigensolver is required, and the optimal sampling strategy is unclear.
+Secondly, with quantum wave function from quantum eigensolvers, it is not wise to achieve
+matching between fragments in the same way as classical BE, as many incoherent samples are
+needed to obtain a good estimation of the 1-RDM elements. Clearly, performing matching
+in a quantum way is desired.
+In the next two sections (Secs. 3 and 4), we present how we address these two challenges
+by an adaptive quantum sampling scheduling algorithm and a quantum coherent matching
+algorithm in detail.
+3
+Quantum Bootstrap Embedding Methods
+In previous sections, we have seen potential advantages of performing bootstrap embedding
+on quantum computers, and discussed two major challenges of doing so. In this section,
+we present the theoretical formulation of our bootstrap embedding method on a quantum
+computer that addresses these challenges.
+Sec. 3.1 first set up notations and discuss a few aspects of locality and global symmetry on
+performing embedding of fermions on quantum computers. Sec. 3.2 discuss a naive extension
+of the classical BE algorithm on quantum computers by matching individual elements of the
+RDMs directly, and highlight the disadvantage of doing so. Sec. 3.3 introduces the SWAP
+test circuit and show that it achieves the matching between two RDMs coherently. In 3.4,
+we discuss some subtleties on why it is impossible to incorporate this coherent matching
+condition into the Lagrange multiplier optimization method, and present an alternative
+quadratic penalty method to perform the optimization.
+18
+
+3.1
+Fermion-Qubit Mapping - Global Symmetry vs. Locality
+When mapping electronic structure problem to qubits on quantum computers, it is well-
+known that the global anti-symmetric property of fermionic wave functions necessarily leads
+to an overhead in operator lengths or qubit counts.62 On the other hand, chemical informa-
+tion is usually local if represented using localized single-particle orbitals.63,64 In the case of
+performing bootstrap embedding, this tension between locality of chemical information and
+global fermionic anti-symmetry is more subtle. Because bootstrap embedding intrinsically
+uses the fermionic occupation number in the local orbitals (LOs) to perform matching, it is
+therefore convenient to preserve such locality when constructing the mapping. Throughout
+the discussion, without loss of generality, we assume a mapping that preserves fermionic local
+occupation number, such as the Jordan-Wigner mapping where each spin-orbital is mapped
+to one qubit. Our discussion equally applies to cases where a non-local mapping is used (such
+as parity mapping). In that case, a unitary transformation from the non-local mapping to
+a local mapping will be required before actually computing the matching conditions. It is
+usually more convenient to work with qubit reduced density matrices (RDMs)65 on quantum
+computers instead of k-electron RDMs.66 Due to this reason, we shall formulate our QBE
+method based on these qubit RDMs. The full density matrix of fragment A is thus provided
+by ρ(A) = |ΨA⟩ ⟨ΨA|. Given an orbital set R ⊂ O(A) for O(A) being set of orbitals in fragment
+A. Let ρ(A)
+R
+signify the RDM obtained from ρ(A) by tracing out the set of qubits not in R.
+Specially, if R only contains orbitals on the edge (center) of fragment A, then ρ(A)
+R
+represents
+information about the density information (for example the occupation number) on the edge
+(center) of A.
+These RDMs can be expanded under an arbitrary set of orthonormal basis {Σα} as follows
+ρ(A)
+R
+= I + �4m−1
+α=1 ⟨Σα⟩A Σα
+2m
+(15)
+where ⟨Σα⟩A = ⟨ΨA| Σα |ΨA⟩ = Tr
+�
+ρ(A) Σα
+�
+, ∀α ∈ [1, 4m − 1], and m = |R| is the number
+19
+
+of orbitals in the set R. One convenient orthonormal basis set is the generalized Gell-Mann
+basis.67 In the special case of a 1-qubit RDM, {Σα} (α = x, y, z) is the familiar Pauli
+matrices.
+3.2
+Naive RDM Linear Matching and its Disadvantage
+A naive implementation of BE on a quantum computer is to simply replace 1-RDM in
+Eq.
+(6) with the qubit RDM in Eq.
+(15) on the fragment overlapping regions.
+Such
+an extension imposes matching constraints on each elements of the RDMs, resulting the
+following constraint vector in analogous to Eq. (8)
+Qlin(ρ(A)
+R ; ρ(B)
+R ) =
+�
+�����
+⟨Σ1⟩A − ⟨Σ1⟩B
+...
+⟨Σ4m−1⟩A − ⟨Σ4m−1⟩B
+�
+�����
+= 0.
+(16)
+It is obvious that ρ(A)
+R
+− ρ(B)
+R
+= 0, if and only if all the (4m − 1) components in the above
+constraint are satisfied.
+Similarly, we can associate a scalar Lagrange multiplier to each constraint in Eq. (16)
+and use this linear RDM constraint in place of the 1-RDM constraint Q1-RDM(Ψ(A); P(B))
+in Eq. (9). Finding the stationary point of this new Lagrangian gives the same eigenvalue
+equation as Eq. (12) with a new BE potential given by
+VBE =
+�
+B̸=A,CB∩EA̸=∅
+λ(A)
+B
+· [I ⊗ Σr ⊗ I]
+(17)
+where Σr =
+�
+Σ1, · · · , Σα, · · · , Σ4m−1
+�
+is a (4m − 1)-dimensional vector of the orthonormal
+basis in Eq. (15), and λ(A)
+B
+is the Lagrange multipliers now modulating the local potentials
+on each qubit basis, and n is the number of overlapping sites between A and B.
+To perform the optimization, the eigenvalue equation Eq. (12) with the above new BE
+20
+
+potential in (17) can be solved on a quantum computer to obtain an updated wave function
+for fragment A. By iteratively solving the eigenvalue equation and updating the Lagrange
+multipliers {λ, µ} using either gradient-based or gradient-free methods,68 an algorithm can
+be formulated to solve the optimization problem. For completeness, we document the algo-
+rithm from the naive linear matching of RDMs in Sec. S8 of the SI.
+The above is a convenient way to impose the constraint on quantum computers, but it
+is computationally costly as the number of constraints in (16) increases exponentially as
+the number of overlapping sites n on neighboring fragments. For each constraint equation,
+the expectation values ⟨Σα⟩ has to be measured on the quantum computer, which therefore
+introduces an exponential overhead on the sampling complexity.
+In the next section, we introduce a simple alternative to evaluate the mismatch between
+two RDMs on a quantum computer much faster based on a SWAP test.
+3.3
+Coherent Quantum Matching from SWAP Test
+The wave functions of two overlapping fragments are stored coherently as many amplitudes
+that suppose with each other. The beauty of quantum computers and algorithms lies at the
+ability to coherently manipulating such amplitudes simultaneously. We may naturally ask:
+are there quantum algorithms or circuits that can coherently achieve matching between an
+exponentially large number of amplitudes, without explicitly measuring each amplitude?
+In quantum information, there is a class of quantum protocols to perform the task of
+estimating the overlap between two wave functions or RDMs under various assumptions.69
+Among these protocols, the SWAP test is widely used.47,70 Such a SWAP test on a quantum
+computer can also be naturally implemented by simple controlled-SWAP operations as in Fig.
+4, showing a SWAP test between two qubits. The essence of a SWAP test is to entangle the
+symmetric and anti-symmetric subspaces of the two quantum states (|φ⟩ and |ψ⟩) to a single
+21
+
+ancillary qubit, such that the quantum state of the system before the final measurement is
+|Ψ⟩ = 1
+2
+�
+|0⟩
+�
+|φ⟩ |ψ⟩ + |ψ⟩ |φ⟩
+�
++ |1⟩
+�
+|φ⟩ |ψ⟩ − |ψ⟩ |φ⟩
+��
+.
+(18)
+By measuring the top single ancillary qubit in the usual computational Z-basis (collapsing
+it to either the |0⟩ or |1⟩ state), the overlap of the two qubit wave function, |⟨φ|ψ⟩|, can be
+directly obtained from the measurement outcome probability:
+Prob[M = 0] = 1 + |⟨φ|ψ⟩|2
+2
+,
+(19)
+without requiring explicit estimation of the density matrix elements of each individual qubit.
+|0⟩
+H
+•
+H
+M
+|φ⟩
+×
+|ψ⟩
+×
+Figure 4: Quantum circuit of a SWAP test between two qubits (lower, with state |φ⟩ and
+|ψ⟩). The circuit is composed of two Hadamard gate (H), a controlled-SWAP operation in
+between, and a final Z-basis measurement M on an additional ancilla qubit (top), where
+M = 0, 1.
+Can we recast the linear matching conditions as linear combination of several SWAP tests?
+Observe that an equivalent condition alternative to Eq. (16) is the following quadratic match-
+ing condition (see Sec. S3 of SI for a proof of the equivalence between the two quantum
+matching conditions)
+Qquad(ρ(A)
+R ; ρ(B)
+R ) = Tr
+��
+ρ(A)
+R
+− ρ(B)
+R
+�2�
+= 0.
+(20)
+Interestingly, the above quadratic constraint can be rewritten as a linear combination of
+three different multi-qubit generalization of the SWAP tests (with each repeated multiple
+times), regardless of the number of overlapping sites (Fig. 1iiiq). Two of the SWAP tests are
+22
+
+to estimate the purity of ρ(A)
+R
+and ρ(B)
+R
+each, while the third one is to estimate the overlap
+between ρ(A)
+R
+and ρ(B)
+R . See Sec. S4 how to generalize the SWAP test on two qubits to a
+multi-qubit setting and how to relate the SWAP test results to the quadratic constraint.
+The reformulation of the quadratic constraint allows us to estimate the mismatch be-
+tween two fragments by measuring only a single ancilla qubit (estimating three different
+amplitudes). As compared to the linear constraint case where an exponentially large num-
+ber of constraints have to be estimated individually (4m − 1 where m = |R| is the number of
+overlapping sites again), the quadratic matching based on SWAP tests achieves an exponential
+saving in the types of measurements required.
+Furthermore, the reduction of the mismatch to the estimation of only a few (three)
+amplitudes in SWAP tests allows an additional quadratic speedup by amplifying the amplitude
+of the ancilla qubit before measure it. We will discuss more details on how to achieve the
+quadratic speedup in Sec. 4.3. Admittedly, such amplitude amplification algorithm may be
+applied even to the naive linear RDM matching by boosting individual RDM amplitude, but
+the resulting quantum circuit will be much more complicated.
+3.4
+Optimization Using the Quadratic Penalty Method
+With an efficient way to estimate the quadratic penalty constraint established in Eq. (20), it
+now appears feasible to use this new constraint in Eq. (9) as in the case of linear constraint.
+However, the nature of the quadratic matching in Eq. (20) makes the same Lagrange mul-
+tiplier optimization method used in the linear case invalid. We first discuss in more detail
+why this approach fails, in Sec. 3.4.1; we then describe an alternative way of treating the
+quadratic constraint as a penalty term to optimize the resulting objective function in Sec.
+3.4.2.
+23
+
+3.4.1
+Violation of the Constraint Qualification
+A necessary condition to use the Lagrange multiplier method for constraint optimization
+is that the gradient of the constraint itself with respect to system variables has to be
+non-zero at the solution point (this guarantees a non-zero effective potential to be added
+to the original Hamiltonian), a.k.a., constraint qualification.71,72 Specifically, we require
+∇Qquad(ρ(A)
+R ; ρ(B)
+R ) ̸= 0 when ρ(A)
+R
+= ρ(B)
+R .
+Unfortunately, in the quadratic case, we have
+∇Qquad(ρ(A)
+R ; ρ(B)
+R ) ∝ ρ(A)
+R
+− ρ(B)
+R
+= 0
+(21)
+when ρ(A)
+R
+and ρ(B)
+R
+matches, which violates the above condition. Note that any high-order
+constraint other than linear order will violate the constraint qualification. The existence
+of such constraint qualification makes sense also from a physical point of view. Because
+the gradient ∇Qquad(ρ(A)
+R ; ρ(B)
+R ) enters the eigenvalue equation (13) as the BE potential VBE
+modulated by the Lagrange multipliers. The vanishing of this potential near the solution
+point means there is no way to modulate VBE by adjusting the Lagrange multipliers, and
+therefore will lead to failure of convergence of the Lagrange multiplier.
+Alternatively, the quadratic constraint can be treated as a penalty by using λ(A)
+B Qquad(ρ(A)
+R ; ρ(B)
+R )
+to substitute the constraint λ(A)
+B
+· Q1-RDM(Ψ(A); P(B)) in Eq. (9). We can then employ the
+quadratic penalty method73 to minimize this cost function. To highlight the distinction
+of quadratic penalty method from the Lagrange multiplier method, we use “cost function”
+instead of “Lagrangian” to refer to the objective function in the quadratic penalty case.
+3.4.2
+Details of the Quadratic Penalty Method
+The idea of the penalty method is to use the constraint as a penalty where the magnitude
+of λ(A)
+B
+serves as a weight to the penalty. Initially, λ(A)
+B
+is set to a small constant, and then
+we treat the resulting cost function as an unconstrained minimization where its minimum
+24
+
+is found by varying the wave functions. The next step is to increase λ(A)
+B
+to a larger value
+leading to a new Lagrangian, which is then minimize again by varying the wave function
+parameters. This procedure is repeated until the penalty parameter λ(A)
+B
+is large enough to
+guarantee a small mismatch Qquad(ρ(A)
+r
+; ρ(B)
+r
+). In our case, we choose all λ(A)
+B
+= λ for all pairs
+of adjacent fragments.
+It is helpful to note that optimization of the wave function is done again using the
+eigenvalue equation as in Eq. (12) by tuning the BE potential VBE. In other words, for a
+fixed penalty parameter λ, the fragment Lagrangian LA({VBE}) is minimized with respect
+to VBE. For a particular parametrization in terms of local potentials {vα} on the edge sites
+of fragment A
+VBE({vα}) =
+M
+�
+α=0
+vα I ⊗ Σα ⊗ I,
+(22)
+where {Σα} is a set of Hermitian generator basis of size M on the edge sites of fragment A
+(can be Pauli operators for a single edge site), and {vα} is the corresponding local potential
+(real numbers). Note that M in Eq. (22) can be much smaller than the total number of
+generators (4m) on the edge sites, because in each bootstrap embedding iteration, only a
+small local potential is added to the Hamiltonian. This perturbative nature of the bootstrap
+embedding iteration allows us to expand the BE potential VBE in each iteration under the
+Hermitian generator basis from the previous iteration, such that the BE potential in each
+iteration is diagonal dominant, i.e., M ≪ 4m where n is the number of edge sites on any
+fragment A.
+To update {vα}, we derive the following gradient
+dL(A)
+dvα
+=
+�
+n′̸=0
+�
+C†(I ⊗ W(n′)
+α
+⊗ I)C(n′)�
+×
+�
+C(n′)† �
+H(A) + E(A)
+0
++ 2λ (I ⊗ (ρEA − ρCB) ⊗ I)
+�
+C
+�
+(23)
+∀α ∈ [0, M], that can, in principle, be used to perform the updating of VBE to minimize
+25
+
+L(A). In the above, C(n) is the eigenvector of the n-th eigenstate (n ≥ 1) while C is the
+eigenvector of the ground state, W(n′)
+α
+is a perturbation matrix between ground state and
+the n′-th eigenstate for the α-th Pauli basis at the edge site of fragment A, whereas ρEA and
+ρCB are the RDM at the edge and center sites of fragment A and B, respectively (see SI Sec.
+S5 for detailed derivation).
+The above gradient in Eq. (23) is only formally useful, but computing it exactly requires
+all the eigenstates to be known (not only the ground state) which is clearly very costly if
+possible. Nevertheless, it serves as a good starting point to develop approximated updating
+scheme or to perform bootstrap embedding for excited states.
+We leave such topics for
+future investigation.
+In the present work, instead of using Eq.
+(23) to update VBE, we
+employ gradient-free schemes to update {vα} and measure the required expectation values
+using SWAP test to obtain the mismatch to evaluate the cost function L(A).
+We note that one additional advantage of this quadratic penalty method is that it can
+be easily integrated with variational eigensolvers34 by treating the quadratic penalty as
+an additional term in the VQE cost function.74 The drawback is that the optimized wave
+function only exactly equals to the true wave function when the penalty goes to infinity
+λ → ∞. Practically, we find that choosing the penalty parameter large enough is sufficient
+to obtain satisfactory results.
+4
+Quantum Bootstrap Embedding Algorithms
+Given the theoretical formulation of QBE method in Sec. 3, we present a general hybrid
+quantum-classical algorithm in this section that can be practically used to solve the BE
+problem on quantum computers to find the BE potentials VBE that satisfies the matching
+condition.
+In our quantum bootstrap embedding algorithm, the electronic structure problem of
+the total system is formulated as a minimization of a composite objective function with a
+26
+
+penalty term constructed from the matching conditions on the full qubit RDMs on overlap-
+ping regions of adjacent fragments. We then design an iterative hybrid quantum-classical
+algorithm to solve the optimization problem, where a quantum subroutine as an eigensolver
+is employed to prepare the ground state of fragment Hamiltonian. The quantum matching
+algorithm employs a SWAP test46,47 between wave functions of two fragments to evaluate the
+matching conditions, which is a dramatic improvement as compared to the straightforward
+method of measuring an exponential number (with respect to the number of qubits on the
+fragment edge) of RDM elements. Additionally, the quantum bootstrap embedding frame-
+work is internally self-consistent without the need to match fragment density matrices to
+external more accurate solutions. The adaptive sampling changes the number of samples as
+the optimization proceeds in order to achieve an increasingly better matching conditions.
+We note that the SWAP test adds only little computational cost to quantum eigensolvers
+which can be readily performed on current NISQ devices. The amplitude amplified coherent
+quantum matching requires iterative application of eigensolvers multiple times which are
+more suitable for small fault-tolerant quantum computers.
+The rest of this section is organized as follows. Sec. 4.1 gives an outline of the QBE
+algorithm with the quadratic penalty method. Sec. 4.2 discusses possible choices of quantum
+eigensolvers with an analysis on sampling complexities. We then present a way to achieve
+an additional quadratic speedup by using coherent amplitude estimating algorithm in Sec.
+4.3.
+4.1
+The Algorithm
+We present a high-level framework of the main algorithm in this section. As a comparison,
+the QBE algorithm with naive linear matching can be found in SI Sec. S8. Code for the
+algorithms and data for generating the plots are available as open source on github.75
+To quantify the mismatch across all fragments, we define ∆ρ to be the root mean square
+density matrix mismatch averaged over all the overlapping sites of all the fragments according
+27
+
+to
+∆ρ =
+�
+�
+�
+�
+1
+Nsites
+�
+A,B
+�
+r∈E(A)∩C(B)
+Tr
+��
+ρ(B)
+r
+− ρ(A)
+r
+�2�
+(24)
+where Tr
+��
+ρ(B)
+r
+− ρ(A)
+r
+�2�
+= Qquad(ρ(A)
+r
+; ρ(B)
+r
+) as in Eq. (20), which may also be recognized
+as the Frobenius norm of (ρ(B)
+r
+− ρ(A)
+r
+). Nsites is the total number of terms in the double sum
+in Eq. (24), Nsites = �
+A̸=B |E(A) ∩ C(B)|, with |S| denoting the number of elements in set S.
+The cost function L(A)(λ) being optimized is discussed in Sec. 3.4.1. For clarity, we write
+it explicitly here
+L(A)(λ) =⟨ ˆH(A)⟩A +
+�
+B
+λQquad(ρ(A)
+R ; ρ(B)
+R ),
+(25)
+with Qquad given by Eq.
+(20).
+We have omitted the term E(A) for simplicity since the
+normalization of the wave function is guaranteed for a fault-tolerant quantum computer.
+However, this term can be important on a noisy quantum computer where the purity of
+the wave function can be contaminated. Note the expectation value in Eq. (25) has to be
+estimated by collecting samples on a quantum computer.
+The quantum bootstrap embedding algorithm with quadratic penalty method is presented
+below in Alg. 1. The algorithm takes as its input the total Hamiltonian of the original system,
+and then perform the fragmentation and parameter initialization, followed by the main
+optimization loop to achieve the matching. Finally, it returns the optimized BE potential
+V (A)
+BE for any fragment A and the final mismatch ∆ρ. Inside the main loop (line 9 of Alg. 1),
+the cost function L(A)(λ) for each fragment A is minimized for a fixed penalty parameter λ
+(line 10 and 11). The penalty λ is then increased geometrically (line 12) until the mismatch
+28
+
+criteria is met, i.e., ∆ρ ≤ ε.
+Algorithm 1:
+Quantum bootstrap embedding algorithm:
+quadratic penalty
+method
+1 Input: Geometry of the total molecular system and the associated ab initio
+Hamiltonian.
+2
+/* Initialization
+*/
+3 Fragmentation: Divide the full molecular system into Nfrag overlapping fragments;
+4 for A = 1 to Nfrag do
+5
+Generate H(A) using Eq. (S1) of SI Sec. S1;
+6
+Set V (A)
+BE = 0;
+7 Parameter initialization: set initial penalty factor λ = 1; set initial mismatch
+∆ρ > ϵ.
+8
+/* Main loop:
+*/
+9 while ∆ρ > ε do
+10
+for A = 1 to Nfrag do
+11
+Minimize L(A)(λ) as in Eq. (25) : Repeatedly generate V (A)
+BE and estimate
+the penalty loss function L(A)(λ) using SWAP test.
+12
+Increase penalty parameter: λ ← γλ, for some fixed γ > 1.
+13
+Update mismatch: for A = 1, Nfrag do
+14
+Estimate Qquad(ρ(A)
+r
+; ρ(B)
+r
+) using N SWAP
+samp (Eq. (27)) samples for each SWAP test.
+15
+Classically compute the mismatch ∆ρ using Eq. (24).
+16 Returns:
+�
+H(A) + V (A)
+BE
+�
+for all A, ∆ρ.
+A key step of the algorithm is the minimization of L(A)(λ) at line 11, which consists of
+repeatedly generating the BE potential V (A)
+BE and estimate the mismatch using SWAP test.
+BE potentials V (A)
+BE are generated differently for different optimization algorithms. In our
+29
+
+implementation, a quasi-Newton method, the L-BFGS-B76 algorithm, is used at line 11 for
+minimizing L(A)(λ), where V (A)
+BE is proposed by the optimizer in order to estimate the inverse
+Hessian matrix to steer the optimization properly. Alternatively, if derivative-free methods
+such as Nelder-Mead77 is used, V (A)
+BE will be generated in a high-dimensional simplex defined
+by the coefficients {vα} in Eq. (22), which is repeatedly refined.
+Once V (A)
+BE is generated, the first term in the cost function in Eq. (25) is estimated by
+invoking the quantum eigensolver for the Hamiltonian
+�
+H(A) + V (A)
+BE
+�
+. The second term, the
+mismatch in Eq. (25) can be estimated by measurement outcomes of the ancilla qubit in
+the SWAP test (SI Sec. S4). The mismatch estimation at line 13 is performed in the same
+way as those in line 11. Note that the number of samples N SWAP
+samp (Eq. (27)) for the SWAP test
+estimation can be changed adaptively in different BE iterations for different accuracy, which
+we discuss in detail in the next section.
+4.2
+Eigensolver Subroutines and Sampling Complexity
+Two major quantum eigensolvers, QPE78 and VQE34 can be used in line 11 and 14 of Alg. 1
+to estimate the cost function. QPE is an exact eigensolver, where the system wave function
+collapses to the exact ground state regardless of the number of evaluation qubits used. In
+contrast to QPE, VQE is an approximate eigensolver and the results depends on the choice
+of ansatz and the optimization algorithm used.
+A crucial feature of a quantum eigensolver is its probabilistic nature, in a sense that
+any measurement collapses the entire quantum state. This perspective allows us to treat a
+quantum eigensolver as a sign-problem-free sampling oracle for correlated electronic structure
+problems where Ref.79 provides a concrete example.
+The stochastic nature also means a more careful treatment on the number of samples is
+required to fully quantify any potential quantum speedup. In general, for typical iterative
+mixed quantum-classical algorithms, some parameters are usually passed from one iteration
+to the next, where the parameters are estimated by repeatedly sampling from a quantum
+30
+
+eigensolver oracle through proper measurement. This means the uncertainty on these pa-
+rameters estimated from one iteration has to be small enough to avoid a divergence of the
+algorithm as iteration continues.
+In particular in the bootstrap embedding case, the sampling accuracy on the fragment
+overlap of each iteration has to be good enough such that the uncertainty of the mismatch
+passed to the next iteration will not spoil the iteration and lead to diverging results as
+iterations continue. When estimating the overlap S to an accuracy ϵ naively by density
+matrix tomography of individual RDM elements, it is shown under mild assumptions that
+the total number of samples required (see Sec. S6 in SI)
+N TMG
+samp (S, ϵ, n) = O(en)
+�D
+ϵ2
+�
+,
+(26)
+where n is the number of qubits on the overlapping region, and D is a system-dependent
+constant as a function of the two RDMs. In contrast, the quantum matching based on SWAP
+test costs
+N SWAP
+samp(S, ϵ) =
+�1 − S2
+8
+� 1
+ϵ2,
+(27)
+which is independent of the size n of the overlapping region of two fragments. This demon-
+strates that our quadratic quantum matching achieves an exponential speedup compared to
+naive tomography of density matrices. This dramatic speedup is perhaps not that surpris-
+ing because we only care about one particular observable (the overlap) instead of the full
+subsystem RDMs. Therefore, if the observable can be mapped to measurement outcome of
+few qubits by some quantum operations (SWAP test in this case), advantages are expected in
+general.
+Moreover, the dependence of N SWAP
+samp(S, ϵ) on the overlap S and estimation accuracy ϵ
+allows an adaptive sampling schedule to be implemented for line 11 and 14 of Alg. 1. For
+example, we may use the overlap S estimated from the previous BE iteration to compute
+31
+
+the required N SWAP
+samp in the current BE iteration. The accuracy ϵ can also be dynamically
+tuned according to the error of the first term in Eq.
+(25), as well as the value of the
+penalty parameter λ. For example, at the beginning BE iterations, the mismatch (∆ρ or
+more precisely Qquad(ρ(A)
+r
+; ρ(B)
+r
+)) is large so that a moderate ϵ suffices. As the BE iteration
+proceeds, the overlap converges exponentially, therefore an exponentially decreasing ϵ has to
+be used as well. A numerical value of ϵ needs be determined from case to case.
+In addition, Eq. (27) suggests an interesting behavior. As the QBE algorithm proceeds
+and the overlap S increases, fewer samples are needed to achieve a target accuracy. If S
+approaches 1 exponentially fast as S ∼ 1 − e−γ·niter for some constant γ, then the required
+number of samples for SWAP will degrees exponentially as BE iteration niter goes N SWAP
+samp ∼
+e−γ·niter/ϵ2. In practice, the overlap of two subsystem can never approach 1 but saturates
+to a constant 0 < c < 1 when matching is achieved, and therefore N SWAP
+samp ∼ (1 − c)/ϵ2 still
+obeys the 1/ϵ2 scaling generally. This, on the other hand, suggests that a larger overlapping
+region is advantageous to reduce N SWAP
+samp because the RDM of a larger subsystem of a pure
+state will have greater purity (hence larger c) in general.
+4.3
+Additional Quadratic Speedup
+The above perspective of treating quantum eigensolver as oracle where some amplitude is
+estimated through proper measurements allows us to achieve an additional quadratic speedup
+in our quantum bootstrap embedding algorithm. The intuition is that instead of directly
+measure a small quantum amplitude to accumulate enough counts to reduce the error bar,
+we may use quantum algorithms to first amplify the amplitude before the measurement.
+There are well-established ways of performing such amplitude amplification task via coherent
+quantum algorithms.48
+In particular, in each iteration of the algorithm, it can be shown (SI Sec.
+S7) that
+by combining oblivious amplitude amplification and a binary search protocol, estimating
+the overlap up to precision ϵ between adjacent fragments takes N SWAP+AE
+samp
+samples (state
+32
+
+preparation and SWAP tests)
+N SWAP+AE
+samp
+=
+√
+2
+2 ln(2)ϵ ln2(1
+ϵ),
+(28)
+regardless of the overlap S.
+Comparing (28) with (27), the above analysis suggests that our coherent quantum match-
+ing algorithm achieves a quadratic speed up (up to a factor of polylog(1
+ϵ)) as compared to the
+SWAP test based quantum matching algorithm, which is consistent with typical behavior of a
+Grover-type of search algorithm. Moreover, in contrast to (26), an exponential advantage is
+present with respect to the size of the overlapping region, indicating the benefit of using our
+quadratic QBE algorithm for fragment matching in the presence of large overlapping region.
+5
+Results and Discussions
+With the theoretical foundation and algorithms discussed in previous sections, we present
+numerical results in this section, demonstrating the convergence of the QBE algorithm in
+Sec. 5.1 with an exact solver (at infinite sampling limit). In Sec. 5.2, we present numerical
+evidence for the sampling advantage of the QBE algorithm by considering its behavior with
+a finite number of samples.
+We use a typical benchmark system in quantum chemistry,
+hydrogen chains under minimal basis, to perform the numerical calculations. More numerical
+results using approximate variational quantum eigensolvers (VQE) on a random spin model
+can be found in Sec. S9 E of SI.
+5.1
+Convergence of QBE in Infinite Sampling Limit
+We focus on demonstrating the convergence of QBE in the infinite sampling limit by using
+exact deterministic solver with the quadratic constraint in Eq. (20) and linear constraint
+in Eq. (16). As a standard benchmark system for electronic structure, we perform QBE
+on a H8 chain under a minimal STO-3G basis, which is fragmented into six overlapping
+33
+
+Figure 5: Convergence of the quantum bootstrap embedding algorithms on (a) density
+mismatch and (b) energy error for the linear constraint (pink) and quadratic penalty
+method (red) in the infinite sample limit for an H8 molecule. The dashed trend lines in
+both panels indicate an exponential fit.
+fragments each with six embedding orbitals. Fig. 5a shows the exponential convergence of
+the density mismatch for an H8 molecule in both linear and quadratic constraint cases. A
+similar convergence is established for a toy spin model and a perturbed H4 molecule using a
+VQE eigensolver with the linear RDM matching (more details can be found in Sec. S9 E of
+the SI).
+To quantify how much energy error the final converged result has, Fig. 5b shows the
+absolute value of the error in energy using the energy in the last (11th) iteration as a reference.
+We can see that the energy errors from both the linear and quadratic constraint algorithm
+exhibit similar exponential convergence as the density mismatch. Moreover, the energy in
+both cases converge to the same value within 10−6 in the last iteration (not shown in the
+figure). We note that the linear constraint case shows a slightly oscillatory convergence,
+while the quadratic case is free of such oscillatory behavior. The fact that quadratic appears
+to converge slightly faster than linear may be coincidence for the system investigated, and
+34
+
+10-
+QBE (Linear)
+Density Mismatch
+QBE (Quadratic)
+10-5
+10-6
+10-7
+0
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+10-
+10-
+10-5
+10-6.
+10-7
+10-8
+2
+0
+3
+8
+9
+10
+1
+6
+7
+Iteration Numberthe convergence rate in general depends on the optimization algorithm chosen. See Sec. S9 D
+of the SI for a detailed description on definition of the energy.
+5.2
+Sampling Advantage of Coherent Quantum Matching
+In the previous section, we have seen that our quantum bootstrap embedding algorithm
+convergence as expected in the infinite sampling limit. It is also seen (in the SI) that the
+approximate VQE leads to biased behavior on the density matching. In practice, only a
+finite number of samples can be collected on a quantum computer, and we will focus on theis
+scenario in this section. In particular, we present numerical data demonstrating the sampling
+advantage of our coherent quantum matching algorithm. Sec. 5.2.1 discusses the sampling
+advantage of the quantum matching algorithm for an overlapping region of increasing size.
+In Sec.
+5.2.2, the additional quadratic speedup in estimating the overlap via amplitude
+amplification and binary search (AE) is presented.
+5.2.1
+Advantage in Fragment Overlap Size
+To perform bootstrap embedding, it is usually advantageous to partition the system into
+fragments with large overlapping region to increase the convergence rate, because a large
+overlapping region necessarily means more information is provided to update the local po-
+tential for the following BE iteration. However, a larger overlapping size also lead to an
+exponentially higher sampling complexity versus the number of qubits in the overlapping
+region if estimating the overlap naively from density matrix tomography as in Eq. (26).
+The quantum matching algorithm implemented by a SWAP test (Fig. 1iiiq) bypass the need
+for density matrix tomography, and therefore leads to a sample complexity as in Eq. (27)
+independent of the size of the overlapping region.
+To validate our theoretical sample complexity, a simulation of the quantum matching
+algorithm with QPE as an eigensolver for two identical H4 chain is performed using a noiseless
+Qiskit AerSimulator (see SI Sec. S9 C for more details) for an increasing overlap region
+35
+
+ranging from 2 to 4, 6, and 8 qubits (schematic in Fig. 6). In the simulation, we first
+use QPE to prepare the ground state for two non-interacting H4 molecules separately. A
+SWAP test is then performed on relevant qubits in the overlapping region between the two
+H4 molecules. The evaluation qubits for QPE and the ancilla qubit for SWAP test are all
+measured afterwards. Post-selection on the QPE evaluation qubits are performed in order to
+select the ground states of H4 molecules. The SWAP test results are processed and converted
+to the estimation on the overlap S.
+Figure 6: Sampling complexity ratio of naive density matrix tomography (TMG) and SWAP
+test versus number of qubits in the overlapping region for a target precision ϵ = 0.001 on
+overlap S. The inset shows a simulated convergence of overlap (S) estimation using
+quantum matching (SWAP) for the case of two overlapping qubits. Data are obtained from a
+non-interacting chain of H4 (see SI Sec. S9 C for details).
+The inset of Fig. 6 shows the estimated overlap S as a function of sample size (number
+of eigensolver calls) in the case of two overlap qubits.
+The estimated overlap converges
+to the exact value (black dashed horizontal line) for roughly four million samples within
+5 × 10−4 (error bar invisible for the last data point). This demonstrates the correctness of
+our quantum matching algorithm.
+By repeating similar estimation as described above for increasingly larger overlapping
+regions, the exponential sampling advantage of the quantum matching algorithm over naive
+36
+
+105
+Exact
+S
+0.550
+亚
+Simulated
+rlap
+Over
+0.525
+104
+des
+0.500
+104
+106
+EigensolverCalls
+103
+102
+2
+:3
+4
+5
+6
+7
+8
+9
+Number of Overlap Qubitsdensity matrix tomography is evident in Fig.
+6.
+As we can see, to achieve a constant
+target precision of ϵ = 0.001 on the overlap S, the ratio between the SWAP test estimation
+and the naive tomography estimation for the required number of eigensolver calls increases
+exponentially as the number of qubits.
+We note that in general, overlaps between density matrices are not low-rank observables,
+so the sampling complexity of estimating it is likely to be high. However, more efficient
+sampling schemes may exist than the naive density matrix tomography as presented in Eq.
+(26). For example, by sampling the differences in the RDMs between the current and the
+previous BE iterations, the sampling complexity could be much better than exponential. We
+leave this for future investigation.
+5.2.2
+Additional Quadratic Speedup in Accuracy
+We have seen in the previous section that the quantum matching implemented by a SWAP
+test shows an exponential sampling advantage in terms of the size of the overlapping region
+as compared to naive density matrix tomography. However, the sample complexity in the
+estimation accuracy ϵ follows the same scaling of 1/ϵ2 as classical sampling based algorithms.
+As is derived in Sec. 4.3, we see that the sample complexity can be reduced to roughly 1/ϵ
+with a coherent quantum matching algorithm, by combining amplitude estimation and a
+binary search protocol, thus achieving a quadratic speedup.
+In this section, we present
+concrete numerical data demonstrating this quadratic speedup.
+Fig. 7 shows that for a single BE iteration, the required number of samples (eigensolver
+calls) on estimating the RDM overlap S between two adjacent fragments as a function of the
+required precision on the overlap, comparing the SWAP test based quantum matching (blue)
+and the coherent overlap estimation combining the SWAP test and amplitude estimation
+(SWAP+AE) (red). We can see that the required number of samples increases quadratically
+as the accuracy ϵ increases for the SWAP test based estimation. In contrast, the slope of the
+SWAP+AE sample complexity is reduced to roughly half of the SWAP test. It is worthwhile to
+37
+
+Figure 7: Number of eigensolver calls required as a function of target precision at overlap
+S = 0.4, comparing incoherent (blue) and coherent (red) estimation. The blue scatter
+points for the incoherent are obtained from classical variational Monte Carlo estimation
+and the blue dashed line shows the incoherent sample as derived in Eq. (27). The red data
+points are obtain from the linear constraint convergence in Fig. 5, while the red dashed
+line shows the complexity as derived in the SI. The inset plots the eigensolver calls as a
+function of the overlap S for a target precision ϵ = 0.001. Note the crossover in both plots.
+The coherent estimation shows a square-root advantage at high target precision.
+note that this quadratic speedup is only advantageous in the high precision (small ϵ) limit,
+as is evident from the existence of a crossing point in Fig. 7 (between 10−4 and 10−2), which
+defines a critical ϵ∗. For ϵ < ϵ∗, SWAP+AE is favored whereas the SWAP test wins when ϵ > ϵ∗.
+Moreover, the dependence of the sampling complexity on the value of the overlap S is
+very different. This difference is clear from the inset of Fig. 7, comparing the SWAP (blue)
+and the SWAP+AE estimation (red). In more detail, the sample complexity for the SWAP
+test decreases quadratically as the overlap S approaches 1 (Eq. (27)). As a comparison,
+the SWAP+AE stays roughly a constant for the coherent quantum matching ((28)), because
+the amplitude amplification process used in the present work is agnostic to the value of the
+amplitude (overlap S), i.e., oblivious amplitude amplification.80,81 The slight drop in sample
+complexity in the SWAP+AE approach (red line, inset of Fig. 7) is due to the discrete bit
+representation of S (see Sec. S7 B of SI for details). The different scaling on S between
+these two algorithms leads to a crossover of the sampling complexity at roughly S = 0.8 for
+38
+
+1015
+100000
+Number of Eigensolver Calls
+1012
+50000
+0
+109
+0000010000
+0.0
+0.5
+1.0
+Overlap S
+106
+103
+SWAP+AE
+SWAP
+100
+10-8
+10-6
+10-4
+10-2
+100
+Target Precisiona target precision of ϵ = 0.001. This crossover suggests again that the plain SWAP test is
+advantageous for a large overlap, while amplitude estimation works better for small overlap
+S.
+In addition, as mentioned in the previous section, as the bootstrap embedding iteration
+proceeds, the exponential convergence of the density mismatch (overlap S) suggests the need
+for an exponentially increasing accuracy ϵ on the overlap estimation. This further means
+the number of samples per iteration in the SWAP test should increases exponentially as the
+the number of iterations. Similarly, SWAP+AE achieves a square-root speedup in the total
+sample numbers (remains exponential). We note that there may exist ways of sampling the
+overlap in the current BE iteration normalized by the previous BE iteration to accelerate
+this requirement on a large number of samples, which we leave for future investigation.
+6
+Conclusion and Outlook
+In conclusion, we have developed a general quantum bootstrap embedding method to find
+the ground state of large electronic structure problems on a quantum computer by taking
+advantage of quantum algorithms. We formulated the original electronic structure problem as
+a optimization problem using a quadratic penalty to impose matching condition of adjacent
+fragments. A coherent quantum matching algorithm based on the SWAP test achieves efficient
+matching with an exponential sampling advantage compared to naive RDM tomography.
+By estimating the amplitude that encodes the overlap information combing an amplitude
+amplification and binary search protocol, an additional quadratic speedup is achieved. In
+addition, an adaptive sampling scheme is used based on previous overlap information and
+the desired target accuracy to improve the sampling efficiency.
+We demonstrate the performance of the QBE algorithm using a linear hydrogen molecule
+under minimal basis.
+Our QBE algorithm is shown to achieve exponential convergence
+in density mismatch and energy error similar to classical bootstrap embedding. However,
+39
+
+instead of the exponential cost of an exact classical solver (full configuration interaction),
+quantum eigensolvers such as quantum phase estimation can solve the fragment electronic
+structure exactly without incurring the exponential cost.
+While we have made progress toward solving electronic structure problems employing
+quantum resources in bootstrap embedding, there are several open questions to explore in
+the future. At the algorithmic level, it is important to reconstruct the total system density
+matrices from subsystem ones82 in order to compute observables other than the energy.
+Ideally, quantum algorithms that can perform the reconstruction process would be desired.
+Moreover, we have established how the bootstrap embedding potential can affect the system
+energy including the excited states in Eq. (23). Future works on developing a QBE algorithm
+targeting excited states83 or finite temperature electronic structures58,84,85 would be of great
+interest. Alternative constraint optimization methods such as the augmented Lagrangian
+method can also be explored to achieve potentially better convergence.16
+In addition, the idea of quantum matching proposed in the present work could also be
+exploited further in other embedding theories to harness quantum computers and resources,
+including but not limited to embedding schemes based on wave functions, density matrices,
+and Green’s functions.9 In these contexts, it is likely that more sophisticated quantum prim-
+itives and algorithms could accomplish quantum matching more efficiently than the simple
+SWAP test we employ. For example, it is possible that higher order matching, or matching
+of derivatives, could be accomplished quantum-mechanically, thus side-stepping sampling
+noise.
+More broadly, these quantum embedding theories and algorithms enabled by quantum
+computation resources open new possibilities in chemistry, physics, and quantum informa-
+tion. For example, large molecular systems in catalysis86,87 and protein-ligand binding com-
+plexes88,89 likely can be simulated at a much higher accuracy by combining state-of-the-
+art quantum and classical computational resources in embedding properly. In condensed
+matter and material science, quantum bootstrap embedding may be adapted to periodic
+40
+
+systems20,90,91 for quantum material design92 and probing phase diagrams of various lattice
+models93 close to the thermodynamic limit.
+Finally, from a viewpoint of quantum information, the concept of embedding is closely
+related to entanglement. Understanding the connection between the performance of quan-
+tum embedding algorithms and fragment-bath entanglement entropy may provide a general
+way to describe and understand the complexity of chemical and physical problems from a
+quantum information perspective.94–96 Current quantum computers are small – we believe
+our quantum bootstrap embedding method provides a general strategy to use multiple small
+quantum machines to solve large problems in chemistry and beyond. We look forward to
+future development in these directions.
+Acknowledgement
+YL thanks Di Luo, Minh Tran, and Daniel Ranard for helpful discussions. The work on
+analysis and numerical simulation was supported by the U.S. Department of Energy, Office
+of Science, National Quantum Information Science Research Centers, Co-Design Center for
+Quantum Advantage, under contract number DE-SC0012704.
+The conceptual algorithm
+development was supported in part by NTT Research.
+Supporting Information Available
+Additional theoretical and numerical details.
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+arXiv:2301.02389v1  [cs.LG]  6 Jan 2023
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+Hoai-An Nguyen
+Ching-An Cheng
+Rutgers University
+Microsoft Research
+Abstract
+Real-world reinforcement learning (RL) is of-
+ten severely limited since typical RL algorithms
+heavily rely on the reset mechanism to sample
+proper initial states. In practice, the reset mech-
+anism is expensive to implement due to the need
+for human intervention or heavily engineered en-
+vironments. To make learning more practical,
+we propose a generic no-regret reduction to sys-
+tematically design reset-free RL algorithms. Our
+reduction turns reset-free RL into a two-player
+game. We show that achieving sublinear regret
+in this two player game would imply learning a
+policy that has both sublinear performance regret
+and sublinear total number of resets in the origi-
+nal RL problem. This means that the agent even-
+tually learns to perform optimally and avoid re-
+sets. By this reduction, we design an instantia-
+tion for linear Markov decision processes, which
+is the first provably correct reset-free RL algo-
+rithm to our knowledge.
+1
+INTRODUCTION
+Reinforcement learning (RL) enables an artificial agent to
+learn problem-solving skills directly through interactions.
+However, RL is notorious for its sample inefficiency, and
+successful stories of RL so far are mostly limited to appli-
+cations where an accurate simulator of the world is avail-
+able (like in games). Real-world RL, such as robot learn-
+ing, remains a challenging open question.
+One key obstacle preventing the collection of a large num-
+ber of samples in real-world RL is the need for reset-
+ting the agent.
+The ability to reset the agent to proper
+initial states plays an important role in typical RL algo-
+rithms, as it affects which region the agent can explore
+and whether the agent can recover from its past mis-
+takes (Kakade and Langford, 2002). In the absence of a
+reset mechanism, agents may get stuck in absorbing states,
+such as those where it has damaged itself or irreparably al-
+tered the learning environment. Therefore, in most settings,
+completely avoiding resets without prior knowledge of the
+reset states or environment is infeasible.
+For instance, a robot learning to walk would inevitably
+fall before perfecting the skill, and timely intervention is
+needed to prevent damaging the hardware and to return
+the robot to a walkable configuration. Another example
+we can consider is a robot manipulator learning to stack
+three blocks on top of each other. Unrecoverable states that
+would require intervention would include the robot knock-
+ing a block off the table, or the robot smashing itself force-
+fully into the table. Reset would then reconfigure the scene
+to a meaningful initial state that is good for the robot to
+learn from.
+Resetting is a necessary part of the real-world learning pro-
+cess if we want an agent to be able to adapt to any en-
+vironment, but it is non-trivial. Unlike in simulation, we
+cannot just set a real-world agent (e.g., a robot) to an ar-
+bitrary initial state with a click of a button. Resetting in
+the real world is usually quite expensive and requires con-
+stant human monitoring and intervention. Consider again
+the example of a robot learning to stack blocks. Normally,
+a person would oversee the entire learning process. During
+the process, they would manually reset the robot to a mean-
+ingful starting state before it enters an unrecoverable state
+where the problem can no longer be solved. Sometimes au-
+tomatic resetting can be implemented by cleverly engineer-
+ing the physical learning environment (Gupta et al., 2021),
+but it is not always feasible.
+An approach we can take to make real-world RL more
+cost-efficient is through reset-free RL. The goal of reset-
+free RL is to have an agent learn how to perform well
+while minimizing the amount of external resets required.
+Some examples of problems that have been approached in
+reset-free RL include agents learning dexterity skills, such
+as picking up an item or inserting a pipe, and learning
+how to walk (Gupta et al., 2021; Ha et al., 2020). While
+there has been numerous works proposing reset-free RL
+algorithms using approaches such as multi-task learning
+(Gupta et al., 2021; Ha et al., 2020), learning a reset pol-
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+icy (Eysenbach et al., 2018; Sharma et al., 2022), and skill-
+space planning (Lu et al., 2020), to our knowledge, there
+has not been any work with provable guarantees.
+In this work, we take the first step to provide a provably
+correct framework to design reset-free RL algorithms. Our
+framework is based on the idea of a no-regret reduction.
+First, we reduce the reset-free RL problem to a sequence of
+safe RL problems with an adaptive initial state sequence,
+where each safe RL problem is modeled as a constrained
+Markov decision process (CMDP) with the states requiring
+resets marked as unsafe. Then we derive our main no-regret
+reduction, which further turns this sequence into a two-
+player game between a primal player (updating the Marko-
+vian policy) and a dual player (updating the Lagrange mul-
+tiplier function of the constrained MDPs). Interestingly,
+we show that such a reduction can be constructed without
+using the typical Slater’s condition for strong duality and
+despite the fact that CMDPs with different initial states in
+general do not share a common Markovian optimal policy.
+We show that if no regret is achieved in this game, then
+the regret of the original RL problem and the total num-
+ber of required resets are both provably sublinear. This
+means that the agent eventually learns to perform optimally
+and avoids resets. Using this reduction, we design a reset-
+free RL algorithm instantiation under the linear MDP as-
+sumption, using Ghosh et al. (2022) as the baseline algo-
+rithm for the primal player and projected gradient descent
+for the dual player. We prove that our algorithm achieves
+˜O(
+√
+d3H4K) regret and ˜O(
+√
+d3H4K) resets with high
+probability, where d is the feature dimension, H is the
+length of an episode, and K is the total number of episodes.
+2
+RELATED WORK
+Reset-free RL is a relatively new concept in the literature,
+and the work thus far, to our knowledge, has been limited to
+non-provable approaches with empirical verification. One
+such approach is by learning a reset policy in addition to the
+main policy (Eysenbach et al., 2018; Sharma et al., 2022).
+The idea is to learn a policy that will bring the agent back
+to a safe initial state if they encounter a reset state concur-
+rently with a policy that maximizes reward. A reset state is
+a state in which human intervention normally would have
+been required. This approach prevents the need for man-
+ual resets; however, there is usually some required assump-
+tions on knowledge of the reset policy reward function and
+therefore knowledge of the reset states (Eysenbach et al.,
+2018). Sharma et al. (2022) avoid this assumption but as-
+sume given demonstrations on how to accomplish the goal
+and a fixed initial state distribution.
+Another popular approach is using multi-task learning.
+This is similar to learning a reset policy, but can be thought
+of as a way to increase the amount of possible actions an
+agent can take to perform a reset. The objective is to learn a
+number of tasks so that a combination of them can achieve
+the main goal, and in addition, some tasks can perform nat-
+ural resets for other tasks. One problem that was tackled
+by Gupta et al. (2021) was that of inserting a light bulb into
+a lamp. The tasks their agent learns is recentering, insert-
+ing, lifting, and flipping the bulb. Here, if the bulb starts
+on the ground, the agent can recenter the bulb, lift it, flip it
+over (if needed), and finally insert it. In addition, many of
+the tasks perform resets for the others. For example, if the
+agent drops the bulb while lifting it, it can recenter the bulb
+and then try lifting it again. This approach breaks down the
+reset process and (possibly) makes it easier to learn. How-
+ever, this approach often requires the order in which tasks
+should be learned to be provided manually (Gupta et al.,
+2021; Ha et al., 2020).
+A related problem is infinite-horizon non-episodic RL
+with provable guarantees (see Wei et al. (2020, 2019);
+Dong et al. (2019) and the references within) as this prob-
+lem is also motivated by not using resets. In this setting,
+there is only one episode that goes on indefinitely. The ob-
+jective is to maximize the cumulative reward, and progress
+is usually measured in terms of regret with the compara-
+tor being an optimal policy. However, compared with the
+reset-free RL setting we study here, extra assumptions,
+such as the absence or knowledge of absorbing states, are
+usually required to achieve sublinear regret. In addition,
+the objective does not necessarily lead to a minimization
+of resets as the agent can leverage reset transitions to max-
+imize rewards. Learning in infinite-horizon CMDPs has
+been studied (Zheng and Ratliff, 2020; Jain et al., 2022),
+but to our knowledge, all such works make strong assump-
+tions such as a fixed initial state distribution or known dy-
+namics. In this paper, we focus on an episodic setting of
+reset-free RL (see Section 3); a non-episodic formulation
+of reset-free RL could be an interesting one for further re-
+search.
+To our knowledge, we propose the first provable reset-
+free RL technique in the literature. By borrowing ideas
+from literature on the much more extensively studied area
+of safe RL, we propose to associate states requiring re-
+sets with the concept of unsafe states in safe RL. Safe
+reinforcement learning involves solving the standard RL
+problem while adhering to some safety constraints. There
+has been a lot of work in safe RL, with approaches
+such as utilizing a baseline safe (but not optimal) policy
+(Huang et al., 2022; Garcia Polo and Fernandez Rebollo,
+2011), pessimism (Amani and Yang, 2022), and shielding
+(Alshiekh et al., 2018; Wagener et al., 2021). These works
+have had promising empirical results but usually require
+extra assumptions such as a given baseline policy or knowl-
+edge of unsafe states.
+There are also provable safe RL algorithms.
+To our
+knowledge, all involve framing safe RL as a CMDP.
+Here, the safety constraints are modeled as a cost, and
+
+Hoai-An Nguyen, Ching-An Cheng
+the overall goal is to maximize performance while keep-
+ing the cost below a threshold.
+The provable guaran-
+tees are commonly either sublinear regret and constraint
+violations, or sublinear regret with zero constraint vi-
+olation (Wei et al., 2021; HasanzadeZonuzy et al., 2021;
+Qiu et al., 2020; Wachi and Sui, 2020; Efroni et al., 2020;
+Ghosh et al., 2022; Ding et al., 2021).
+However, most
+works (including all the aforementioned ones), consider the
+episodic case where the initial state distribution of each
+episode is fixed. This prevents a very natural extension
+to reset-free learning as human intervention would be re-
+quired to reset the environment at the end of each episode.
+Works that allow for arbitrary initial state require fairly
+strong assumptions, such as knowledge (and the existence)
+of safe actions from each state (Amani et al., 2021).
+In our work, we utilize techniques from provable safe RL
+for reset-free RL, but weaken the typical assumptions to
+allow for arbitrary initial states. This relaxation is neces-
+sary for the reset-free RL problem and also allows for eas-
+ier extensions to both lifelong and multi-task learning. We
+achieve this relaxation with a key observation that identifies
+a shared Markovian-policy saddle-point across CMDPs of
+perfectly safe RL with different initial states (that is, the
+constraint in the CMDP imposes perfect safety). This ob-
+servation is new to our knowledge, and it is derived from
+the particular structure of perfectly safe RL, which is a sub-
+problem used in our reset-free RL reduction. We note that
+general CMDPs with different initial states do not generally
+admit shared Markovian-policy saddle-points. Therefore,
+on the technical side, our algorithm can also be viewed as
+the first safe RL algorithm that allows for arbitrary initial
+state sequences without strong assumptions.
+While we propose a generic reduction technique to de-
+sign reset-free RL algorithms, our regret and constraint
+violation bounds are still comparable to the above works
+when specialized to their setting. Under the linear MDP
+assumption, our algorithm achieves ˜O(
+√
+d3H4K) regret
+and violation (equivalently, the number of resets in reset-
+free RL), which is asymptotically equivalent to Ghosh et al.
+(2022) and comparable to the bounds of ˜O
+√
+d2H6K from
+Ding et al. (2021) for a fixed initial state.
+3
+PRELIMINARY
+We consider episodic reset-free RL: in each episode, the
+agent aims to optimize for a fixed-horizon return starting
+from the last state of the previous episode or some state
+that the agent was reset to in the previous episode if reset
+occurs (e.g., due to the robot falling over).
+Problem Setup and Notation
+Formally, we can de-
+fine episodic reset-free RL as a Markov decision process
+(MDP), (S, A, P, r, H), where S is the state space, A is
+the action space, P = {Ph}H
+h=1 is the transition dynamics,
+r = {rh}H
+h=1 is the reward function, and H is the task hori-
+zon. We assume P and r are unknown. We allow S to be
+large or continuous but assume A is relatively small so that
+maxa∈A can be performed. We designate the set of reset
+states as Sreset ⊆ S; we do not assume that the agent has
+knowledge of Sreset. We also do not assume that there is a
+reset-free action for each state, as opposed to (Amani et al.,
+2021). Therefore, the agent needs to plan for the long-term
+to avoid resets. We assume rh : S × A → [0, 1], and
+for simplicity, we assume rh is deterministic. However, we
+note that it would be easy to extend this to the setting where
+rewards are stochastic.
+The agent interacts with the environment for K total
+episodes. Following the convention of episodic problems,
+we suppose the state space S is layered, and a state sτ ∈ S
+at time τ is factored into two components sτ = (¯s, τ)
+where ¯s denotes the time-invariant part. Reset happens at
+time τ if ¯s ∈ Sreset (which we also write as sτ ∈ Sreset),
+and the initial state of the next episode will be s1 = (¯s′, 1)
+where ¯s′ is sampled from a fixed but unknown state distri-
+bution. Otherwise, the initial state of the next episode is
+the last state of the current episode, i.e., for episode k + 1,
+sk+1
+1
+= (¯s, 1) if sk
+H = (¯s, H) in episode k.1
+We denote the set of Markovian policies as ∆, and a policy
+π ∈ ∆ as π = {πh(ah|sh)}H
+h=1. We define the state value
+function and the state-action value function under π as2
+V π
+r,h(s) := Eπ
+� min(H,τ)
+�
+t=h
+rt(st, at)|sh = s
+�
+(1)
+Qπ
+r,h(s, a) := rh(s, a) + E
+�
+V π
+r,h+1(sh+1)|sh = s, ah = a
+�
+,
+where h ≤ τ, and we recall τ is the time step when the
+agent enters Sreset (if at all).
+Objective
+The overall goal is for the agent to learn a
+Markovian policy to maximize its cumulative reward while
+avoiding resets. Therefore, our performance measures are
+1We can extend this setup to reset-free multi-task or lifelong
+RL problems that are modeled as contextual MDPs since our algo-
+rithm can work with any initial state sequence. In this case, we can
+treat each state here as sτ = (¯s, c, τ), where c denotes the context
+that stays constant within an episode. If no reset happens, the ini-
+tial state of episode k + 1 can be written as sk+1
+1
+= (¯s, ck+1, 1)
+if sk
+H = (¯s, ck, H) in episode k, where the new context ck+1 can
+follow any distribution and may depend on the current context ck.
+2This value function definition is the same as the H-step cu-
+mulative reward in an MDP formulation where we place the agent
+into a fictitious zero-reward absorbing state (i.e., a mega-state ab-
+stracting Sreset) after the agent enters Sreset. We choose the cur-
+rent formulation to make the definition of resets more transparent.
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+as follows (we seek to minimize both quantities):
+Regret(K) =
+max
+π∈∆0(K)
+K
+�
+k=1
+V π
+r,1(sk
+1) − V πk
+r,1 (sk
+1)
+(2)
+Resets(K) =
+K
+�
+k=1
+Eπk
+
+
+min(H,τ)
+�
+h=1
+1[sh ∈ Sreset]
+���s1 = sk
+1
+
+
+(3)
+where ∆0(K) ⊆ ∆ is the set of Markovian policies that
+avoid resets for all episodes, and πk is the policy used by
+the agent in episode k. Note that by the reset mechanism
+�min(H,τ)
+h=1
+1[sh ∈ Sreset] ∈ {0, 1}.
+Notice that the initial states in our regret and reset measures
+are determined by the learner, not the optimal policy like in
+some classical definitions of regret. Given the motivation
+behind reset-free RL (see Section 1), we can expect that the
+initial states here are by construction meaningful for perfor-
+mance comparison; otherwise, a reset would have occurred
+to set the learner to a meaningful state. A following impli-
+cation is that all bad absorbing states are in Sreset; hence,
+the agent cannot use the trivial solution of hiding in a bad
+absorbing state to achieve small regret.
+To make the problem feasible, we assume that achieving no
+resets is possible. We state this formally in the assumption
+below.
+Assumption 1. For any sequence {sk
+1}K
+k=1, the set ∆0(K)
+is not empty. That is, there is a Markovian policy π ∈ ∆
+such that Eπ[�H
+h=1 1[sh ∈ Sreset]|s1 = sk
+1] = 0.
+This is a reasonable assumption in practice. If reset hap-
+pens, the agent should be set to a state that the agent can
+continue to operate in without reset; if the agent is at a state
+where no such reset-free policy exists, reset should happen.
+This assumption is similar to the assumption on the exis-
+tence of a perfectly safe policy in safe RL literature, which
+is a common and relatively weak assumption (Ghosh et al.,
+2022; Ding et al., 2021). If there were to be initial states
+that inevitably lead to a reset, the problem would be infea-
+sible.
+4
+A NO-REGRET REDUCTION FOR
+RESET-FREE RL
+We present our main reduction of reset-free RL to regret
+minimization in a two-player game. In the following, we
+first show that reset-free RL can be framed as a sequence of
+CMDPs of safe RL problems with an adaptive initial state
+sequence. Then we design a two-player game based on a
+primal-dual analysis of this sequence of CMDPs. Finally,
+we show achieving sublinear regret in this two-player game
+implies sublinear regret and resets in the original reset-free
+RL problem in (2). The complete proofs for this section
+can be found in Appendix A.1.
+4.1
+Reset-free RL as a Sequence of CMDPs
+The first step of our reduction is to cast the reset-free RL
+problem in Section 3 to a sequence of CMDP problems
+which share the same rewards, constraints, and dynamics,
+but have different initial states. Each problem instance in
+this sequence corresponds to an episode of the reset-free
+RL problem, and its constraint describes the probability of
+the agent entering a state that requires reset.
+Specifically, we denote these constrained MDPs3 as
+{(S, A, P, r, H, c, sk
+1)}K
+k=1:
+in episode k, the CMDP problem is defined as
+max
+π∈∆ V π
+r,1(sk
+1), s.t. V π
+c,1(sk
+1) ≤ 0
+(4)
+where we define the cost as
+ch(s, a) := 1[s ∈ Sreset]
+and V π
+c,1, defined similarly to (1), is the state value func-
+tion with respect to the cost c . We note that the initial
+state, sk
+1, depends on the past behaviors of the agent, and
+Assumption 1 ensures each CMDP in (4) is a feasible prob-
+lem (i.e., there is a Markovian policy satisfying the con-
+straint). We can interpret each CMDP in (4) as a safe RL
+problem by treating Sreset as the unsafe states that a safe
+RL agent should avoid. From this perspective, the con-
+straint in (4) can be viewed as the probability of a trajectory
+entering an unsafe state.
+Since CMDPs are typically defined without early episode
+termination unlike (1), with abuse of notation, we extend
+the definitions of P, S, r, c as follows so that the CMDP
+definition above is consistent with the common literature.
+We introduce a fictitious absorbing state denoted as s† in
+S, where rh(s†, a) = 0 and ch(s†, a) = 0; once the agent
+enters s†, it stays there until the end of the episode. We
+extend the definition P such that, after the agent is in a state
+s ∈ Sreset, any action it takes brings it to s† in the next
+time step. In this way, we can write the value function,
+e.g. for reward, as V π
+r,h(s) = Eπ
+� �H
+t=h rt(st, at)|sh = s
+�
+in terms of this extended dynamics. We note that these
+two formulations are mathematically the same for the pur-
+pose of learning; when the agent enters s†, it means that the
+agent is reset in the episode.
+By the construction above, we can write
+Resets(K) =
+K
+�
+k=1
+V πk
+c,1 (sk
+1)
+which is the same as the number of total constraint vio-
+lations across the CMDPs. Because we do not make any
+3In general, the solution (i.e., optimal policy) to a CMDP
+depends on its initial state, unlike in MDPs (see remark 2.2 in
+Altman (1999)).
+
+Hoai-An Nguyen, Ching-An Cheng
+assumptions about the agent’s knowledge of the constraint
+function (e.g., the agent does not know states ∈ Sreset), we
+allow the agent to reset during learning while minimizing
+the total number of resets over all K episodes.
+4.2
+Reduction to Two-Player Game
+From the problem formulation above, we see that the ma-
+jor difficulty of reset-free RL is the coupling between an
+adaptive initial state sequence and the constraint on reset
+probability. If we were to remove either of them, we can
+use standard algorithms, since the problem will become a
+single CMDP problem (Altman, 1999) or an episodic RL
+problem with varying initial states (Jin et al., 2019).
+We propose a reduction to systematically design algorithms
+for this sequence of CMDPs and therefore for reset-free
+RL. The main idea is to approximately solve the saddle
+point problem of each CMDP in (4), i.e.,
+max
+π∈∆ min
+λ≥0 V π
+r,1(sk
+1) − λV π
+c,1(sk
+1)
+(5)
+where λ denotes the dual variable (i.e.
+the La-
+grange multiplier).
+Each CMDP can be framed as
+a linear program (Hern´andez-Lerma and Lasserre, 2002)
+whose primal and dual optimal values match (see
+section 8.1 in Hazan et al. (2016)).
+Therefore, for
+each CMDP, maxπ∈∆ minλ≥0 V π
+r,1(sk
+1) − λV π
+c,1(sk
+1) =
+minλ≥0 maxπ∈∆ V π
+r,1(sk
+1) − λV π
+c,1(sk
+1).
+While using a primal-dual algorithm to solve for the sad-
+dle point of a single CMDP is straightforward and known,
+using this approach for a sequence of CMDPs is not obvi-
+ous. Each CMDP’s optimal policy and Lagrange multiplier
+can be a function of the initial state (Altman, 1999), and
+in general, a common saddle point of Markovian polices
+and Lagrange multipliers does not necessarily exist for a
+sequence of CMDPs with varying initial states.4 As a re-
+sult, it is unclear if there exists a primal-dual algorithm to
+solve this sequence, especially given that the initial states
+here are adaptively chosen.
+Existence of a Shared Saddle-Point
+Fortunately, there
+is a shared saddle-point with a Markovian policy across all
+the CMDPs considered here due to the special structure of
+reset-free RL. It is a proof that does not use Slater’s condi-
+tion for strong duality, unlike similar literature, but attains
+the desired property. Instead we use Assumption 1 and the
+fact that the cost c is non-negative. We formalize this be-
+low.
+Theorem 1. There exist a function ˆλ(·) where for each s,
+ˆλ(s) ∈ arg min
+y≥0
+�
+max
+π∈∆ V π
+r,1(s) − yV π
+c,1(s)
+�
+,
+4A shared saddle-point with a non-Markovian policy always
+exists on the other hand.
+and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a
+saddle-point to the CMDPs
+max
+π∈∆ V π
+r,1(s1), s.t. V π
+c,1(s1) ≤ 0
+for all initial states s1 ∈ S such that the CMDP is feasible.
+That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S,
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) ≥ V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1)
+≥ V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1).
+(6)
+Corollary 1.
+For π∗
+in
+Theorem 1,
+it holds that
+Regret(K) = �K
+k=1 V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1).
+We prove for ease of construction that the pair (π∗, λ∗)
+where λ∗(·) = ˆλ(·) + 1 is also a saddle-point.
+Corollary 2. For any saddle-point to the CMDPs
+max
+π∈∆ V π
+r,1(s1), s.t. V π
+c,1(s1) ≤ 0
+of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also
+a saddle-point as defined in eq (6).
+Therefore, the pair (π∗, λ∗) in Corollary 2 is a saddle-point
+to all the CMDPs the agent faces. This makes potentially
+designing a two-player game reduction possible. In the
+next section, we give the details of our construction.
+Two-Player Game
+Our two-player game proceeds itera-
+tively in the following manner: in episode k, a dual player
+determines a state value function λk : S → R, and a
+primal player determines a policy πk which can depend
+on λk.
+Then the primal and dual player receive losses
+Lk(πk, λ) and −Lk(π, λk), respectively, where Lk(π, λ)
+is a Lagrangian function defined as
+Lk(π, λ) := V π
+r,1(sk
+1) − λ(sk
+1)V π
+c,1(sk
+1).
+(7)
+The regret of these two players are defined as follows.
+Definition 1. Let πc and λc be comparators. The regret of
+the primal and the dual players are
+Rp({πk}K
+k=1, πc) :=
+K
+�
+k=1
+Lk(πc, λk) − Lk(πk, λk) (8)
+Rd({λk}K
+k=1, λc) :=
+K
+�
+k=1
+Lk(πk, λk) − Lk(πk, λc). (9)
+We present our main reduction theorem for reset-free RL
+below. By Theorem 2, if both players have sublinear re-
+gret in the two-player game, then the resulting policy se-
+quence will have sublinear performance regret and a sub-
+linear number of resets in the original RL problem. Since
+there are many standard techniques (Hazan et al., 2016)
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+from online learning to solve such a two-player game, we
+can leverage them to systematically design reset-free RL
+algorithms. In the next section, we will give an example
+algorithm of this reduction for linear MDPs.
+Theorem 2. Under Assumption 1, for any sequences
+{πk}K
+k=1 and {λk}K
+k=1 , it holds that
+Regret(K) ≤ Rp({πk}K
+k=1, π∗) + Rd({λk}K
+k=1, 0)
+Resets(K) ≤ Rp({πk}K
+k=1, π∗) + Rd({λk}K
+k=1, λ∗)
+where (π∗, λ∗) is the saddle-point defined in Corollary 2.
+Proof
+Sketch
+of
+Theorem 1
+Let
+Q∗
+c(s, a)
+=
+minπ∈∆ Qπ
+c (s, a) and V ∗
+c (s)
+=
+minπ∈∆ V π
+c (s).
+We
+define π∗ in Theorem 1 as the optimal policy to the
+following MDP: (S, A, P, r, H),
+where we define a
+state-dependent action space A as
+As = {a ∈ A : Q∗
+c(s, a) ≤ V ∗
+c (s)}.
+By definition, As is non-empty for all s.
+We also define a shorthand notation: we write π ∈ A(s) if
+Eπ[�H
+t=1 1{at /∈ Ast}|s1 = s] = 0. We have the follow-
+ing lemma, which is an application of the performance dif-
+ference lemma (see Lemma 6.1 in (Kakade and Langford,
+2002) and Lemma A.1 in (Cheng et al., 2021)).
+Lemma 1. For any s1 ∈ S such that V ∗
+c (s1) = 0 and any
+π ∈ ∆, it is true that π ∈ A(s1) if and only if V π
+c (s1) = 0.
+We prove our main claim, (6), below. Because V π∗
+c,1 (s1) =
+0, the first inequality is trivial: V π∗
+r,1 (s1)−λ(s1)V π∗
+c,1 (s1) =
+V π∗
+r,1 (s1) = V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1).
+To prove the second inequality, we use Lemma 1:
+V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1)
+≤ max
+π∈∆ V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1)
+= min
+y≥0 max
+π∈∆ V π
+r,1(s1) − yV π
+c,1(s1)
+=
+max
+π∈Ac(s1)
+V π
+r,1(s1)
+(By Lemma 1 )
+=V π∗
+r,1 (s1) = V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1).
+Proof Sketch of Theorem 2
+We first establish the fol-
+lowing intermediate result that will help us with our de-
+composition.
+Lemma 2. For any primal-dual sequence {πk, λk}K
+k=1,
+�K
+k=1(Lk(π∗, λ′) − Lk(πk, λk))
+≤
+Rp({π}K
+k=1, π∗),
+where (π∗, λ′) is the saddle-point defined in either
+Theorem 1 or Corollary 2.
+Then we upper bound Regret(K) and Resets(K) by
+Rp({πk}K
+k=1, πc) and Rd({λk}K
+k=1, λc) for suitable com-
+parators. This decomposition is inspired by the techniques
+used in Ho-Nguyen and Kılınc¸-Karzan (2018).
+We first bound Resets(K).
+Lemma 3. For any primal-dual sequence {πk, λk}K
+k=1,
+�K
+k=1 V πk
+c,1 (sk
+1) ≤ Rp({π}K
+k=1, π∗) + Rd({λ}K
+k=1, λ∗),
+where (π∗, λ∗) is the saddle-point defined in Corollary 2.
+Proof. Notice �K
+k=1 V πk
+c,1 (sk
+1)
+=
+�K
+k=1 Lk(πk, ˆλ) −
+Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined
+in Theorem 1.
+By (6), and adding and subtracting
+�K
+k=1 Lk(πk, λk), we can bound this difference by
+K
+�
+k=1
+Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗).
+Using Lemma 2 and Definition 1 to upper bound the above,
+we get the desired result.
+Lastly, we bound Regret(K) with the lemma below and
+Corollary 1.
+Lemma 4. For any primal-dual sequence {πk, λk}K
+k=1,
+�K
+k=1(V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1))
+≤
+Rp({π}K
+k=1, π∗) +
+Rd({λ}K
+k=1, 0), where (π∗, λ∗) is the saddle-point defined
+in Corollary 2.
+Proof. Note that L(π∗, λ∗) = L(π∗, 0) since V π∗
+c,1 = 0 for
+all k ∈ [K] = {1, ..., K}. Since by definition, for any π,
+Lk(π, 0) = V π
+r,1(sk
+1), we have the following:
+K
+�
+k=1
+V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1) =
+K
+�
+k=1
+Lk(π∗, λ∗) − Lk(πk, 0)
+=
+K
+�
+k=1
+Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0)
+≤Rp({π}K
+k=1, π∗) + Rd({λ}K
+k=1, 0)
+where the last inequality follows from Lemma 2 and
+Definition 1.
+5
+RESET-FREE LEARNING FOR
+LINEAR MDP
+To demonstrate the utility of our reduction, we design a
+provably correct algorithm instantiation for reset-free RL.
+We consider a linear MDP setting, which is common in the
+RL theory literature (Jin et al., 2019).
+Assumption 2. We assume (S, A, P, r, c, H) is linear with
+a known feature map φ : S × A → Rd: for any h ∈ [H],
+there exists d unknown signed measures µh = {µ1
+h, ..., µd
+h}
+over S such that for any (s, a, s′) ∈ S × A × S, we have
+Ph(s′|a) = ⟨φ(s, a), µh(s′)⟩,
+
+Hoai-An Nguyen, Ching-An Cheng
+and there exists unknown vectors ωr,h, ωc,h ∈ Rd such that
+for any (s, a) ∈ S × A,
+rh(s, a) = ⟨φ(s, a), ωr,h⟩
+ch(s, a) = ⟨φ(s, a), ωc,h⟩.
+We assume, for all (s, a, h) ∈ S×A×[H], ||φ(s, a)||2 ≤ 1,
+and max{||µh(s)||2, ||ωr,h||2, ||ωc,h||2} ≤
+√
+d.
+In addition, we make a linearity assumption on the function
+λ∗ defined in Theorem 1.
+Assumption 3. We assume the knowledge of a feature ξ :
+S → Rd such that ∀s ∈ S, ||ξ(s)||2 ≤ 1 and λ∗(s) =
+⟨ξ(s), θ∗⟩, for some unknown vector θ∗ ∈ Rd. In addition,
+we assume the knowledge of a convex set5 U ⊆ Rd such
+that θ∗, 0 ∈ U and ∀θ ∈ U, ||θ||2 ≤ B and ⟨ξ(s), θ⟩ ≥ 0. 6
+5.1
+Algorithm
+The basis of our algorithm lies between the interaction be-
+tween the primal and dual players. We let the dual player
+perform projected gradient descent and the primal player
+update policies based on upper confidence bound with the
+knowledge of the decision of the dual player. This sequen-
+tial strategy resembles the optimistic style update in online
+learning (Mertikopoulos et al., 2018).
+Specifically, in each episode, upon receiving the initial
+state, we execute actions according to the policy based on
+a softmax (lines 5-8). Then, we perform the dual update
+through projected gradient descent. The dual player plays
+for the next round, k + 1, after observing its loss after the
+primal player plays for the current round, k. The projec-
+tion is to a l2 ball containing λ∗(·) (lines 9-11). Finally, we
+perform the update of the primal player by computing the
+Q-functions for both the reward and cost with a bonus to
+encourage exploration (lines 12-20).
+This algorithm builds upon Ghosh et al. (2022). However,
+notably, we extend it to handle the adaptive initial state se-
+quence seen in reset-free RL by Theorems 1 and 2.
+5.2
+Analysis
+We show below that our algorithm achieves regret and
+number of resets that are sublinear in the total number of
+time steps, KH, using Theorem 2. This result is asymptot-
+ically equivalent to Ghosh et al. (2022) and comparable to
+the bounds of ˜O
+√
+d2H6K from Ding et al. (2021).
+5Such a set can be constructed by upper bounding the values
+using scaling and ensuring non-negativity using a sum of squares
+approach.
+6From the previous section, we can see that the optimal func-
+tion for the dual player is not necessarily unique.
+So, we as-
+sume bounds on at least one optimal function that we designate
+as λ∗(s).
+Theorem 3. Under Assumptions 1, 2, and 3, with high
+probability, Regret(K)
+≤
+˜O((B + 1)
+√
+d3H4K) and
+Resets(K) ≤ ˜O((B + 1)
+√
+d3H4K).
+Proof Sketch of Theorem 3
+We provide a proof sketch
+here and defer the complete proof to Appendix A.2. We
+first bound the regret of {πk}K
+k=1 and {λk}K
+k=1, and then
+use this to prove the bounds on our algorithm’s regret and
+number of resets with Theorem 2.
+We first bound the regret of {λk}K
+k=1.
+Lemma 5. Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈
+U. Then it holds that Rd({λk}K
+k=1, λc) ≤ 1.5B
+√
+K +
+�K
+k=1(λk(sk
+1) − λc(sk
+1))(V k
+c,1(sk
+1) − V πk
+c,1 (sk
+1)).
+Proof. We notice first an equality.
+Rd({λk}K
+k=1, λc) =
+K
+�
+k=1
+Lk(πk, λk) − Lk(πk, λc)
+=
+K
+�
+k=1
+λc(sk
+1)V πk
+c,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)
+=
+K
+�
+k=1
+(λk(sk
+1) − λc(sk
+1))(−V k
+c,1(sk
+1))
++
+K
+�
+k=1
+(λk(sk
+1) − λc(sk
+1))(V k
+c,1(sk
+1) − V πk
+c,1 (sk
+1)).
+We observe that the first term is an online linear problem
+for θk (the parameter of λk(·)).
+In episode k ∈ [K],
+λk is played, and then the loss is revealed.
+Since the
+space of θk is convex, we use standard results (Lemma
+3.1 (Hazan et al., 2016)) to show that updating θk through
+projected gradient descent results in an upper bound for
+�K
+k=1(λk(sk
+1) − λc(sk
+1))(−V k
+c,1(sk
+1)).
+We now bound the regret of {π}K
+k=1.
+Lemma 6. Consider any πc.
+With high probability,
+Rp({π}K
+k=1, πc) ≤ 2H(1 + B + H) + �K
+k=1 V k
+r,1(sk
+1) −
+V πk
+r,1 (sk
+1) + λk(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1)).
+Proof. First we expand the regret into two terms.
+Rp({π}K
+k=1, πc) =
+K
+�
+k=1
+Lk(πc, λk) − Lk(πk, λk)
+=
+K
+�
+k=1
+V πc
+r,1 (sk
+1) − λk(sk
+1)V πc
+c,1(sk
+1) − [V πk
+r,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)]
+=
+K
+�
+k=1
+V πc
+r,1 (sk
+1) − λk(sk
+1)V πc
+c,1(sk
+1) − [V k
+r,1(sk
+1) − λk(sk
+1)V k
+c,1(sk
+1)]
++
+K
+�
+k=1
+V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) + λk(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1)).
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+Algorithm 1 Primal-Dual Reset-Free RL Algorithm for Linear MDP with Adaptive Initial States
+1: Input: Feature maps φ and ξ. Failure probability p. Some universal constant c.
+2: Initialization: θ1 = 0, wr,h = 0, wc,h = 0, α =
+log(|A|)K
+2(1 + B + H), β = cdH
+�
+log(4 log |A|dKH/p)
+3: for episodes k = 1, ...K do
+4:
+Observe the initial state sk
+1.
+5:
+for step h = 1, ..., H do
+6:
+Compute πh,k(a|·) ←
+exp(α(Qk
+r,h(·, a) − λk(sk
+1)Qk
+c,h(·, a)))
+�
+a exp(α(Qk
+r,h(·, a) − λk(sk
+1)Qk
+c,h(·, a))).
+7:
+Take action ak
+h ∼ πh,k(·|sk
+h) and observe sk
+h+1.
+8:
+end for
+9:
+ηk ← B
+√
+k
+10:
+Update θk+1 ← ProjU(θk + ηk · ξ(sk
+1)V k
+c,1(sk
+1))
+11:
+λk+1(·) ← ⟨θk+1, ξ(·)⟩
+12:
+for step h = H, ..., 1 do
+13:
+Λk+1
+h
+←
+k�
+i=1
+φ(si
+h, ai
+h)φ(si
+h, ai
+h)T + λI.
+14:
+wk+1
+r,h ← (Λk+1
+h
+)−1[
+k�
+i=1
+φ(si
+h, ai
+h)[rh(si
+h, ai
+h) + V k+1
+r,h+1(si
+h+1)]]
+15:
+wk+1
+c,h ← (Λk+1
+h
+)−1[
+k�
+i=1
+φ(si
+h, ai
+h)[ch(si
+h, ai
+h) + V k+1
+c,h+1(si
+h+1)]]
+16:
+Qk+1
+r,h (·, ·) ← max{min{⟨wk+1
+r,h , φ(·, ·)⟩ + β(φ(·, ·)T (Λk+1
+h
+)−1φ(·, ·))1/2, H − h + 1}, 0}
+17:
+Qk+1
+c,h (·, ·) ← max{min{⟨wk+1
+c,h , φ(·, ·)⟩ − β(φ(·, ·)T (Λk+1
+h
+)−1φ(·, ·))1/2, 1}, 0}
+18:
+V k+1
+r,h (·) = �
+a πh,k(a|·)Qk+1
+r,h (·, a)
+19:
+V k+1
+c,h (·) = �
+a πh,k(a|·)Qk+1
+c,h (·, a)
+20:
+end for
+21: end for
+To bound the first term, we use Lemma 3 from Ghosh et al.
+(2022), which characterizes the property of upper confi-
+dence bound.
+Lastly,
+we derive a bound on Rd({λk}K
+k=1, λc) +
+Rp({πk}K
+k=1, πc), which directly implies our final up-
+per bound on Regret(K) and Resets(K) in Theorem 3 by
+Theorem 2. Combining the upper bounds in Lemma 5 and
+Lemma 6, we have a high-probability upper bound
+Rd({λk}K
+k=1, λc) + Rp({πk}K
+k=1, πc)
+≤ 1.5B
+√
+K + 2H(1 + B + H)+
++
+K
+�
+k=1
+V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) + λc(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1))
+where the last term is the overestimation error due to opti-
+mism. Note that for all k ∈ [K], V k
+r,1(sk
+1) and V k
+c,1(sk
+1) are
+as defined in Algorithm 1 and are optimistic estimates of
+V π∗
+r,1 (sk
+1) and V π∗
+c,1 (sk
+1). To bound this term, we use Lemma
+4 from (Ghosh et al., 2022).
+6
+CONCLUSION
+We propose a generic no-regret reduction for designing
+provable reset-free RL algorithms.
+Our reduction casts
+reset-free RL into the regret minimization problem of a
+two-player game, for which many existing no-regret al-
+gorithms are available. As a result, we can reuse these
+techniques to systematically build new reset-free RL algo-
+rithms. In particular, we design a reset-free RL algorithm
+for linear MDPs using our new reduction techniques, taking
+the first step towards designing provable reset-free RL al-
+gorithms. Extending these techniques to nonlinear function
+approximators and verifying their effectiveness empirically
+are important future research directions.
+Acknowledgements
+Part of this work was done during Hoai-An Nguyen’s in-
+ternship at Microsoft Research.
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+Learning infinite-horizon average-reward mdps with lin-
+ear function approximation. CoRR, abs/2007.11849.
+Wei, C., Jafarnia-Jahromi, M., Luo, H., Sharma, H., and
+Jain, R. (2019). Model-free reinforcement learning in
+infinite-horizon average-reward markov decision pro-
+cesses. CoRR, abs/1910.07072.
+Wei, H., Liu, X., and Ying, L. (2021). A provably-efficient
+model-free algorithm for constrained markov decision
+processes. arXiv preprint arXiv:2106.01577.
+Zheng, L. and Ratliff, L. (2020). Constrained upper con-
+fidence reinforcement learning. In Bayen, A. M., Jad-
+babaie, A., Pappas, G., Parrilo, P. A., Recht, B., Tomlin,
+C., and Zeilinger, M., editors, Proceedings of the 2nd
+Conference on Learning for Dynamics and Control, vol-
+ume 120 of Proceedings of Machine Learning Research,
+pages 620–629. PMLR.
+
+Hoai-An Nguyen, Ching-An Cheng
+A
+Appendix
+A.1
+Missing Proofs for Section 4
+A.1.1
+Proof of Theorem 1
+Theorem 1. There exist a function ˆλ(·) where for each s,
+ˆλ(s) ∈ arg min
+y≥0
+�
+max
+π∈∆ V π
+r,1(s) − yV π
+c,1(s)
+�
+,
+and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a saddle-point to the CMDPs
+max
+π∈∆ V π
+r,1(s1), s.t. V π
+c,1(s1) ≤ 0
+for all initial states s1 ∈ S such that the CMDP is feasible. That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S,
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) ≥ V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1)
+≥ V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1).
+(6)
+For policy π∗, we define it by the following construction (we ignore writing out the time dependency for simplicity): first,
+we define a cost-based MDP Mc = (S, A, P, c, H). Let Q∗
+c(s, a) = minπ∈∆ Qπ
+c (s, a) and V ∗
+c (s) = minπ∈∆ V π
+c (s) be
+the optimal values, where we recall V π
+c and Qπ
+c are the state and state-action values under policy π with respect to the cost.
+Now we construct another reward-based MDP M = (S, A, P, r, H), where we define the state-dependent action space A
+as
+As = {a ∈ A : Q∗
+c(s, a) ≤ V ∗
+c (s)}.
+By definition, As is non-empty for all s. We define a shorthand notation: we write π ∈ A(s) if Eπ[�H
+t=1 1{at /∈
+Ast}|s1 = s] = 0. Then we have the following lemma, which is a straightforward application of the performance
+difference lemma.
+Lemma 1. For any s1 ∈ S such that V ∗
+c (s1) = 0 and any π ∈ ∆, it is true that π ∈ A(s1) if and only if V π
+c (s1) = 0.
+Proof. By performance difference lemma (Kakade and Langford, 2002), we can write
+V π
+c (s1) − V ∗
+c (s1) = Eπ
+� H
+�
+t=1
+Q∗
+c(st, at) − V ∗
+c (st)|s1 = s1
+�
+.
+If for some s1 ∈ S, π ∈ A(s1), then Eπ
+��H
+t=1 Q∗
+c(st, at) − V ∗
+c (st)
+�
+≤ 0, which implies V π
+c (s1) ≤ V ∗
+c (s1). But since V ∗
+c
+is optimal, V π
+c (s1) = V ∗
+c (s1). On the other hand, suppose V π
+c (s1) = 0. It implies Eπ
+��H
+t=1 Q∗
+c(st, at) − V ∗
+c (st)
+�
+= 0
+since V ∗
+c (s1) = 0. Because by definition of optimality Q∗
+c(st, at) − V ∗
+c (st) ≥ 0, this implies π ∈ A(s1).
+We set our candidate policy π∗ as the optimal policy of this M. By Lemma 1, we have V π∗
+c
+(s) = V ∗
+c (s), so it is also an
+optimal policy to Mc. We prove our main claim of Theorem 1 below:
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) ≥ V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1) ≥ V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1).
+Proof. Because V π∗
+c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial:
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) = V π∗
+r,1 (s1) = V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1).
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+For the second inequality, we use Lemma 1:
+V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1) ≤ max
+π∈∆ V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1)
+= min
+y≥0 max
+π∈∆ V π
+r,1(s1) − yV π
+c,1(s1)
+=
+max
+π∈Ac(s1)
+V π
+r,1(s1)
+(By Lemma 1 )
+= V π∗
+r,1 (s1)
+= V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1).
+A.1.2
+Proof of Corollary 1
+Corollary 1. For π∗ in Theorem 1, it holds that Regret(K) = �K
+k=1 V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1).
+Proof. To
+prove
+Regret(K)
+=
+�K
+k=1 V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1),
+it
+suffices
+to
+prove
+�K
+k=1 V π∗
+r,1 (sk
+1)
+=
+maxπ∈∆0(K)
+�K
+k=1 V π
+r,1(sk
+1).
+By Lemma 1 and under Assumption 1, we notice that maxπ∈∆0(K)
+�K
+k=1 V π
+r,1(sk
+1) =
+maxπ∈A(sk
+1 ),∀k∈[K]
+�K
+k=1 V π
+r,1(sk
+1). This is equal to �K
+k=1 V π∗
+r,1 (sk
+1) by the definition of π∗ in the proof of Theorem 1.
+A.1.3
+Proof of Corollary 2
+Corollary 2. For any saddle-point to the CMDPs
+max
+π∈∆ V π
+r,1(s1), s.t. V π
+c,1(s1) ≤ 0
+of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also a saddle-point as defined in eq (6).
+Proof. We prove that eq (6) holds for (π∗, λ∗), that is
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) ≥ V π∗
+r,1 (s1) − λ∗(s1)V π∗
+c,1 (s1) ≥ V π
+r,1(s1) − λ∗(s1)V π
+c,1(s1).
+Because V π∗
+c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial:
+V π∗
+r,1 (s1) − λ(s1)V π∗
+c,1 (s1) = V π∗
+r,1 (s1) = V π∗
+r,1 (s1) − λ∗(s1)V π∗
+c,1 (s1).
+For the second inequality, we use Theorem 1:
+V π
+r,1(s1) − λ∗(s1)V π
+c,1(s1) ≤V π
+r,1(s1) − ˆλ(s1)V π
+c,1(s1)
+≤V π∗
+r,1 (s1) − ˆλ(s1)V π∗
+c,1 (s1)
+=V π∗
+r,1 (s1) − λ∗(s1)V π∗
+c,1 (s1)
+where the first step is because V π
+c,1(s1) by definition is in [0, 1] and λ∗ = ˆλ + 1, and the second step is by Theorem 1.
+A.1.4
+Proof of Theorem 2
+Theorem 2. Under Assumption 1, for any sequences {πk}K
+k=1 and {λk}K
+k=1 , it holds that
+Regret(K) ≤ Rp({πk}K
+k=1, π∗) + Rd({λk}K
+k=1, 0)
+Resets(K) ≤ Rp({πk}K
+k=1, π∗) + Rd({λk}K
+k=1, λ∗)
+where (π∗, λ∗) is the saddle-point defined in Corollary 2.
+We first establish the following intermediate result that will help us with our decomposition.
+
+Hoai-An Nguyen, Ching-An Cheng
+Lemma 2. For any primal-dual sequence {πk, λk}K
+k=1, �K
+k=1(Lk(π∗, λ′) − Lk(πk, λk)) ≤ Rp({π}K
+k=1, π∗), where
+(π∗, λ′) is the saddle-point defined in either Theorem 1 or Corollary 2.
+Proof. We derive this lemma by Theorem 1 and Corollary 2. First notice by Theorem 1 and Corollary 2 that for λ′ = λ∗, ˆλ,
+K
+�
+k=1
+Lk(π∗, λ′) =
+K
+�
+k=1
+V π∗
+r,1 (sk
+1) − λ′(sk
+1)V π∗
+c,1 (sk
+1)
+≤
+K
+�
+k=1
+V π∗
+r,1 (sk
+1) − λk(sk
+1)V π∗
+c,1 (sk
+1) =
+K
+�
+k=1
+Lk(π∗, λk).
+Then we can derive
+K
+�
+k=1
+(Lk(π∗, λ′) − Lk(πk, λk)) =
+K
+�
+k=1
+Lk(π∗, λ′) − Lk(π∗, λk) + Lk(π∗, λk) − Lk(πk, λk)
+≤
+K
+�
+k=1
+Lk(π∗, λk) − Lk(πk, λk) = Rp({π}K
+k=1, π∗)
+which finishes the proof.
+Then we upper bound Regret(K) and Resets(K) by Rp({πk}K
+k=1, πc) and Rd({λk}K
+k=1, λc) for suitable comparators. This
+decomposition is inspired by the techniques used in Ho-Nguyen and Kılınc¸-Karzan (2018).
+We first bound Resets(K).
+Lemma 3. For any primal-dual sequence {πk, λk}K
+k=1, �K
+k=1 V πk
+c,1 (sk
+1) ≤ Rp({π}K
+k=1, π∗) + Rd({λ}K
+k=1, λ∗), where
+(π∗, λ∗) is the saddle-point defined in Corollary 2.
+Proof. Notice �K
+k=1 V πk
+c,1 (sk
+1) = �K
+k=1 Lk(πk, ˆλ) − Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined in Theorem 1.
+This is because, as defined, λ∗ = ˆλ + 1. Therefore, we bound the RHS. We have
+K
+�
+k=1
+Lk(πk, ˆλ) − Lk(πk, λ∗) =
+K
+�
+k=1
+Lk(πk, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗)
+≤
+K
+�
+k=1
+Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗)
+≤Rp({π}K
+k=1, π∗) + Rd({λ}K
+k=1, λ∗)
+where second inequality is because �K
+k=1 Lk(π∗, ˆλ) ≥ �K
+k=1 Lk(πk, ˆλ) by Theorem 1, and the first inequality follows
+from Lemma 2 and Definition 1.
+Lastly, we bound Regret(K) with the lemma below and Corollary 1.
+Lemma 4. For any primal-dual sequence {πk, λk}K
+k=1, �K
+k=1(V π∗
+r,1 (sk
+1)−V πk
+r,1 (sk
+1)) ≤ Rp({π}K
+k=1, π∗)+Rd({λ}K
+k=1, 0),
+where (π∗, λ∗) is the saddle-point defined in Corollary 2.
+Proof. Note that L(π∗, λ∗) = L(π∗, 0) since V π∗
+c,1 (sk
+1) = 0 for all k ∈ [K]. Since by definition, for any π, Lk(π, 0) =
+V π
+r,1(sk
+1), we have the following:
+K
+�
+k=1
+V π∗
+r,1 (sk
+1) − V πk
+r,1 (sk
+1) =
+K
+�
+k=1
+Lk(π∗, λ∗) − Lk(πk, 0)
+=
+K
+�
+k=1
+Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0)
+≤Rp({π}K
+k=1, π∗) + Rd({λ}K
+k=1, 0)
+where the last inequality follows from Lemma 2 and Definition 1.
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+A.2
+Missing Proofs for Section 5
+A.2.1
+Proof of Theorem 3
+Theorem 3. Under Assumptions 1, 2, and 3, with high probability, Regret(K) ≤ ˜O((B + 1)
+√
+d3H4K) and Resets(K) ≤
+˜O((B + 1)
+√
+d3H4K).
+We first bound the regret of {πk}K
+k=1 and {λk}K
+k=1, and then use this to prove the bounds on our algorithm’s regret and
+number of resets with Theorem 2.
+We first bound the regret of {λk}K
+k=1.
+Lemma 5. Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈ U.
+Then it holds that Rd({λk}K
+k=1, λc) ≤ 1.5B
+√
+K +
+�K
+k=1(λk(sk
+1) − λc(sk
+1))(V k
+c,1(sk
+1) − V πk
+c,1 (sk
+1)).
+Proof. We notice first an equality.
+Rd({λk}K
+k=1, λc) =
+K
+�
+k=1
+Lk(πk, λk) − Lk(πk, λc)
+=
+K
+�
+k=1
+λc(sk
+1)V πk
+c,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)
+=
+K
+�
+k=1
+λc(sk
+1)V πk
+c,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)
++
+K
+�
+k=1
+λc(sk
+1)V k
+c,1(sk
+1) − λc(sk
+1)V k
+c,1(sk
+1) + λk(sk
+1)V k
+c,1(sk
+1) − λk(sk
+1)V k
+c,1(sk
+1)
+=
+K
+�
+k=1
+(λk(sk
+1) − λc(sk
+1))(−V k
+c,1(sk
+1)) +
+K
+�
+k=1
+(λk(sk
+1) − λc(sk
+1))(V k
+c,1(sk
+1) − V πk
+c,1 (sk
+1)).
+We observe that the first term is an online linear problem for θk (the parameter of λk(·)).
+In episode k ∈ [K],
+λk is played, and then the loss is revealed.
+Since the space of θk is convex, we use standard results (Lemma
+3.1 (Hazan et al., 2016)) to show that updating θk through projected gradient descent results in an upper bound for
+�K
+k=1(λk(sk
+1) − λc(sk
+1))(−V k
+c,1(sk
+1)). We restate the lemma here.
+Lemma 7 (Lemma 3.1 (Hazan et al., 2016)). Let S ⊆ Rd be a bounded convex and closed set in Euclidean space.
+Denote D as an upper bound on the diameter of S, and G as an upper bound on the norm of the subgradients of convex
+cost functions fk over S. Using online projected gradient descent to generate sequence {xk}K
+k=1 with step sizes {ηk =
+D
+G
+√
+k, k ∈ [K]} guarantees, for all K ≥ 1:
+RegretK = max
+x∗∈K
+K
+�
+k=1
+fk(xk) − fk(x∗) ≤ 1.5GD
+√
+K.
+Let us bound D. By Assumption 3, λ∗ = ⟨ξ(s), θ∗⟩ and ||θ∗||2 ≤ B. Since the comparator we use is λ∗, we can set D to
+be B. To bound G, we observe that the subgradient of our loss function is ξ(s)V k
+c,1(sk
+1) for each k ∈ [K]. Therefore, since
+V k
+c,1(sk
+1) ∈ [0, 1] and ||ξ(s)||2 ≤ 1 by Assumption 3, we can set G to be 1.
+We now bound the regret of {π}K
+k=1.
+Lemma 6. Consider any πc. With high probability, Rp({π}K
+k=1, πc) ≤ 2H(1 + B + H) + �K
+k=1 V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) +
+λk(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1)).
+
+Hoai-An Nguyen, Ching-An Cheng
+Proof. First we expand the regret into two terms.
+Rp({π}K
+k=1, πc) =
+K
+�
+k=1
+Lk(πc, λk) − Lk(πk, λk)
+=
+K
+�
+k=1
+V πc
+r,1(sk
+1) − λk(sk
+1)V πc
+c,1(sk
+1) − [V πk
+r,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)]
+=
+K
+�
+k=1
+V πc
+r,1(sk
+1) − λk(sk
+1)V πc
+c,1(sk
+1) − [V πk
+r,1 (sk
+1) − λk(sk
+1)V πk
+c,1 (sk
+1)]
++
+K
+�
+k=1
+[V k
+r,1(sk
+1) − λk(sk
+1)V k
+c,1(sk
+1)] − [V k
+r,1(sk
+1) − λk(sk
+1)V k
+c,1(sk
+1)]
+=
+K
+�
+k=1
+V πc
+r,1(sk
+1) − λk(sk
+1)V πc
+c,1(sk
+1) − [V k
+r,1(sk
+1) − λk(sk
+1)V k
+c,1(sk
+1)]
++
+K
+�
+k=1
+V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) + λk(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1)).
+To bound the first term, we use Lemma 3 from Ghosh et al. (2022), which characterize the property of upper confidence
+bound. For completeness, we re-write the lemma here. 7
+Lemma 8 (Lemma 3 (Ghosh et al., 2022)). With probability 1−p/2, it holds that T1 = �K
+k=1
+�
+V πc
+r,1(sk
+1)−λkV πc
+c,1(sk
+1)
+�
+−
+�
+V k
+r,1(sk
+1) − λkV k
+c,1(sk
+1)
+�
+≤ KH log(|A|)/α. Hence, for α =
+log(|A|)K
+2(1 + C + H), T1 ≤ 2H(1 + C + H), where C is such
+that λk ≤ C.
+In our problem setting, we can set C = B in the lemma above. Therefore, the first term is bounded by 2H(1+B+H).
+Lastly, we derive a bound on Rd({λk}K
+k=1, λc) + Rp({πk}K
+k=1, πc), which directly implies our final upper bound on
+Regret(K) and Resets(K) in Theorem 3 by Theorem 2.
+Lemma 9. For any πc and λc(s) = ⟨ξ(s), θc⟩ such that ∥θc∥ ≤ B, we have with probability 1 − p, Rd({λk}K
+k=1, λc) +
+Rp({πk}K
+k=1, πc) ≤ 1.5B
+√
+K + 2H(1 + B + H) + O((B + 1)
+√
+d3H4Kι2) where ι = log[log(|A|)4dKH/p].
+Proof. Combining the upper bounds in Lemma 5 and Lemma 6, we have an upper bound of
+Rd({λk}K
+k=1, λc) + Rp({πk}K
+k=1, πc) =1.5B
+√
+K +
+K
+�
+k=1
+(λk(sk
+1) − λc(sk
+1))(V k
+c,1(sk
+1) − V πk
+c,1 (sk
+1))
++ 2H(1 + B + H) +
+K
+�
+k=1
+V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) + λk(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1))
+=1.5B
+√
+K + 2H(1 + B + H)+
++
+K
+�
+k=1
+V k
+r,1(sk
+1) − V πk
+r,1 (sk
+1) + λc(sk
+1)(V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1))
+where the last term is the overestimation error due to optimism. To bound this term, we use Lemma 4 from Ghosh et al.
+(2022). We re-write the lemma here.
+Lemma 10 (Lemma 4 (Ghosh et al., 2022)). WIth probability at least 1 − p/2, for any λ ∈ [0, C], �K
+k=1
+�
+V k
+r,1(sk
+1) −
+V πk
+r,1 (sk
+1)
+�
++ λ �K
+k=1
+�
+V πk
+c,1 (sk
+1) − V k
+c,1(sk
+1)
+�
+≤ O((λ + 1)
+√
+d3H4Kι2) where ι = log[log(|A|)4dKH/p].
+7Note that Ghosh et al. (2022) use a utility function rather than a cost function to denote the constraint on the MDP (cost is just −1×
+utility). Also note that their Lemma 3 is proved for an arbitrary initial state sequence and for any comparator (which includes π∗).
+
+Provable Reset-free Reinforcement Learning by No-Regret Reduction
+Since we have a bound on all λc(sk
+1) of B for all k ∈ [K], we have a bound of O((B + 1)
+√
+d3H4Kι2).
+

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+arXiv:2301.05512v1  [math.PR]  13 Jan 2023
+Almost sure invariance principle for the Kantorovich distance
+between the empirical and the marginal distributions of strong
+mixing sequences
+J´erˆome Dedecker∗, Florence Merlev`ede †
+January 16, 2023
+Abstract
+We prove a strong invariance principle for the Kantorovich distance between the empirical
+distribution and the marginal distribution of stationary α-mixing sequences.
+Running head. ASIP for the empirical W1 distance.
+Keywords. Empirical process, Wasserstein distance, Almost sure invariance principle, Compact
+law of the iterated logarithm, Bounded law of the iterated logarithm, Conditional Value at Risk
+Mathematics Subject Classification (2010). 60F15, 60G10, 60B12
+1
+Introduction and notations
+Let (Xi)i∈Z be a strictly stationary sequence of real-valued random variables. Define the two
+σ-algebras F0 = σ(Xi, i ≤ 0) and Gk = σ(Xi, i ≥ k), and recall that the strong mixing coefficients
+(α(k))k≥0 of Rosenblatt [13] are defined by
+α(k) =
+sup
+A∈F0,B∈Gk
+|P(A ∩ B) − P(A)P(B)| .
+(1.1)
+Let µ be the common distribution of the Xi’s, and let
+µn = 1
+n
+n
+�
+k=1
+δXk
+be the empirical measure based on X1, . . . , Xn.
+In this paper, we prove a strong invariance
+principle for the Kantorovich distance W1(µn, µ) between µn and µ under a condition on the
+mixing coefficients α(k). Recall that the Kantorovich distance (also called Wasserstein distance
+of order 1) between two probability measures µ and ν is defined by
+W1(µ, ν) =
+inf
+π∈M(µ,ν)
+�
+|x − y|π(dx, dy) ,
+where M(µ, ν) is the set of probability measures on R2 with marginals µ and ν. We shall use the
+following well known representation for probabilities on the real line:
+W1(µ, ν) =
+�
+|Fµ(x) − Fν(x)|dx ,
+(1.2)
+∗J´erˆome Dedecker, Universit´e de Paris, CNRS, MAP5, UMR 8145, 45 rue des Saints-P`eres, F-75006 Paris, France.
+†Florence Merlev`ede, Universit´e Gustave Eiffel, LAMA, UMR 8050 CNRS, F-77454 Marne-La-Vall´ee, France.
+1
+
+where Fµ is the cumulative distribution function of µ.
+Let H : t → P([X0| > t) be the tail function of |X0|. In the case where (Xi)i∈Z is a sequence
+of independent and identically distributed (i.i.d.) random variables, del Barrio et al. [2] used the
+representation (1.2) and a general result of Jain [7] for Banach-valued random variables to prove a
+central limit theorem for √nW1(µn, µ). More precisely, they showed that √nW1(µn, µ) converges
+in distribution to the L1(dt) norm of an L1(dt)-valued Gaussian random variable, provided that
+� ∞
+0
+�
+H(t) dt < ∞ .
+(1.3)
+They also proved that √nW1(µn, µ) is stochastically bounded iff (1.3) holds, proving that this
+condition is necessary and sufficient for the weak convergence of √nW1(µn, µ).
+Still in the i.i.d. case, we easily deduce from Chapters 8 and 10 in Ledoux and Talagrand [8]
+that: if (1.3) holds, then the sequence
+√n
+√2 log log nW1(µn, µ)
+(1.4)
+satisfies a compact law of the iterated logarithm.
+For strongly mixing sequences in the sense of Rosenblatt [13], we proved in [6] the central limit
+theorem for √nW1(µn, µ) under the condition
+� ∞
+0
+�
+�
+�
+�
+∞
+�
+k=0
+(α(k) ∧ H(t)) dt < ∞
+(1.5)
+(where a ∧ b means the minimum between two reals a and b), and we give sufficient conditions
+for (1.5) to hold. Note that, in [6], we used a weaker version of the α-mixing coefficients, that
+enables to deal with a large class of non-mixing processes in the sense of Rosenblatt [13].
+In Section 2 of this paper, we prove a strong invariance principle for W1(µn, µ) under the
+condition (1.5). The compact law of the iterated logarithm for (1.4) easily follows from this strong
+invariance principle. In Section 3, we apply our general result to derive the almost sure rate of
+convergence of the empirical estimator of the Conditional Value at Risk (CV aR) for stationary
+α-mixing sequences.
+In the rest of the paper, we shall use the following notation: for two sequences (an)n≥1 and
+(bn)n≥1 of positive reals, an ≪ bn means there exists a positive constant C not depending on n
+such that an ≤ Cbn for any n ≥ 1.
+2
+Main result
+Our main result is the following strong invariance principle for W1(µn, µ).
+Theorem 2.1. Assume that (1.5) is satisfied. Then, enlarging the probability space if necessary,
+there exists a sequence of i.i.d. L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with
+covariance function defined as follows: for any f, g ∈ L∞(dt),
+Γ(f, g) = Cov
+��
+f(t)Z1(t) dt,
+�
+g(t)Z1(t) dt
+�
+=
+�
+k∈Z
+��
+f(t)g(s)Cov(1X0≤t, 1Xk≤s) ds dt ,
+(2.1)
+and such that
+nW1(µn, µ) −
+� �����
+n
+�
+k=1
+Zk(t)
+����� dt = o(
+�
+n log log n)
+almost surely.
+2
+
+Remark 2.2. In [4], Cuny proved a strong invariance principle for W1(µn, µ). under the condition
+∞
+�
+k=0
+1
+√
+k + 1
+� ∞
+0
+�
+α(k) ∧ H(t) dt < ∞
+(2.2)
+(in fact, he proved the result for a weaker version of the α-mixing coefficient, the same as that
+used in [6] for the central limit theorem). It follows from Section 5 of [6], that the condition (1.5)
+is always less restrictive than (2.2).
+As a consequence of Theorem 2.1, we get the compact law of the iterated logarithm. Let K
+be the unit ball of the reproducing kernel Hilbert space (RKHS) associated with Γ, and C be the
+image of K by the L1(dt) norm. The following corollary holds:
+Corollary 2.1. Assume that (1.5) is satisfied. Then the sequence
+√n
+√2 log log nW1(µn, µ)
+is almost surely relatively compact, with limit set C.
+The proof of Theorem 2.1 is based on two ingredients: a martingale approximation in L1(dt),
+as in [6], and the following version of the bounded law of the iterated logarithm, which has an
+interest in itself.
+Proposition 2.1. Assume that (1.5) holds, and let
+V =
+� ∞
+0
+�
+�
+�
+�
+∞
+�
+k=0
+(α(k) ∧ H(t)) dt .
+(2.3)
+Then, there exists a universal constant η such that for any ε > 0,
+�
+n≥2
+1
+nP
+�
+max
+1≤k≤n kW1(µk, µ) > (ηV + ε)
+�
+n log log n
+�
+< ∞ .
+(2.4)
+Remark 2.3. (The bivariate case). Let (Xi, Yi)i∈Z be a stationary sequence of R2-valued random
+variables, and define the coefficients α(k) as in (1.1), with the two σ-algebras F0 = σ(Xi, Yi, i ≤ 0)
+and Gk = σ(Xi, Yi, i ≥ k). Let µX (resp. µY ) be the common distribution of the Xi’s (resp. the
+Yi’s), and let
+µn,X = 1
+n
+n
+�
+k=1
+δXk
+and
+µn,Y = 1
+n
+n
+�
+k=1
+δYk .
+Combining the arguments in [3] and the proof of Theorem 2.1, one can prove the following strong
+invariance principle for n (W1(µn,X, µn,Y ) − W1(µX, µY )).
+Let ϕ be the continuous function from L1(dt) to R defined by
+ϕ(x) =
+� �
+sign{FX(t) − FY (t)} x(t)1FX(t)̸=FY (t) + |x(t)|1FX(t)=FY (t)
+�
+dt ,
+where FX (resp. FY ) is the cumulative distribution function of µX (resp. µY ). Assume that
+� ∞
+0
+�
+�
+�
+�
+∞
+�
+k=0
+(α(k) ∧ HX(t)) dt < ∞
+and
+� ∞
+0
+�
+�
+�
+�
+∞
+�
+k=0
+(α(k) ∧ HY (t)) dt < ∞ .
+Then, enlarging the probability space if necessary, there exists a sequence of i.i.d. L1(dt)-valued
+centered Gaussian random variables (Zi)i≥1 with covariance function given by: for any f, g ∈
+L∞(dt),
+�Γ(f, g) = Cov
+��
+f(t)Z1(t) dt,
+�
+g(t)Z1(t) dt
+�
+=
+�
+k∈Z
+��
+f(t)g(s)Cov(1X0≤t − 1Y0≤t, 1Xk≤s − 1Yk≤s) ds dt ,
+3
+
+and such that
+n (W1(µn,X, µn,Y ) − W1(µX, µY )) − ϕ
+� n
+�
+k=1
+Zk
+�
+= o(
+�
+n log log n)
+almost surely.
+3
+Rates of convergence of the empirical estimator of
+the Conditional Value at Risk
+The Conditional Value at Risk at level u ∈ (0, 1] of a real-valued integrable random variable X
+(CV aRu(X)) is a “risk measure” (according to the definition of Acerbi and Tasche [1]), which is
+widely used in mathematical finance. It is sometimes called Expected Shortfall of Average Value
+at Risk. We refer to the paper [1] for a clear definition of that indicator, and for its relation with
+other well known measures, such as the Value at Risk, the Worst Conditional Expectation, the
+Tail Conditional Expectation... According to Acerbi and Tasche [1], CV aRu(X) can be expressed
+as
+CV aRu(X) = − 1
+u
+� u
+0
+F −1
+X (x)dx ,
+where FX is the cumulative distribution function of the variable X, and F −1
+X
+is its usual cadlag
+inverse: F −1
+X (u) = inf{x ∈ R : FX(x) ≥ u}.
+Concerning the difference between the Conditional Value at Risk of two random variables X
+and Y , the following elementary inequality holds (see for instance [12]):
+|CV aRu(X) − CV aRu(Y )| ≤ 1
+u
+� 1
+0
+|F −1
+X (x) − F −1
+Y (x)|dx = 1
+uW1(µX, µY ) ,
+(3.1)
+where µX (resp. µY ) is the distribution of X (resp. Y ).
+Consider now the problem of estimating CV aRu(X) from the random variables X1, ..., Xn,
+where (Xi)i∈Z is a stationary sequence of α-mixing random variables with common distribution
+µ = µX. A natural estimator is then
+�
+CV aRu,n = − 1
+u
+� u
+0
+F −1
+n (x)dx ,
+where Fn is the empirical distribution function based on X1, . . . , Xn. From (3.1), we get the upper
+bound
+���CV aRu(X) − �
+CV aRu,n
+��� ≤ 1
+u
+� 1
+0
+|F −1
+X (x) − F −1
+n (x)|dx = 1
+uW1(µn, µ) ,
+From Corollary 2.1, we obtain the almost sure rate of convergence of �
+CV aRu,n: if (1.5) holds,
+then
+lim sup
+n→∞
+√n
+√2 log log n
+���CV aRu(X) − �
+CV aRu,n
+��� ≤ κ(Γ)
+u
+almost surely,
+where κ(Γ) is the largest value of the compact set C of Corollary 2.1 (recall that the covariance
+function Γ is defined in (2.1)). It is well known (see for instance Section 8 in [8]) that the constant
+κ(Γ) can be expressed as
+κ(Γ) =
+sup
+f:∥f∥∞≤1
+�
+Var
+��
+f(t)Z(t)dt
+��1/2
+≤
+����
+�
+|Z(t)|dt
+����
+2
+,
+where Z is an L1(dt)-valued centered random variable with covariance function Γ.
+4
+
+4
+Proofs
+4.1
+Proof of Theorem 2.1
+Let (Ω, A, P) be the underlying probability space.
+By a standard argument, one may assume
+that Xi = X0 ◦ T , where T : Ω �→ Ω is a bijective, bi-measurable transformation, preserving the
+probability P. Let also Fi = σ(Xk, k ≤ i).
+Let Y0(t) = 1X0≤t − F(t), and Yk(t) = Y0(t) ◦ T k = 1Xk≤t − F(t). With these notations and
+the representation (1.2) one has that
+nW1(µn, µ) =
+� �����
+n
+�
+k=1
+Yk(t)
+����� dt .
+(4.1)
+From Section 4 in [6], we know that, if (1.5) holds, then
+Y0(t) = D0(t) + A(t) − A(t) ◦ T,
+(4.2)
+where D0 is such that E(D1(t)|F−1) = 0 almost surely and
+�
+∥D0(t)∥2 dt < ∞, and A is such that
+�
+∥A(t)∥1 dt < ∞. Moreover, the covariance operator of D0 is exactly Γ: for any f, g ∈ L∞(dt),
+Γ(f, g) = Cov
+��
+f(t)D0(t) dt,
+�
+g(t)D0(t) dt
+�
+.
+(4.3)
+Let Dk(t) = D0 ◦ T k. From (4.2), it follows that
+n
+�
+k=1
+Yk =
+n
+�
+k=1
+Dk + A ◦ T − A ◦ T n .
+(4.4)
+From [4, Proposition 3.3], we know that, enlarging the probability space if necessary, there exists
+a sequence of i.i.d. L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance
+function Γ such that
+� �����
+n
+�
+k=1
+Dk(t) −
+n
+�
+k=1
+Zk(t)
+����� dt = o
+��
+n log log n
+�
+almost surely.
+(4.5)
+Hence, the result will follow from (4.1), (4.4) and (4.5) if we can prove that
+lim
+n→∞
+1
+√n log log n
+�
+|A(t) ◦ T n| dt = 0
+almost surely.
+(4.6)
+To prove (4.6), we start by considering the integral over [−M, M]c, for M > 0. Applying again
+[4, Proposition 3.3], we infer that
+lim sup
+n→∞
+1
+√2n log log n
+�
+[−M,M]c
+�����
+n
+�
+k=1
+Dk(t)
+����� dt ≤
+�
+[−M,M]c ∥D0(t)∥2 dt
+almost surely.
+(4.7)
+Now, as will be clear from the proof, Proposition 2.1 also holds on the space L1([−M, M]c, dt),
+and implies that there exists a universal constant η such that, for any positive ε,
+lim sup
+n→∞
+1
+√n log log n
+�
+[−M,M]c
+�����
+n
+�
+k=1
+Yk(t)
+����� dt ≤ ε+η
+� ∞
+M
+�
+�
+�
+�
+∞
+�
+k=0
+min {α(k), H(t)} dt
+almost surely.
+(4.8)
+From (4.7) and (4.8), we infer that
+lim
+M→∞ lim sup
+n→∞
+1
+√n log log n
+�
+[−M,M]c |A(t) ◦ T n| dt = 0
+almost surely.
+5
+
+Hence the proof of (4.6) will be complete if we prove that, for any M > 0,
+lim sup
+n→∞
+1
+√n log log n
+� M
+−M
+|A(t) ◦ T n| dt = 0
+almost surely.
+(4.9)
+To prove (4.9), we work in the space H = L2([−M, M], dt), and we denote by ∥ · ∥H and ⟨·, ·⟩
+the usual norm and scalar product on H. Since E(∥D0∥2
+H) < ∞, we know from [4] that �n
+k=1 Dk
+satisfies the compact law of the iterated logarithm in H. Since �
+k≥0 α(k) < ∞ and Y0 is bounded
+in H, we infer from [5] that �n
+k=1 Yk satisfies also the compact law of the iterated logarithm in H.
+Now, arguing exactly as in the end of the proof of [5, Theorem 4], one has: for any f in H
+lim
+n→∞
+⟨f, A ◦ T n⟩
+√n log log n = 0
+almost surely.
+(4.10)
+Let (ei)i≥1 be a complete orthonormal basis of H and PN(f) = �N
+k=1⟨f, ek⟩ek be the projection
+of f on the space spanned by the first N elements of the basis. From (4.10), we get that
+lim
+n→∞
+PN(A ◦ T n)
+√n log log n = 0
+almost surely.
+(4.11)
+On another hand, applying again [4, Proposition 3.3] (as done in (4.7)), we get
+lim
+N→∞ lim sup
+n→∞
+1
+√n log log n
+�����(I − PN)
+� n
+�
+k=1
+Dk
+������
+H
+= 0
+almost surely,
+(4.12)
+and applying [5, Theorem 4],
+lim
+N→∞ lim sup
+n→∞
+1
+√n log log n
+�����(I − PN)
+� n
+�
+k=1
+Yk
+������
+H
+= 0
+almost surely.
+(4.13)
+From (4.4), (4.12) and (4.13), we infer that
+lim
+N→∞ lim sup
+n→∞
+∥(I − PN)A ◦ T n∥H
+√n log log n
+= 0
+almost surely,
+which, together with (4.11), implies (4.9). The proof of Theorem 2.1 is complete. ⋄
+4.2
+Proof of Proposition 2.1
+For any n ∈ N, let us introduce the following notations:
+R(u) = min{q ∈ N∗ : α(q) ≤ u}Q(u)
+and
+R−1(x) = inf{u ∈ [0, 1] : R(u) ≤ x} .
+For a positive real a that will be specified later, let
+mn = a
+�
+n
+log log n ,
+vn = R−1(mn) ,
+Mn = Q(vn) .
+(4.14)
+For any M > 0, let gM(y) = (y ∧ M) ∨ (−M). For any integer i, define
+X′
+i = gMn(Xi) and X′′
+i = Xi − X′
+i .
+(4.15)
+We first recall that, by the dual expression of W1(µn, µ),
+nW1(µn, µ) = sup
+f∈Λ1
+n
+�
+i=1
+(f(Xi) − E(f(Xi))) .
+6
+
+where Λ1 is the set of Lipschitz functions such that |f(x) − f(y)| ≤ |x − y|. Hence,
+nW1(µn, µ) ≤ sup
+f∈Λ1
+n
+�
+i=1
+�
+f(X′
+i) − E(f(X′
+i))
+�
++ sup
+f∈Λ1
+n
+�
+i=1
+�
+f(Xi) − f(X′
+i) − E(f(Xi) − f(X′
+i))
+�
+.
+Therefore, setting,
+F ′
+n(t) = 1
+n
+n
+�
+k=1
+1{X′
+k≤t}
+and
+F ′(t) = P(X′
+1 ≤ t) ,
+and noticing that
+k∥F ′
+k − F ′∥1 = sup
+f∈Λ1
+k
+�
+i=1
+�
+f(X′
+i) − E(f(X′
+i)
+�
+,
+we get
+max
+1≤k≤n kW1(µk, µ) ≤ max
+1≤k≤n k∥F ′
+k − F ′∥1 +
+n
+�
+i=1
+(|X′′
+i | + E(|X′′
+i |) .
+(4.16)
+Now, note that
+�
+n≥2
+1
+√n log log nE(|X′′
+n|) ≤
+�
+n≥2
+1
+√n log log n
+� +∞
+0
+P
+�
+|X0|1|X0|>Q(vn) > t
+�
+dt
+≤
+�
+n≥2
+1
+√n log log n
+� +∞
+Q(vn)
+H(t)dt ≤
+�
+n≥2
+1
+√n log log n
+� vn
+0
+Q(u)du
+≤
+�
+n≥2
+1
+√n log log n
+� 1
+0
+Q(u)1mn≤R(u)du ≪
+� 1
+0
+R(u)Q(u)du.
+But, according to Propositions 5.1 and 5.2 in [6], condition (1.5) implies that
+� 1
+0
+R(u)Q(u)du < ∞ .
+(4.17)
+Hence, to prove (2.4) it suffices to show that there exists an universal constant η such that for
+any ε > 0,
+�
+n≥2
+1
+nP
+�
+max
+1≤k≤n k∥F ′
+k − F ′∥1 > ηV
+�
+n log log n
+�
+< ∞ .
+(4.18)
+For this purpose, let
+qn = min{k ∈ N∗ : α(k) ≤ vn} ∧ n .
+(4.19)
+Since R is right continuous, we have R(R−1(w)) ≤ w for any w, hence
+qnMn = R(vn) = R(R−1(mn)) ≤ mn .
+(4.20)
+Assume first that qn = n. Bounding f(X′
+i) − E(f(X′
+i)) by 2Mn, we obtain
+max
+1≤k≤n k∥F ′
+k − F ′∥1 ≤ 2nMn = 2qnMn ≤ 2mn .
+(4.21)
+Taking into account the definition of mn, it follows that there exists n0 depending on a, V and η,
+such that for any n ≥ n0, 8mn ≤ κV √n log log n. This proves the proposition in the case where
+qn = n.
+From now on, we assume that qn < n. Therefore qn = min{k ∈ N∗ : α(k) ≤ vn} and then
+α(qn) ≤ vn. For any integer i, define
+Ui(t) =
+iqn
+�
+k=(i−1)qn+1
+�
+1X′
+k≤t − E
+�
+1X′
+k≤t
+��
+.
+7
+
+and notice that
+max
+1≤k≤n k∥F ′
+k − F ′∥1 ≤ 2qnMn +
+� Mn
+−Mn
+max
+1≤j≤[n/qn]
+�����
+j
+�
+i=1
+Ui(t)
+����� dt .
+Let kn = [n/qn]. For any t, applying Rio’s coupling lemma (see [11, Lemma 5.2]) recursively,
+we can construct random variables (U ∗
+i (t))1≤i≤kn such that
+• U ∗
+i (t) has the same distribution as U ′
+i for all 1 ≤ i ≤ kn,
+• the random variables (U ∗
+2i(t))2≤2i≤kn are independent, as well as the random variables
+(U ∗
+2i−1(t))1≤2i−1≤kn,
+• we can suitably control ∥Ui(t) − U ∗
+i (t)∥1 as follows: for any i ≥ 1,
+∥Ui(t) − U ∗
+i (t)∥1 ≤ 4qnα(qn) .
+(4.22)
+Substituting U ∗
+i (t) to Ui(t), we obtain
+max
+1≤k≤n k∥F ′
+k − F ′∥1 ≤ 2qnMn +
+max
+2≤2j≤[n/qn]
+�����
+j
+�
+i=1
+U ∗
+2i(t)
+�����
++
+max
+1≤2j−1≤[n/qn]
+�����
+j
+�
+i=1
+U ∗
+2i−1(t)
+�����+
+[n/qn]
+�
+i=1
+|Ui(t) − U ∗
+i (t)| .
+(4.23)
+Therefore, setting κ = η/4, for n ≥ n0,
+P
+�
+max
+1≤k≤n k∥F ′
+k − F ′∥1 ≥ 4V κ
+�
+n log log n
+�
+≤ I1(n) + I2(n) + I3(n) ,
+(4.24)
+where
+I1(n) = P
+
+
+� Mn
+−Mn
+[n/qn]
+�
+i=1
+|Ui(t) − U ∗
+i (t)| dt ≥ V κ
+�
+n log log n
+
+
+I2(n) = P
+�� Mn
+−Mn
+max
+2≤2j≤[n/qn]
+�����
+j
+�
+i=1
+U ∗
+2i(t)
+����� dt ≥ V κ
+�
+n log log n
+�
+I3(n) = P
+�� Mn
+−Mn
+max
+1≤2j−1≤[n/qn]
+�����
+j
+�
+i=1
+U ∗
+2i−1(t)
+����� dt ≥ V κ
+�
+n log log n
+�
+.
+Using Markov’s inequality and (4.22), we get
+I1(n) ≪
+n
+√n log log nMnα(qn) ≪
+n
+√n log log nvnQ(vn) ≪
+n
+√n log log n
+� R−1(mn)
+0
+Q(u)du .
+Hence, by (4.17),
+�
+n≥2
+1
+nI1(n) ≪
+�
+n≥2
+1
+√n log log n
+� R−1(mn)
+0
+Q(u)du ≪
+� 1
+0
+R(u)Q(u)du < ∞ .
+To handle now the term I2(n) (as well as I3(n)) in the decomposition (4.24), we shall use
+again Markov’s inequality but this time at the order p ≥ 2. Hence for p ≥ 2, taking into account
+the stationarity, we get
+I2(n) ≤
+1
+(V κ)p(n log log n)p/2
+
+
+� Q(vn)
+−Q(vn)
+�����
+max
+2≤2j≤[n/qn]
+�����
+j
+�
+i=1
+˜U2i(t)
+�����
+�����
+p
+dt
+
+
+p
+.
+8
+
+Applying Rosenthal’s inequality (see for instance [9, Theorem 4.1]) and taking into account the
+stationarity, there exist two positive universal constants c1 and c2 not depending on p such that
+�����
+max
+2≤2j≤[n/qn]
+�����
+j
+�
+i=1
+U ∗
+2i(t)
+�����
+�����
+p
+p
+≤ cp
+1pp/2(n/qn)p/2∥U2(t)∥p
+2 + cp
+2pp(n/qn)∥U2(t)∥p
+p := J1(t) + J2(t) .
+(4.25)
+Using similar arguments as to handle the quantity I2(n) in the proof of [6, Proposition 3.4], we
+have
+� Q(vn)
+−Q(vn)
+∥U2(t)∥2dt =
+� Q(vn)
+−Q(vn)
+�
+Var
+� qn
+�
+i=1
+1{X′
+i≤t}
+��1/2
+dt
+≤ 2
+√
+2√qn
+� Q(vn)
+0
+�qn−1
+�
+k=0
+α(k) ∧ H(t)
+�1/2
+dt ≤ 2V
+�
+2qn .
+(4.26)
+Hence
+�
+n≥2
+1
+n(V κ)p(n log log n)p/2
+�� Q(vn)
+−Q(vn)
+J1(t)1/pdt
+�p
+≤
+�
+n≥2
+(2
+√
+2c1√p)p
+nκp(log log n)p/2 .
+Let now
+p = pn = max{c log log n, 2},
+where c will be specified later. Set n1 = min{n ≥ 2 : c log log n ≥ 2}. It follows that
+�
+n≥n1
+1
+n(V κ)p(n log log n)p/2
+�� Q(vn)
+−Q(vn)
+J1(t)1/pdt
+�p
+≤
+�
+n≥n1
+1
+n
+�2c1
+√
+2c
+κ
+�c log log n
+,
+which is finite provided we take κ such that 2c1
+√
+2c
+κ
+= α−1 with α > 1 and c > (log α)−1.
+On another hand, proceeding as in (4.26), we deduce that, for any t > 0,
+∥U2(t)∥p
+p =
+�����
+qn
+�
+i=1
+�
+1{X′
+i≤t} − P(X′
+i ≤ t)
+������
+p
+p
+≤ qp−2
+n
+�����
+qn
+�
+i=1
+�
+1{X′
+i≤t} − P(X′
+i ≤ t)
+������
+2
+2
+≤ 2qp−1
+n
+qn−1
+�
+k=0
+(α(k) ∧ H(t)).
+In addition
+� Q(vn)
+0
+�qn−1
+�
+k=0
+α(k) ∧ H(t)
+�1/p
+dt =
+� Q(vn)
+0
+�� H(t)
+0
+(α−1(u) ∧ qn)du
+�1/p
+dt
+≤
+� Q(vn)
+0
+�
+vnqn +
+� H(t)
+vn
+(α−1(u) ∧ qn)du
+�1/p
+dt .
+Note that u < H(t) ⇐⇒ t < Q(u). Consequently u < H(t) implies that Q−2(u) < t−2. Hence
+� Q(vn)
+0
+�qn−1
+�
+k=0
+α(k) ∧ H(t)
+�1/p
+dt
+≤ (vnqn)1/pQ(vn) +
+� Q(vn)
+0
+�
+t−2
+� H(t)
+vn
+(α−1(u) ∧ qn)Q2(u)du
+�1/p
+≤ (vnqn)1/pQ(vn) +
+�� 1
+vn
+R(u)Q(u)du
+�1/p � Q(vn)
+0
+t−2/pdt
+≤ (vnqn)1/pQ(vn) +
+�� 1
+0
+R(u)Q(u)du
+�1/p
+p(p − 2)−1Q(vn)1−2/p .
+9
+
+Set n2 = min{n ≥ 2 : c log log n ≥ 4}. It follows that
+�
+n≥n2
+1
+n(V κ)p(n log log n)p/2
+�� Q(vn)
+−Q(vn)
+J2(t)1/pdt
+�p
+≤ 2
+�
+n≥n2
+(4c2p)p
+(κV )p(n log log n)p/2 qp−2
+n
+�
+vnqnQp(vn) + 2pQ(vn)p−2
+� 1
+0
+R(u)Q(u)du
+�
+.
+Note that
+vnqnQ2(vn) = vnα−1(vn)Q2(vn) ≤
+� 1
+0
+R(u)Q(u)du .
+Hence, since qnMn ≤ mn, we get
+�
+n≥n2
+1
+n(V κ)p(n log log n)p/2
+�� Q(vn)
+−Q(vn)
+J2(t)1/pdt
+�p
+≤ 4
+� 1
+0
+R(u)Q(u)du
+�
+n≥n2
+(8c2p)p
+(κV )p(n log log n)p/2 mp−2
+n
+≤ 4a−2
+� 1
+0
+R(u)Q(u)du
+�
+n≥n2
+�8ac2c
+κV
+�p log log n
+n
+,
+which is finite by taking into account (4.17), and if we choose a = (c1κV )/(2c2
+√
+2c). Indeed, in
+this case,
+8ac2c
+κV
+= 2c1
+√
+2c
+κ
+× 2ac2
+√
+2c
+c1κV
+= α−1 .
+This ends the proof of the proposition. ⋄
+References
+[1] C. Acerbi and D. Tasche (2002), On the coherence of Expected Shortfall. Journal of Banking
+and Finance 26 1487-1503.
+[2] E. del Barrio, E. Gin´e and C. Matr´an (1999), Central limit theorems for the Wasserstein
+distance between the empirical and the true distributions. Ann. Probab. 27 1009-1071.
+[3] P. Berthet, J. Dedecker, and F. Merlev`ede, Central limit theorem and almost sure results for
+bivariate empirical W1 distances. (2020) https://hal.archives-ouvertes.fr/hal-02881842
+[4] C. Cuny (2017), Invariance principles under the Maxwell-Woodroofe condition in Banach
+spaces. Ann. Probab. 45 1578–1611.
+[5] J. Dedecker and F. Merlev`ede (2010), On the almost sure invariance principle for stationary
+sequences of Hilbert-valued random variables. Dependence in probability, analysis and number
+theory, 157–175, Kendrick Press, Heber City, UT.
+[6] J. Dedecker and F. Merlev`ede (2017), Behavior of the Wasserstein distance between the empir-
+ical and the marginal distributions of stationary α-dependent sequences. Bernoulli 23 2083–
+2127.
+[7] N. C. Jain (1977), Central limit theorems and related questions in Banach space. Proceedings
+of Symposium in Pure and Applied Mathematics 31 55-65. Amer. Math. Soc. Providence, RI.
+[8] M. Ledoux and M. Talagrand (1991), Probability in Banach spaces. Isoperimetry and pro-
+cesses. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23 Springer-Verlag, Berlin,
+xii+ 480 pp.
+[9] I. Pinelis (1994), Optimum bounds for the distributions of martingales in Banach spaces. Ann.
+Probab. 22 1679–1706.
+10
+
+[10] E. Rio (1995), The functional law of the iterated logarithm for stationary α-mixing sequences.
+Ann. Probab. 23 1188-1203.
+[11] E. Rio (2000), Th´eorie asymptotique des processus al´eatoires faiblement d´ependants. Math.
+Appl. 31 Berlin.
+[12] E. Rio (2017), About the conditional value at risk of partial sums. C. R. Math. Acad. Sci.
+Paris 355 1190-1195.
+[13] M. Rosenblatt (1956), A central limit theorem and a strong mixing condition, Proc. Nat.
+Acad. Sci. U.S.A. 42 43-47.
+11
+

79E5T4oBgHgl3EQfQQ7U/content/tmp_files/load_file.txt

@@ -0,0 +1,339 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf,len=338
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='05512v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='PR]  13 Jan 2023  Almost sure invariance principle for the Kantorovich distance  between the empirical and the marginal distributions of strong  mixing sequences  J´erˆome Dedecker∗, Florence Merlev`ede †  January 16, 2023  Abstract  We prove a strong invariance principle for the Kantorovich distance between the empirical  distribution and the marginal distribution of stationary α-mixing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Running head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' ASIP for the empirical W1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Empirical process, Wasserstein distance, Almost sure invariance principle, Compact  law of the iterated logarithm, Bounded law of the iterated logarithm, Conditional Value at Risk  Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' 60F15, 60G10, 60B12  1  Introduction and notations  Let (Xi)i∈Z be a strictly stationary sequence of real-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Define the two  σ-algebras F0 = σ(Xi, i ≤ 0) and Gk = σ(Xi, i ≥ k), and recall that the strong mixing coefficients  (α(k))k≥0 of Rosenblatt [13] are defined by  α(k) =  sup  A∈F0,B∈Gk  |P(A ∩ B) − P(A)P(B)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1)  Let µ be the common distribution of the Xi’s, and let  µn = 1  n  n  �  k=1  δXk  be the empirical measure based on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  In this paper, we prove a strong invariance  principle for the Kantorovich distance W1(µn, µ) between µn and µ under a condition on the  mixing coefficients α(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Recall that the Kantorovich distance (also called Wasserstein distance  of order 1) between two probability measures µ and ν is defined by  W1(µ, ν) =  inf  π∈M(µ,ν)  �  |x − y|π(dx, dy) ,  where M(µ, ν) is the set of probability measures on R2 with marginals µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' We shall use the  following well known representation for probabilities on the real line:  W1(µ, ν) =  �  |Fµ(x) − Fν(x)|dx ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2)  ∗J´erˆome Dedecker, Universit´e de Paris, CNRS, MAP5, UMR 8145, 45 rue des Saints-P`eres, F-75006 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  †Florence Merlev`ede, Universit´e Gustave Eiffel, LAMA, UMR 8050 CNRS, F-77454 Marne-La-Vall´ee, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  1  where Fµ is the cumulative distribution function of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Let H : t → P([X0| > t) be the tail function of |X0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' In the case where (Xi)i∈Z is a sequence  of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=') random variables, del Barrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' [2] used the  representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2) and a general result of Jain [7] for Banach-valued random variables to prove a  central limit theorem for √nW1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' More precisely, they showed that √nW1(µn, µ) converges  in distribution to the L1(dt) norm of an L1(dt)-valued Gaussian random variable, provided that  � ∞  0  �  H(t) dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3)  They also proved that √nW1(µn, µ) is stochastically bounded iff (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3) holds, proving that this  condition is necessary and sufficient for the weak convergence of √nW1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Still in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' case, we easily deduce from Chapters 8 and 10 in Ledoux and Talagrand [8]  that: if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3) holds, then the sequence  √n  √2 log log nW1(µn, µ)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4)  satisfies a compact law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  For strongly mixing sequences in the sense of Rosenblatt [13], we proved in [6] the central limit  theorem for √nW1(µn, µ) under the condition  � ∞  0  �  �  �  �  ∞  �  k=0  (α(k) ∧ H(t)) dt < ∞  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5)  (where a ∧ b means the minimum between two reals a and b), and we give sufficient conditions  for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Note that, in [6], we used a weaker version of the α-mixing coefficients, that  enables to deal with a large class of non-mixing processes in the sense of Rosenblatt [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  In Section 2 of this paper, we prove a strong invariance principle for W1(µn, µ) under the  condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' The compact law of the iterated logarithm for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4) easily follows from this strong  invariance principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' In Section 3, we apply our general result to derive the almost sure rate of  convergence of the empirical estimator of the Conditional Value at Risk (CV aR) for stationary  α-mixing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  In the rest of the paper, we shall use the following notation: for two sequences (an)n≥1 and  (bn)n≥1 of positive reals, an ≪ bn means there exists a positive constant C not depending on n  such that an ≤ Cbn for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  2  Main result  Our main result is the following strong invariance principle for W1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Then, enlarging the probability space if necessary,  there exists a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with  covariance function defined as follows: for any f, g ∈ L∞(dt),  Γ(f, g) = Cov  ��  f(t)Z1(t) dt,  �  g(t)Z1(t) dt  �  =  �  k∈Z  ��  f(t)g(s)Cov(1X0≤t, 1Xk≤s) ds dt ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1)  and such that  nW1(µn, µ) −  � �����  n  �  k=1  Zk(t)  ����� dt = o(  �  n log log n)  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  2  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' In [4], Cuny proved a strong invariance principle for W1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' under the condition  ∞  �  k=0  1  √  k + 1  � ∞  0  �  α(k) ∧ H(t) dt < ∞  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2)  (in fact, he proved the result for a weaker version of the α-mixing coefficient, the same as that  used in [6] for the central limit theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' It follows from Section 5 of [6], that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5)  is always less restrictive than (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  As a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1, we get the compact law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Let K  be the unit ball of the reproducing kernel Hilbert space (RKHS) associated with Γ, and C be the  image of K by the L1(dt) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' The following corollary holds:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Then the sequence  √n  √2 log log nW1(µn, µ)  is almost surely relatively compact, with limit set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1 is based on two ingredients: a martingale approximation in L1(dt),  as in [6], and the following version of the bounded law of the iterated logarithm, which has an  interest in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) holds, and let  V =  � ∞  0  �  �  �  �  ∞  �  k=0  (α(k) ∧ H(t)) dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3)  Then, there exists a universal constant η such that for any ε > 0,  �  n≥2  1  nP  �  max  1≤k≤n kW1(µk, µ) > (ηV + ε)  �  n log log n  �  < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' (The bivariate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Let (Xi, Yi)i∈Z be a stationary sequence of R2-valued random  variables, and define the coefficients α(k) as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1), with the two σ-algebras F0 = σ(Xi, Yi, i ≤ 0)  and Gk = σ(Xi, Yi, i ≥ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Let µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' µY ) be the common distribution of the Xi’s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' the  Yi’s), and let  µn,X = 1  n  n  �  k=1  δXk  and  µn,Y = 1  n  n  �  k=1  δYk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Combining the arguments in [3] and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1, one can prove the following strong  invariance principle for n (W1(µn,X, µn,Y ) − W1(µX, µY )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Let ϕ be the continuous function from L1(dt) to R defined by  ϕ(x) =  � �  sign{FX(t) − FY (t)} x(t)1FX(t)̸=FY (t) + |x(t)|1FX(t)=FY (t)  �  dt ,  where FX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' FY ) is the cumulative distribution function of µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' µY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Assume that  � ∞  0  �  �  �  �  ∞  �  k=0  (α(k) ∧ HX(t)) dt < ∞  and  � ∞  0  �  �  �  �  ∞  �  k=0  (α(k) ∧ HY (t)) dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Then, enlarging the probability space if necessary, there exists a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' L1(dt)-valued  centered Gaussian random variables (Zi)i≥1 with covariance function given by: for any f, g ∈  L∞(dt),  �Γ(f, g) = Cov  ��  f(t)Z1(t) dt,  �  g(t)Z1(t) dt  �  =  �  k∈Z  ��  f(t)g(s)Cov(1X0≤t − 1Y0≤t, 1Xk≤s − 1Yk≤s) ds dt ,  3  and such that  n (W1(µn,X, µn,Y ) − W1(µX, µY )) − ϕ  � n  �  k=1  Zk  �  = o(  �  n log log n)  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  3  Rates of convergence of the empirical estimator of  the Conditional Value at Risk  The Conditional Value at Risk at level u ∈ (0, 1] of a real-valued integrable random variable X  (CV aRu(X)) is a “risk measure” (according to the definition of Acerbi and Tasche [1]), which is  widely used in mathematical finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' It is sometimes called Expected Shortfall of Average Value  at Risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' We refer to the paper [1] for a clear definition of that indicator, and for its relation with  other well known measures, such as the Value at Risk, the Worst Conditional Expectation, the  Tail Conditional Expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' According to Acerbi and Tasche [1], CV aRu(X) can be expressed  as  CV aRu(X) = − 1  u  � u  0  F −1  X (x)dx ,  where FX is the cumulative distribution function of the variable X, and F −1  X  is its usual cadlag  inverse: F −1  X (u) = inf{x ∈ R : FX(x) ≥ u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Concerning the difference between the Conditional Value at Risk of two random variables X  and Y , the following elementary inequality holds (see for instance [12]):  |CV aRu(X) − CV aRu(Y )| ≤ 1  u  � 1  0  |F −1  X (x) − F −1  Y (x)|dx = 1  uW1(µX, µY ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1)  where µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' µY ) is the distribution of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Consider now the problem of estimating CV aRu(X) from the random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=', Xn,  where (Xi)i∈Z is a stationary sequence of α-mixing random variables with common distribution  µ = µX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' A natural estimator is then  �  CV aRu,n = − 1  u  � u  0  F −1  n (x)dx ,  where Fn is the empirical distribution function based on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1), we get the upper  bound  ���CV aRu(X) − �  CV aRu,n  ��� ≤ 1  u  � 1  0  |F −1  X (x) − F −1  n (x)|dx = 1  uW1(µn, µ) ,  From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1, we obtain the almost sure rate of convergence of �  CV aRu,n: if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) holds,  then  lim sup  n→∞  √n  √2 log log n  ���CV aRu(X) − �  CV aRu,n  ��� ≤ κ(Γ)  u  almost surely,  where κ(Γ) is the largest value of the compact set C of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1 (recall that the covariance  function Γ is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' It is well known (see for instance Section 8 in [8]) that the constant  κ(Γ) can be expressed as  κ(Γ) =  sup  f:∥f∥∞≤1  �  Var  ��  f(t)Z(t)dt  ��1/2  ≤  ����  �  |Z(t)|dt  ����  2  ,  where Z is an L1(dt)-valued centered random variable with covariance function Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  4  4  Proofs  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1  Let (Ω, A, P) be the underlying probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  By a standard argument, one may assume  that Xi = X0 ◦ T , where T : Ω �→ Ω is a bijective, bi-measurable transformation, preserving the  probability P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Let also Fi = σ(Xk, k ≤ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Let Y0(t) = 1X0≤t − F(t), and Yk(t) = Y0(t) ◦ T k = 1Xk≤t − F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' With these notations and  the representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2) one has that  nW1(µn, µ) =  � �����  n  �  k=1  Yk(t)  ����� dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1)  From Section 4 in [6], we know that, if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) holds, then  Y0(t) = D0(t) + A(t) − A(t) ◦ T,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2)  where D0 is such that E(D1(t)|F−1) = 0 almost surely and  �  ∥D0(t)∥2 dt < ∞, and A is such that  �  ∥A(t)∥1 dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Moreover, the covariance operator of D0 is exactly Γ: for any f, g ∈ L∞(dt),  Γ(f, g) = Cov  ��  f(t)D0(t) dt,  �  g(t)D0(t) dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3)  Let Dk(t) = D0 ◦ T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2), it follows that  n  �  k=1  Yk =  n  �  k=1  Dk + A ◦ T − A ◦ T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4)  From [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3], we know that, enlarging the probability space if necessary, there exists  a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance  function Γ such that  � �����  n  �  k=1  Dk(t) −  n  �  k=1  Zk(t)  ����� dt = o  ��  n log log n  �  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5)  Hence, the result will follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) if we can prove that  lim  n→∞  1  √n log log n  �  |A(t) ◦ T n| dt = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='6)  To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='6), we start by considering the integral over [−M, M]c, for M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Applying again  [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3], we infer that  lim sup  n→∞  1  √2n log log n  �  [−M,M]c  �����  n  �  k=1  Dk(t)  ����� dt ≤  �  [−M,M]c ∥D0(t)∥2 dt  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='7)  Now, as will be clear from the proof, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1 also holds on the space L1([−M, M]c, dt),  and implies that there exists a universal constant η such that, for any positive ε,  lim sup  n→∞  1  √n log log n  �  [−M,M]c  �����  n  �  k=1  Yk(t)  ����� dt ≤ ε+η  � ∞  M  �  �  �  �  ∞  �  k=0  min {α(k), H(t)} dt  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='8)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='8), we infer that  lim  M→∞ lim sup  n→∞  1  √n log log n  �  [−M,M]c |A(t) ◦ T n| dt = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  5  Hence the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='6) will be complete if we prove that, for any M > 0,  lim sup  n→∞  1  √n log log n  � M  −M  |A(t) ◦ T n| dt = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='9)  To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='9), we work in the space H = L2([−M, M], dt), and we denote by ∥ · ∥H and ⟨·, ·⟩  the usual norm and scalar product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Since E(∥D0∥2  H) < ∞, we know from [4] that �n  k=1 Dk  satisfies the compact law of the iterated logarithm in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Since �  k≥0 α(k) < ∞ and Y0 is bounded  in H, we infer from [5] that �n  k=1 Yk satisfies also the compact law of the iterated logarithm in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Now, arguing exactly as in the end of the proof of [5, Theorem 4], one has: for any f in H  lim  n→∞  ⟨f, A ◦ T n⟩  √n log log n = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='10)  Let (ei)i≥1 be a complete orthonormal basis of H and PN(f) = �N  k=1⟨f, ek⟩ek be the projection  of f on the space spanned by the first N elements of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='10), we get that  lim  n→∞  PN(A ◦ T n)  √n log log n = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='11)  On another hand, applying again [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='3] (as done in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='7)), we get  lim  N→∞ lim sup  n→∞  1  √n log log n  �����(I − PN)  � n  �  k=1  Dk  ������  H  = 0  almost surely,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='12)  and applying [5, Theorem 4],  lim  N→∞ lim sup  n→∞  1  √n log log n  �����(I − PN)  � n  �  k=1  Yk  ������  H  = 0  almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='13)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='13), we infer that  lim  N→∞ lim sup  n→∞  ∥(I − PN)A ◦ T n∥H  √n log log n  = 0  almost surely,  which, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='11), implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' ⋄  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1  For any n ∈ N, let us introduce the following notations:  R(u) = min{q ∈ N∗ : α(q) ≤ u}Q(u)  and  R−1(x) = inf{u ∈ [0, 1] : R(u) ≤ x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  For a positive real a that will be specified later, let  mn = a  �  n  log log n ,  vn = R−1(mn) ,  Mn = Q(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='14)  For any M > 0, let gM(y) = (y ∧ M) ∨ (−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' For any integer i, define  X′  i = gMn(Xi) and X′′  i = Xi − X′  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='15)  We first recall that, by the dual expression of W1(µn, µ),  nW1(µn, µ) = sup  f∈Λ1  n  �  i=1  (f(Xi) − E(f(Xi))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  6  where Λ1 is the set of Lipschitz functions such that |f(x) − f(y)| ≤ |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Hence,  nW1(µn, µ) ≤ sup  f∈Λ1  n  �  i=1  �  f(X′  i) − E(f(X′  i))  �  + sup  f∈Λ1  n  �  i=1  �  f(Xi) − f(X′  i) − E(f(Xi) − f(X′  i))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Therefore, setting,  F ′  n(t) = 1  n  n  �  k=1  1{X′  k≤t}  and  F ′(t) = P(X′  1 ≤ t) ,  and noticing that  k∥F ′  k − F ′∥1 = sup  f∈Λ1  k  �  i=1  �  f(X′  i) − E(f(X′  i)  �  ,  we get  max  1≤k≤n kW1(µk, µ) ≤ max  1≤k≤n k∥F ′  k − F ′∥1 +  n  �  i=1  (|X′′  i | + E(|X′′  i |) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='16)  Now, note that  �  n≥2  1  √n log log nE(|X′′  n|) ≤  �  n≥2  1  √n log log n  � +∞  0  P  �  |X0|1|X0|>Q(vn) > t  �  dt  ≤  �  n≥2  1  √n log log n  � +∞  Q(vn)  H(t)dt ≤  �  n≥2  1  √n log log n  � vn  0  Q(u)du  ≤  �  n≥2  1  √n log log n  � 1  0  Q(u)1mn≤R(u)du ≪  � 1  0  R(u)Q(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  But, according to Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2 in [6], condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='5) implies that  � 1  0  R(u)Q(u)du < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='17)  Hence, to prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4) it suffices to show that there exists an universal constant η such that for  any ε > 0,  �  n≥2  1  nP  �  max  1≤k≤n k∥F ′  k − F ′∥1 > ηV  �  n log log n  �  < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='18)  For this purpose, let  qn = min{k ∈ N∗ : α(k) ≤ vn} ∧ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='19)  Since R is right continuous, we have R(R−1(w)) ≤ w for any w, hence  qnMn = R(vn) = R(R−1(mn)) ≤ mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='20)  Assume first that qn = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Bounding f(X′  i) − E(f(X′  i)) by 2Mn, we obtain  max  1≤k≤n k∥F ′  k − F ′∥1 ≤ 2nMn = 2qnMn ≤ 2mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='21)  Taking into account the definition of mn, it follows that there exists n0 depending on a, V and η,  such that for any n ≥ n0, 8mn ≤ κV √n log log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' This proves the proposition in the case where  qn = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  From now on, we assume that qn < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Therefore qn = min{k ∈ N∗ : α(k) ≤ vn} and then  α(qn) ≤ vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' For any integer i, define  Ui(t) =  iqn  �  k=(i−1)qn+1  �  1X′  k≤t − E  �  1X′  k≤t  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  7  and notice that  max  1≤k≤n k∥F ′  k − F ′∥1 ≤ 2qnMn +  � Mn  −Mn  max  1≤j≤[n/qn]  �����  j  �  i=1  Ui(t)  ����� dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Let kn = [n/qn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' For any t, applying Rio’s coupling lemma (see [11, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='2]) recursively,  we can construct random variables (U ∗  i (t))1≤i≤kn such that  U ∗  i (t) has the same distribution as U ′  i for all 1 ≤ i ≤ kn,  the random variables (U ∗  2i(t))2≤2i≤kn are independent, as well as the random variables  (U ∗  2i−1(t))1≤2i−1≤kn,  we can suitably control ∥Ui(t) − U ∗  i (t)∥1 as follows: for any i ≥ 1,  ∥Ui(t) − U ∗  i (t)∥1 ≤ 4qnα(qn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='22)  Substituting U ∗  i (t) to Ui(t), we obtain  max  1≤k≤n k∥F ′  k − F ′∥1 ≤ 2qnMn +  max  2≤2j≤[n/qn]  �����  j  �  i=1  U ∗  2i(t)  �����  +  max  1≤2j−1≤[n/qn]  �����  j  �  i=1  U ∗  2i−1(t)  �����+  [n/qn]  �  i=1  |Ui(t) − U ∗  i (t)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='23)  Therefore, setting κ = η/4, for n ≥ n0,  P  �  max  1≤k≤n k∥F ′  k − F ′∥1 ≥ 4V κ  �  n log log n  �  ≤ I1(n) + I2(n) + I3(n) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='24)  where  I1(n) = P  \uf8eb  \uf8ed  � Mn  −Mn  [n/qn]  �  i=1  |Ui(t) − U ∗  i (t)| dt ≥ V κ  �  n log log n  \uf8f6  \uf8f8  I2(n) = P  �� Mn  −Mn  max  2≤2j≤[n/qn]  �����  j  �  i=1  U ∗  2i(t)  ����� dt ≥ V κ  �  n log log n  �  I3(n) = P  �� Mn  −Mn  max  1≤2j−1≤[n/qn]  �����  j  �  i=1  U ∗  2i−1(t)  ����� dt ≥ V κ  �  n log log n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Using Markov’s inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='22), we get  I1(n) ≪  n  √n log log nMnα(qn) ≪  n  √n log log nvnQ(vn) ≪  n  √n log log n  � R−1(mn)  0  Q(u)du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='17),  �  n≥2  1  nI1(n) ≪  �  n≥2  1  √n log log n  � R−1(mn)  0  Q(u)du ≪  � 1  0  R(u)Q(u)du < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  To handle now the term I2(n) (as well as I3(n)) in the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='24), we shall use  again Markov’s inequality but this time at the order p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Hence for p ≥ 2, taking into account  the stationarity, we get  I2(n) ≤  1  (V κ)p(n log log n)p/2  \uf8eb  \uf8ed  � Q(vn)  −Q(vn)  �����  max  2≤2j≤[n/qn]  �����  j  �  i=1  ˜U2i(t)  �����  �����  p  dt  \uf8f6  \uf8f8  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  8  Applying Rosenthal’s inequality (see for instance [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='1]) and taking into account the  stationarity, there exist two positive universal constants c1 and c2 not depending on p such that  �����  max  2≤2j≤[n/qn]  �����  j  �  i=1  U ∗  2i(t)  �����  �����  p  p  ≤ cp  1pp/2(n/qn)p/2∥U2(t)∥p  2 + cp  2pp(n/qn)∥U2(t)∥p  p := J1(t) + J2(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='25)  Using similar arguments as to handle the quantity I2(n) in the proof of [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='4], we  have  � Q(vn)  −Q(vn)  ∥U2(t)∥2dt =  � Q(vn)  −Q(vn)  �  Var  � qn  �  i=1  1{X′  i≤t}  ��1/2  dt  ≤ 2  √  2√qn  � Q(vn)  0  �qn−1  �  k=0  α(k) ∧ H(t)  �1/2  dt ≤ 2V  �  2qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='26)  Hence  �  n≥2  1  n(V κ)p(n log log n)p/2  �� Q(vn)  −Q(vn)  J1(t)1/pdt  �p  ≤  �  n≥2  (2  √  2c1√p)p  nκp(log log n)p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Let now  p = pn = max{c log log n, 2},  where c will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Set n1 = min{n ≥ 2 : c log log n ≥ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' It follows that  �  n≥n1  1  n(V κ)p(n log log n)p/2  �� Q(vn)  −Q(vn)  J1(t)1/pdt  �p  ≤  �  n≥n1  1  n  �2c1  √  2c  κ  �c log log n  ,  which is finite provided we take κ such that 2c1  √  2c  κ  = α−1 with α > 1 and c > (log α)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  On another hand, proceeding as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='26), we deduce that, for any t > 0,  ∥U2(t)∥p  p =  �����  qn  �  i=1  �  1{X′  i≤t} − P(X′  i ≤ t)  ������  p  p  ≤ qp−2  n  �����  qn  �  i=1  �  1{X′  i≤t} − P(X′  i ≤ t)  ������  2  2  ≤ 2qp−1  n  qn−1  �  k=0  (α(k) ∧ H(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  In addition  � Q(vn)  0  �qn−1  �  k=0  α(k) ∧ H(t)  �1/p  dt =  � Q(vn)  0  �� H(t)  0  (α−1(u) ∧ qn)du  �1/p  dt  ≤  � Q(vn)  0  �  vnqn +  � H(t)  vn  (α−1(u) ∧ qn)du  �1/p  dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Note that u < H(t) ⇐⇒ t < Q(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Consequently u < H(t) implies that Q−2(u) < t−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Hence  � Q(vn)  0  �qn−1  �  k=0  α(k) ∧ H(t)  �1/p  dt  ≤ (vnqn)1/pQ(vn) +  � Q(vn)  0  �  t−2  � H(t)  vn  (α−1(u) ∧ qn)Q2(u)du  �1/p  ≤ (vnqn)1/pQ(vn) +  �� 1  vn  R(u)Q(u)du  �1/p � Q(vn)  0  t−2/pdt  ≤ (vnqn)1/pQ(vn) +  �� 1  0  R(u)Q(u)du  �1/p  p(p − 2)−1Q(vn)1−2/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  9  Set n2 = min{n ≥ 2 : c log log n ≥ 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' It follows that  �  n≥n2  1  n(V κ)p(n log log n)p/2  �� Q(vn)  −Q(vn)  J2(t)1/pdt  �p  ≤ 2  �  n≥n2  (4c2p)p  (κV )p(n log log n)p/2 qp−2  n  �  vnqnQp(vn) + 2pQ(vn)p−2  � 1  0  R(u)Q(u)du  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Note that  vnqnQ2(vn) = vnα−1(vn)Q2(vn) ≤  � 1  0  R(u)Q(u)du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  Hence, since qnMn ≤ mn, we get  �  n≥n2  1  n(V κ)p(n log log n)p/2  �� Q(vn)  −Q(vn)  J2(t)1/pdt  �p  ≤ 4  � 1  0  R(u)Q(u)du  �  n≥n2  (8c2p)p  (κV )p(n log log n)p/2 mp−2  n  ≤ 4a−2  � 1  0  R(u)Q(u)du  �  n≥n2  �8ac2c  κV  �p log log n  n  ,  which is finite by taking into account (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='17), and if we choose a = (c1κV )/(2c2  √  2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Indeed, in  this case,  8ac2c  κV  = 2c1  √  2c  κ  × 2ac2  √  2c  c1κV  = α−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' ⋄  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Acerbi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Tasche (2002), On the coherence of Expected Shortfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' del Barrio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Gin´e and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' Dedecker, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' (2020) https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' Merlev`ede (2010), On the almost sure invariance principle for stationary  sequences of Hilbert-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Dependence in probability, analysis and number  theory, 157–175, Kendrick Press, Heber City, UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' Merlev`ede (2017), Behavior of the Wasserstein distance between the empir-  ical and the marginal distributions of stationary α-dependent sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Bernoulli 23 2083–  2127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Jain (1977), Central limit theorems and related questions in Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Proceedings  of Symposium in Pure and Applied Mathematics 31 55-65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Providence, RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
+page_content=' Ledoux and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'}
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+Quality Indicators for Preference-based Evolutionary
+Multi-objective Optimization Using a Reference Point:
+A Review and Analysis∗
+Ryoji Tanabe1 and Ke Li2
+1Faculty of Environment and Information Sciences, Yokohama National University, Yokohama, Japan
+2Department of Computer Science, University of Exeter, EX4 4QF, Exeter, UK
+∗Email: [email protected], [email protected]
+Abstract:
+Some quality indicators have been proposed for benchmarking preference-based evolu-
+tionary multi-objective optimization algorithms using a reference point. Although a systematic review
+and analysis of the quality indicators are helpful for both benchmarking and practical decision-making,
+neither has been conducted. In this context, first, this paper reviews existing regions of interest and
+quality indicators for preference-based evolutionary multi-objective optimization using the reference
+point. We point out that each quality indicator was designed for a different region of interest. Then,
+this paper investigates the properties of the quality indicators. We demonstrate that an achievement
+scalarizing function value is not always consistent with the distance from a solution to the reference
+point in the objective space. We observe that the regions of interest can be significantly different
+depending on the position of the reference point and the shape of the Pareto front. We identify un-
+desirable properties of some quality indicators. We also show that the ranking of preference-based
+evolutionary multi-objective optimization algorithms significantly depends on the choice of quality
+indicators.
+Keywords:
+Preference-based evolutionary multi-objective optimization, quality indicators,
+benchmarking
+1
+Introduction
+The ultimate goal of multi-objective optimization is to facilitate multi-criterion decision-making
+(MCDM) that finds the Pareto-optimal solution(s) satisfying the decision maker’s aspirations [1].
+Partially due to the population-based property, evolutionary algorithms (EAs) have been widely rec-
+ognized as an effective approach for multi-objective optimization, as known as evolutionary multi-
+objective optimization (EMO). Conventional EMO algorithms, such as NSGA-II [2], IBEA [3], and
+MOEA/D [4], are designed to search for a set of trade-off alternatives that approximate the Pareto-
+optimal front (PF) without considering any preference information [5]. Thereafter, this solution set
+is handed over to the decision maker (DM) for an a posteriori MCDM to choose the solution(s) of
+interest (SOI). On the other hand, if the DM’s preference information is available a priori, it can be
+used to navigate an EMO algorithm, also known as preference-based EMO (PBEMO) algorithm [6–8],
+to search for a set of “preferred” trade-off solutions lying in a region of interest (ROI), i.e., a subregion
+of the PF specified according to the DM’s preference information [9]. From the perspective of EMO,
+approximating an ROI can be relatively easier than approximating the complete PF, especially when
+having many objectives. From the perspective of MCDM, using the preference information can reduce
+the DM’s workload since she/he is only asked to investigate her/his potentially preferred solutions.
+Reference point, also known as an aspiration level vector [10], which consists of desirable objective
+values specified by the DM, is one of the most popular approaches for expressing the preference in-
+formation in the EMO literature [11,12]. Comparing to the other preference formats [7,8], specifying
+∗This manuscript is submitted for potential publication. Reviewers can use this version in peer review.
+1
+arXiv:2301.12148v1  [cs.NE]  28 Jan 2023
+
+a reference point is relatively more intuitive and easier for the DM to elicit her/his preference infor-
+mation. Many conventional EMO algorithms have been extended to PBEMO using a reference point,
+such as R-NSGA-II [13], PBEA [14], and MOEA/D-NUMS [15].
+Besides algorithm development, in the EMO literature, quality indicators play a vital role in
+quantitatively benchmarking EMO algorithms for approximating the whole PF [16–18]. Representative
+quality indicators are the hypervolume (HV) [19], the additive ϵ-indicator (Iϵ+) [17], the generational
+distance (GD) [20], and the inverted GD (IGD) [21]. It is worth noting that none of these quality
+indicators take any preference information into account in quality assessment. Thus, they are not
+suitable for evaluating the performance of PBEMO algorithms for approximating the ROI(s).
+In
+fact, quality assessment on PBEMO algorithms have not received significant attention in the EMO
+community until [22]. Early studies mainly relied on visual comparisons which are neither reliable nor
+scalable to many objectives [13,23]. On the other hand, some studies around 2010 (e.g., [24,25]) directly
+applied conventional quality indicators thus are likely to lead to some misleading conclusions [26]. To
+the best of our knowledge, the first quality indicator for PBEMO was proposed in [22]. Although it
+has several technical flaws, this quality indicator had a significant impact on the quality assessment
+for PBEMO as discussed in [26] and [27].
+Motivation for a review. Although there have been a number of preference-based quality indicators
+proposed since [22] in 2010, there is no systematic survey along this line of research. Some survey
+papers on quality indicators [16–18] are available, but they are hardly about preference-based ones.
+Afsar et al. [12] conducted a survey on how to evaluate the performance of interactive preference-
+based multi-objective optimizers, but they focused on experimental conditions rather than quality
+indicators. Bechikh et al. [8] presented an exhaustive review of PBEMO algorithms yet on quality
+indicators. In addition, some previous studies implicitly proposed quality indicators. For example,
+Ruiz et al. [9] proposed WASF-GA. In [9], they also designed a new quality indicator called HVz to
+evaluate the performance of WASF-GA. However, they did not clearly state that the design of HVz
+was their contribution. For this reason, most previous studies on preference-based quality indicators
+(e.g., [26,28,29]) overlooked HVz.
+Motivation for analysis.
+The properties of quality indicators are not obvious, including which
+point set a quality indicator prefers and which quality indicators are consistent/inconsistent with
+each other. Thus, it is likely to incorrectly evaluate the performance of EMO algorithms when using a
+particular quality indicator. To address this issue, some previous studies analyzed quality indicators in
+various ways [30]. Nevertheless, little is known about the properties of quality indicators for PBEMO.
+Although some previous studies (e.g., [29,31]) analyzed a few quality indicators for PBEMO, the scale
+of their experiments is relatively small.
+Apart from this issue of quality indicators, Li et. al. [11] reported the pathological behavior of
+some PBEMO algorithms when setting the reference point far from the PF. They showed that R-
+NSGA-II [13], r-NSGA-II [24], and R-MEAD2 [32] unexpectedly obtain points on the edge of the
+PF, which are far from the reference point. They also showed that only MOEA/D-NUMS [15] works
+expectedly even in this case. Since the DM does not know any information about the PF in real-world
+applications, these undesirable behavior can be observed in practice. However, the previous study [11]
+could not determine what caused these undesirable behavior.
+Contributions.
+Motivated by the above discussion, first, we review ROIs and preference-based
+quality indicators proposed in the literature. We clarify the quality indicators based on their target
+ROIs. Then, we analyze the quality indicators. Through an analysis, we address the following four
+research questions:
+RQ1: Does a Pareto-optimal point with the minimum achievement scalarizing function (ASF) value
+always minimize the distance from the reference point?
+RQ2: What are the differences of the definitions of ROIs considered in previous studies? How do these
+differences influence the behavior of EMO algorithms?
+RQ3: What are the properties of existing quality indicators for PBEMO?
+RQ4: How does the choice of quality indicator affect the ranking of PBEMO algorithms?
+2
+
+Outline. The rest of this paper is organized as follows. Section 2 provides some preliminary knowledge
+pertinent to this paper. Section 3 reviews and analyzes three ROIs considered in previous studies.
+Section 4 reviews 14 preference-based quality indicators developed in the literature. Our experimental
+settings are provided in Section 5 while the results are analyzed in Section 6. Section 7 concludes this
+paper.
+Supplementary file. This paper has a supplementary file. Figure S.∗ and Table S.∗ indicate a figure
+and a table in the supplementary file, respectively.
+Code availability. The Python implementation of all preference-based quality indicators investigated
+in this work is available at https://github.com/ryojitanabe/prefqi.
+2
+Preliminaries
+2.1
+Multi-objective optimization
+The multi-objective optimization problem (MOP) considered in this paper is formulated as:
+minimize
+F(x) = (f1(x), . . . , fm(x))⊤
+subject to
+x ∈ Ω
+,
+(1)
+where x = (x1, . . . , xn)⊤ is an n-dimensional decision vector, and F(x) is an m-dimensional objective
+vector. Ω is the feasible set in the decision space Rn and F : Ω → Rm is the corresponding attainable set
+in the objective space Rm. A solution x1 is said to Pareto dominate x2 if and only if fi(x1) ≤ fi(x2) for
+all i ∈ {1, . . . , m} and fi(x1) < fi(x2) for at least one index i. We denote x1 ≺ x2 when x1 dominates
+x2. In addition, x1 is said to weakly Pareto dominate x2 if fi(x1) ≤ fi(x2) for all i ∈ {1, . . . , m}. A
+solution x∗ is a Pareto-optimal solution if x∗ is not dominated by any solution in Ω. The set of all
+Pareto-optimal solutions in Ω is called the Pareto-optimal set (PS) X ∗ = {x∗ ∈ Ω | ∄x ∈ Ωs.t.x ≺ x∗}.
+The image of the PS in Rm is also called the PF F = F(X ∗). The ideal point pideal ∈ Rm consists
+of the minimum values of the PF for m objective functions. The nadir point pnadir ∈ Rm consists
+of the maximum values of the PF for m objective functions. Thus, for each i ∈ {1, . . . , m}, pideal
+i
+=
+minx∈X ∗{fi(x)} and pnadir
+i
+= maxx∈X ∗{fi(x)}. For the sake of simplicity, we refer F(x) as a point
+p = (p1, . . . , pm)⊤ ∈ Rm in the rest of this paper.
+2.2
+Quality indicators
+A quality indicator is a metric I : Rm → R, I : P �→ I(P) that quantitatively evaluates the quality
+of a point set P = {pi}µ
+i=1 of size µ in terms of at least one of the following four aspects [18]: i)
+convergence: the closeness of the points in P to the PF; ii) uniformity: the distribution of the points
+in P; iii) spread: the range of the points in P along the PF; and iv) cardinality: the number of non-
+dominated points in P. Note that the cardinality has not received much attention in multi-objective
+numerical optimization. As discussed in [33], a quality indicator I is said to be Pareto-compliant if
+I(P1) < I(P2)1 for any pair of point sets P1 and P2 in Rm, where ∃p ∈ P1, ∀˜p ∈ P2 we have p ≺ ˜p.
+Given K > 1 point sets, a unary quality indicator evaluates each one exclusively whereas a K-nary
+quality indicator evaluates the K point sets relatively. As discussed in [17] and [18], both unary and
+K-nary quality indicators have pros and cons. For example, K-nary quality indicators generally do
+not require information about the PF. This is attractive for real-world problems with unknown PFs.
+However, K-nary quality indicators only provide information about the relative quality of the K point
+sets. That is to say we have to re-calculate the quality indicator values of the K + 1 point sets when
+comparing a new point set to the previous K point sets. This might be disadvantageous from the
+perspective of sustainable benchmarking of EMO algorithms.
+Below, we describe two representative quality indicators widely used in the EMO community.
+1In this case, we assume the quality indicator is to be minimized. Otherwise, we have I(P1) > I(P2) instead.
+3
+
+2.2.1
+Hypervolume (HV) [19]
+It measures the volume of the region dominated by the points in P and bounded by the HV-reference
+point y ∈ Rm:
+HV(P) = Λ
+� �
+p∈P
+{q ∈ Rm | p ≺ q ≺ y}
+�
+,
+(2)
+Λ(·) in (2) is the Lebesgue measure. HV(P) can evaluate the quality of P in terms of both convergence
+and diversity. HV is to be maximized.
+2.2.2
+Inverted generational distance (IGD) [21]
+Let S be a set of IGD-reference points uniformly distributed on the PF, IGD measures the average
+distance between each IGD-reference point s ∈ S and its nearest point p ∈ P:
+IGD(P) = 1
+|S|
+��
+s∈S
+min
+p∈P
+�
+dist(s, p)
+��
+,
+(3)
+where dist(·, ·) returns the Euclidean distance between two inputs. IGD in (3) is to be minimized.
+In general, IGD-reference points in S are uniformly distributed on the PF. Like HV, IGD can also
+measure the convergence and diversity of P while it prefers a uniform distribution of points [30].
+Remark 1. The term reference point has been used in various contexts in the EMO literature. To
+avoid confusion, we use the term HV-reference point to indicate the reference point for HV. Similarly,
+we use the term IGD-reference point to indicate a reference point for IGD.
+Remark 2. Note that Pareto-compliant is an important, yet hardly met, characteristic of a quality
+indicator.
+To the best of our knowledge, HV is the only Pareto-compliant indicator in the EMO
+community. This partially explains that HV has been one of the most popular quality indicators.
+2.3
+Achievement scalarizing function
+Wierzbicki [10] proposed the ASF s : Rm → R, p �→ s(p) in the context of MCDM. Although a number
+of scalarizing functions have been proposed for preference-based multi-objective optimization [34], the
+ASF is one of the most popular scalarizing functions. Previous studies on PBEMO (e.g., [9, 14, 26])
+used the following two variants of the ASF:
+s(p) =
+max
+i∈{1,...,m}
+pi − zi
+wi
+,
+(4)
+s(p) =
+max
+i∈{1,...,m} wi(pi − zi),
+(5)
+where z ∈ Rm is the reference point specified by the DM. In (4) and (5), w = (w1, . . . , wm)⊤ is the
+weight vector that represents the relative importance of each objective function, where �m
+i=1 wi = 1
+and wi ≥ 0 for any i. Like in most previous studies, we set w to (1/m, . . . , 1/m)⊤ throughout this
+paper. The ASF is order-preserving in terms of the Pareto dominance relation [10], i.e., s(p1) < s(p2)
+if p1 ≺ p2. A point with the minimum ASF value is also weakly Pareto optimal with respect to z and
+w.
+Only the Pareto-optimal point with respect to z and w can be obtained by minimizing the following
+augmented version of the ASF (AASF) [34]:
+saug(p) = s(p) + ρ
+m
+�
+i=1
+(pi − zi),
+(6)
+where s in (6) can be either one of the ASFs in (4) and (5). In (6), ρ is a small positive value, e.g,
+ρ = 10−6.
+4
+
+2.4
+PBEMO algorithms
+To be self-contained, we give a briefing of six representative PBEMO algorithms considered in our
+experiments2: R-NSGA-II [13], r-NSGA-II [24], g-NSGA-II [23], PBEA [14], R-MEAD2 [32], and
+MOEA/D-NUMS [15]. As their names suggest, R-NSGA-II, r-NSGA-II, and g-NSGA-II are extended
+versions of NSGA-II for preference-based multi-objective optimization. PBEA is a variant of IBEA
+while RMEAD2 and MOEA/D-NUMS are scalarizing function-based approaches. Although R-NSGA-
+II, r-NSGA-II, and PBEA can handle multiple reference points, we only introduce the case when using
+a single reference point. As in [11], we focus on preference-based multi-objective optimization using
+a single reference point as the first step. Below, we use the terms “point set” and “population” syn-
+onymously. We use the term “preferred region” to describe a sub-region of the PF approximated by a
+PBEMO algorithm in the best case. While the ROI is defined by the DM, the preferred region depends
+on the PBEMO algorithm. Although some previous studies used these two regions interchangeably,
+we strictly distinguish them.
+2.4.1
+R-NSGA-II
+As in NSGA-II, the primary criterion in environmental selection in R-NSGA-II is based on the non-
+domination level of each point p. While the secondary criterion in NSGA-II is based on the crowding
+distance, that of R-NSGA-II is based on the following weighted distance to the reference point z:
+dR(p) =
+�
+�
+�
+�
+m
+�
+i=1
+wi
+�
+pi − zi
+pmax
+i
+− pmin
+i
+�
+,
+(7)
+where pmax
+i
+and pmin
+i
+are the maximum and minimum values of the i-th objective function fi in the
+population P = {pi}µ
+i=1 of size µ. The weight vector w in (7) plays a similar role in w in the ASF.
+When comparing individuals in the same non-domination level, ties are broken by their dR values.
+Thus, non-dominated individuals close to z are likely to survive to the next iteration.
+In addition, R-NSGA-II performs ϵ-clearing to maintain the diversity in the population. If the
+distance between two individuals in the objective space is less than ϵ, a randomly selected one is
+removed from the population.
+2.4.2
+r-NSGA-II
+It is an extended version of NSGA-II by replacing the Pareto dominance relation with the r-dominance
+relation. For two points p1 and p2 in P, p1 is said to r-dominate p2 if one of the following two criteria
+is met: 1) p1 ≺ p2; 2) p1 ⊀ p2, p1 ⊁ p2, and dr(p1, p2) < −δ. Here, dr(p1, p2) is defined as follows:
+dr(p1, p2) =
+dR(p1) − dR(p2)
+maxp∈P{dR(p)} − minp∈P{dR(p)},
+(8)
+where the definition of dR in (8) can be found in (7). The threshold δ ∈ [0, 1] determines the spread
+of individuals in the objective space. When δ = 1, the r-dominance relation is the same as the Pareto
+dominance relation. When δ = 0, the r-dominance relation between two non-dominated points p1 and
+p2 is determined by their dR values.
+2.4.3
+g-NSGA-II
+It uses the g-dominance relation instead of the Pareto dominance relation. Let Q be the set of all points
+in Rm that dominate the reference point z or are dominated by z, i.e., Q = {p ∈ Rm | p ≺ z or p ≻ z}.
+A point p1 is said to g-dominate p2 if one of the following three criteria is met: 1) p1 ∈ Q and p2 /∈ Q;
+2) p1, p2 ∈ Q and p1 ≺ p2; 3) p1, p2 /∈ Q and p1 ≺ p2.
+Unlike other PBEMO algorithms, g-NSGA-II does not have a control parameter that adjusts the
+size of the preferred region. However, g-NSGA-II can obtain only points in a very small region when
+2Their behavior was also investigated in [11].
+5
+
+z is close to the PF [24]. In contrast, g-NSGA-II is equivalent to NSGA-II when z is very far the
+PF [11]. This is because the preferred region Q covers the whole PF when z dominates the ideal point
+or is dominated by the nadir point.
+2.4.4
+PBEA
+It is a variant of IBEA using the binary additive ϵ-indicator (Iϵ+) [17]. For a point set P, the Iϵ+
+value of a point p ∈ P to another point q ∈ P \ {p} is defined as:
+Iϵ+(p, q) =
+max
+i∈{1,...,m}{p′
+i − q′
+i},
+(9)
+where p′ and q′ in (9) are normalized versions of p and q based on the maximum and minimum
+values of P. The Iϵ+ value is the minimum objective value such that p′ dominates q′. PBEA uses the
+following preference-based indicator Ip, which takes into account the AASF value in (6):
+Ip(p, q) = Iϵ+(p, q)
+s′(p)
+,
+(10)
+s′(p) = saug(p) + δ − min
+u∈P{saug(u)},
+(11)
+where the previous study [14] used saug with s in (4). In (11), s′(p) is the normalized AASF value of
+p by the minimum AASF value of P. In (11), δ controls the extent of the preferred region. A large
+Ip value indicates that the corresponding p is preferred. As acknowledged in [14], one drawback of
+PBEA is the difficulty in determining the δ value.
+2.4.5
+R-MEAD2
+R-MEAD2 [32] is a decomposition-based EMO algorithm using a set of µ weight vectors W = {wi}µ
+i=1.
+Similar to MOEA/D [4], R-MEAD2 aims to approximate µ Pareto optimal points by simultaneously
+minimizing µ scalar optimization problems with W.
+R-MEAD2 adaptively adjusts the µ weight
+vectors so that the corresponding individuals move toward z. At the beginning of the search, R-
+MEAD2 initializes the weight vector set W randomly.
+For each iteration, R-MEAD2 selects the
+weight vector wc from W, where the corresponding point pc is closest to the reference point z, i.e.,
+pc = argmin
+p∈P
+{dist(p, z)}. Then, R-MEAD2 randomly reinitializes W in an m-dimensional hypersphere
+of radius r centered at wc.
+2.4.6
+MOEA/D-NUMS
+It is featured by a nonuniform mapping scheme (NUMS) that shifts µ uniformly distributed weight
+vectors toward the reference point z. In particular, the distribution of the µ shifted weight vectors,
+denoted as W′, is expected to be biased toward z. In NUMS, a parameter r controls the extent of
+W′. In contrast to R-MEAD2, NUMS adjusts the weight vectors in an offline manner. In theory,
+NUMS can be incorporated into any decomposition-based EMO algorithm by using W′ instead of the
+original W, while MOEA/D-NUMS proposed in [15] is built upon the vanilla MOEA/D. In addition,
+MOEA/D-NUMS uses the AASF in (6) with s in (5) instead of a general scalarizing function (e.g.,
+the Tchebycheff function).
+3
+Review of region of interests
+Conventional EMO algorithms (e.g., NSGA-II [2]) aim to find a set of µ non-dominated points that
+approximate the PF. In contrast, preference-based EMO algorithms (e.g., R-NSGA-II [13]) are de-
+signed to search for a set of µ non-dominated points that approximate the ROI. However, as pointed
+out in [15], the ROI has been loosely defined in the EMO community. According to the definition
+in [15], we define the ROI as a subset of the PF, denoted as R ⊆ F. We assume that the DM is
+interested in not only the closest Pareto-optimal point pc∗ to the reference point z but also a set of
+6
+
+Pareto-optimal points around pc∗. In some cases, the extent of R is defined by a parameter given by
+the DM.
+Below, Section 3.1 describes three ROIs addressed in previous studies. The reference point z is
+said to be feasible if it cannot dominate any Pareto-optimal point. Otherwise, it is said to be infeasible
+if z can dominate at least one Pareto-optimal point. Then, Section 3.2 discusses the three ROIs.
+3.1
+Definitions of three ROIs
+3.1.1
+ROI based on the closest point
+This might be the most intuitive ROI that consists of a set of Pareto-optimal points closest to z in
+terms of the Euclidean distance (e.g., [27] and [32]). Mathematically, it is defined as:
+ROIC =
+�
+p∗ ∈ F | dist(p∗, pc∗) < ζ
+�
+,
+(12)
+where pc∗ = argmin
+p∗∈F
+{dist(p∗, z)} is the closest Pareto-optimal point to z, and ζ is the radius of the
+ROIC. As the example shown in Fig. 1(a), the ROIC is a set of points in a hypersphere of a radius ζ
+centered at pc∗ while the extent of the ROIC depends on ζ. We believe that R-NSGA-II, r-NSGA-II,
+and R-MEAD2 were designed for the ROIC implicitly.
+3.1.2
+ROI based on the ASF
+As studied in [14, 15] and [26], this ROI consists of a set of the Pareto-optimal points closest to the
+one with the minimum ASF value. Mathematically, it is defined as:
+ROIA = {p∗ ∈ F | dist(p∗, pa∗) < ζ},
+(13)
+where pa∗ = argmin
+p∗∈F
+{s(p∗)} is the Pareto-optimal point pa∗ having the minimum ASF value, and
+s is the same as in (4). We believe that PBEA and MOEA/D-NUMS were designed for the ROIA
+implicitly.
+3.1.3
+ROI based on the Pareto dominance relation
+This ROI is defined by an extension of the Pareto dominance relation with regard to the DM specified
+reference point z (e.g., [9,35–37]). When z is feasible, the ROIP is a set of Pareto-optimal points that
+dominate z, i.e., ROIP = {p∗ ∈ F | p∗ ≺ z}. Otherwise, the ROIP is a set of Pareto-optimal points
+dominated by z, i.e., ROIP = {p∗ ∈ F | p∗ ≻ z} when z is infeasible. We believe g-NSGA-II was
+designed for the ROIP.
+3.2
+Discussions
+To have an intuitive understanding of these three ROIs, Fig. 1 shows distributions of Pareto-optimal
+points in the aforementioned ROIs on the 2-objective DTLZ2 problem with a non-convex PF. In
+particular, z0.5 = (0.5, 0.5)⊤ is used as the reference point (denoted as ▲), and we set ζ = 0.1 in the
+ROIC and ROIA. As shown in Figs. 1(a) and (b), the ROIC and ROIA are sets of the points in the
+hyper-spheres centered at pc∗ and pa∗ (denoted as �), respectively. They are equivalent if the closest
+point to z and the point with the minimum ASF value are the same. For this reason, the ROIC and
+ROIA may have been considered as the same ROI in the literature. In contrast, as shown in Fig. 1(c),
+the ROIP is a set of points dominated by z0.5 and its extent is larger than that of the ROIC and ROIA.
+However, it is worth noting that the ROIP does not have any parameter to control its extent as done
+in the ROIC and ROIA. Instead, the size of the ROIP depends on the position of z. If it is too close
+to the PF, the size of the ROIP can be very small; otherwise it can be very large if z is too far away
+from the PF. In the extreme case, if z dominates the ideal point or is dominated by the nadir point,
+the ROIP is the same as the PF. Since a DM has little knowledge of the shape of the PF a priori, it
+is not recommended to use the ROIP in real-world black-box applications.
+7
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(a) ROIC
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(b) ROIA
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(c) ROIP
+Figure 1: Distributions of Pareto-optimal points in the three ROIs on the DTLZ2 problem when using
+z0.5.
+Table 1: Properties of the 14 quality indicators for PBEMO, including the number of point sets K, the type
+of target ROI, the convergence to the PF (C-PF), the convergence to the reference point z (C-z), the diversity
+(Div), the ability to handle point sets outside a preferred region (Out), no use of information about the PF
+(U-PF), and control parameters.
+Indicators
+K
+ROI
+C-PF C-z Div Out U-PF
+Param.
+MASF [14]
+unary
+ROIA
+�
+�
+�
+�
+w
+MED [38]
+unary
+ROIC
+�
+�
+IGD-C [32]
+unary
+ROIC
+�
+�
+�
+�
+r, S
+IGD-A
+unary
+ROIA
+�
+�
+�
+�
+w, r, S
+IGD-P [36]
+unary
+ROIP
+�
+�
+�
+�
+S
+HVz [9]
+unary
+ROIP
+�
+�
+�
+�
+PR [39]
+unary
+ROIP
+�
+PMOD [28]
+unary Unclear
+�
+�
+�
+�
+r, α
+IGD-CF [27] K-nary
+ROIC
+�
+�
+�
+�
+r
+HV-CF [27]
+K-nary
+ROIC
+�
+�
+�
+�
+r, y
+PMDA [31]
+K-nary Unclear
+�
+�
+�
+�
+α, γ
+R-IGD [26]
+K-nary
+ROIA
+�
+�
+�
+�
+r, zw, w, S
+R-HV [26]
+K-nary
+ROIA
+�
+�
+�
+�
+�
+r, zw, w
+EH [29]
+K-nary Unclear
+�
+�
+�
+�
+4
+Review of quality indicators
+This section reviews 14 quality indicators proposed in the literature for assessing the performance
+of PBEMO algorithms. Their properties are summarized in Table 1. According to the definitions
+in Section 3.1, we classify the target ROIs of the 14 quality indicators based on their preferred regions.
+In particular since the target ROIs of PMOD, PMDA, and EH do not belong to any of the three
+ROIs defined in Section 3.1, their ROIs are labeled as unclear. Note that the target ROIs of R-IGD
+and R-HV are slightly different from the ROIA, where they are based on a hypercube, instead of a
+hypersphere. As shown in Table 1, the previous studies assumed different ROIs. This suggests that
+the ROI has not been standardized in the EMO community.
+In the following paragraphs, Section 4.1 first discusses the desirable properties as a quality indicator
+for PBEMO. Then, Sections 4.2 to 4.11 delineate the underlying mechanisms of 14 quality indicators,
+respectively. In particular, the technical details of some quality indicators, including iIGD [40], F-
+HV [41], the referential cluster variance indicator [42], and the hull volume indicator [42], are missing.
+In addition, the HV-based indicator developed in [22] and the spread-based indicator proposed in [43]
+do not consider the preference information from the DM. Therefore, we do not intend to elaborate
+them in this paper.
+8
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+P9
+(a) Distributions of P1 to P9
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+P10
+(b) Distribution of P10
+Figure 2: Distributions of the 10 point sets on the PF of the DTLZ2 problem when m = 2.
+4.1
+Desirable properties of quality indicators
+To facilitate our discussion, we generate 10 synthetic point sets, each of which consists of 20 uni-
+formly distributed points, along the PF of the 2-objective DTLZ2 problem as shown in Fig. 2. More
+specifically, P1 to P5 are distributed on five different subregions of the PF. P6, P7 and P8 are the
+shifted versions of P2, P3 and P4 by adding 0.1 to all elements, respectively. Thus, P6, P7, P8 are
+dominated by P2, P3, P4, respectively. P9 is on the PF, but the extent of P9 is worse than that of
+P3. Unlike the other point sets, the points in P10 are uniformly distributed on the whole PF. Given
+z = (0.5, 0.5)⊤ as the DM specified reference point as shown in Fig. 2, P3 is the best point set in Fig. 2
+with regard to the ROIC, ROIA, and ROIP. In this paper, we argue that a desirable preference-based
+quality indicator is required to assess the four aspects including i) the convergence to the PF; ii) the
+convergence to z; iii) the diversity of trade-off alternatives in a point set; and iv) the ability to handle
+point sets outside an ROI.
+Remark 3. The term “convergence” of a point set P has not been specified in the context of PBEMO.
+As shown in Table 1, we distinguish the convergence to the PF and the convergence to z. In Fig. 2,
+P1, . . . , P5, and P9 have a good convergence to the PF. In contrast, only P3 and P9 have a good
+convergence to z.
+Remark 4. The diversity of a point set in the preferred region is also an important evaluation cri-
+terion. If an ROI contains both P3 and P9, a quality indicator should evaluate P3 as having higher
+diversity than P9.
+Remark 5. Li et al. [26] pointed out that quality indicators should be able to distinguish point sets
+outside the ROI. In Fig. 2, P1 and P2 are outside the ROI. However, P2 is closer to the ROI than P1.
+The same is true for the relation between P4 and P5. In this case, a quality indicator should evaluate
+P2 and P4 as having better quality than P1 and P5. Mohammadi et al. [27] pointed out the importance
+of not using information about the PF, which is generally unavailable in real-world problems. As shown
+in Table 1, 9 out of the 14 indicators satisfy this criterion. Note that an approximation of the PF or
+the ROI found by PBEMO algorithms is available in practice. We believe that the remaining 5 out of
+the 14 indicators can address the issue by simply using the approximation.
+4.2
+MASF
+As done in some previous studies, e.g., [14,44] and [26], the basic idea of this quality indicator is to
+use the minimum ASF (MASF) value of P to evaluate the closeness of P to z:
+MASF(P) = min
+p∈P {s(p)} ,
+(14)
+9
+
+where we use s in (4). MASF can evaluate only the two types of convergence. Since MASF does not
+consider the other µ−1 points in P, MASF cannot evaluate the diversity of P. As the example shown
+in Fig. 2, MASF prefers P9 to P3.
+4.3
+MED
+As its name suggests, the mean Euclidean distance (MED) measures the average of Euclidean distance
+between each point in P to z [38]:
+MED(P) =
+1
+|P|
+�
+p∈P
+�
+�
+�
+�
+m
+�
+i=1
+�
+pi − zi
+pnadir
+i
+− pideal
+i
+�2
+.
+(15)
+MED can evaluate how close all points in P are to z and the PF when z is infeasible. Otherwise, it
+cannot evaluate the convergence to the PF if z is feasible. This is because MED prefers the non-Pareto
+optimal points close to z than the Pareto optimal points. As the example shown in Fig. 2, MED prefers
+the dominated P7 over the non-dominated P3 when z = (1.0, 1.0)⊤.
+4.4
+IGD-based indicators
+Here, we introduce three quality indicators developed upon the IGD metric. In particular, since they
+are designed to deal with the ROIC, ROIA, and ROIP defined in Section 3, respectively, they are thus
+denoted as IGD-C, IGD-A, and IGD-P accordingly in this paper. Note that both IGD-C and IGD-
+P were used in some previous studies [32, 45], and [36], respectively, whereas IGD-A is deliberately
+designed in this paper to facilitate our analysis.
+In practice, the only difference between the original IGD and its three extensions is the choice of
+the IGD-reference point set S. In IGD, IGD-reference points in S are uniformly distributed on the
+whole PF. In contrast, IGD-C, IGD-A, and IGD-P use a subset S′ ⊆ S. S′ can also be a subset of
+each ROI. In the example in Fig. 1, S′ of IGD-C, IGD-A, and IGD-P are in the ROIC, ROIA, and
+ROIP, respectively. Below, for each indicator, we describe how to select S′ from S.
+• For IGD-C, we first find the closest point pc to z from S, i.e., pc = argminp∈S{dist(p, z)}.
+Then, S′ is a set of all points in the region of a hypersphere of radius r centered at pc, i.e.,
+S′ = {p ∈ S | dist(p, pc) < r}.
+• The only difference between IGD-C and IGD-A is the choice of the center point. First, a point
+with the minimum ASF value pa is selected from S, i.e., pa = argminp∈S{s(p)}. We use the
+ASF s in (4) in this study. Then, S′ is a set of all points in the region of a hypersphere of radius
+r centered at pa, i.e., S′ = {p ∈ S | dist(p, pa) < r}.
+• In IGD-P, S′ is selected from S based on the Pareto dominance relation as in the ROIP. If z
+is feasible, S′ = {p ∈ S | p ≺ z}. Otherwise, S′ = {p ∈ S | p ≻ z}. Note that IGD-P does not
+require the radius r.
+4.5
+HVz
+This quality indicator was originally named HVq in [9], where q represents the reference point in [9].
+Since this paper denotes the reference point as z, we use the term “HVz” to make the consistency. It
+computes the HV value of P in the ROIP. The only difference between HV and HVz is the choice of the
+HV-reference point y ∈ Rm as follows. If z is feasible, y = z. If z is infeasible, yi = maxp∗∈ROIP{p∗
+i }
+for each i ∈ {1, . . . , m}. Fig. 3 shows the HV-reference point y in HVz when setting z to (0.9, 0.9)⊤
+and (0.5, 0.5)⊤, where the former is feasible while the latter is infeasible.
+HVz can evaluate the convergence and diversity of a point set in terms of the ROIP. However, it
+cannot handle the point sets outside the ROIP. This is because HV does not consider points dominated
+by the HV-reference point y. In the example in Fig. 2, the HVz values of P1, P2, P4, and P5 are 0.
+10
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+y = z
+(a) Feasible z = (0.9, 0.9)⊤
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+z
+y
+(b) Infeasible z = (0.5, 0.5)⊤
+Figure 3: Examples of the HV-reference point y in HVz.
+4.6
+PR
+The percentage of points in the ROI (PR) evaluates the cardinality of P [39] lying in the DM specified
+ROI:
+PR(P) = |{p ∈ P ∩ R}|
+|P|
+× 100%,
+(16)
+where R is the ROIP defined in [39], though it can be any type of ROI in principle. Note that PR
+is the only cardinality-based indicator considered in our study. A large PR means that many points
+in the corresponding P are in the ROIP. Like HVz, it is clear that PR cannot distinguish point sets
+outside the ROI.
+4.7
+PMOD
+PMOD consists of two algorithmic steps [28]. First, it maps each point p ∈ P onto a hyperplane
+passing through z as:
+p′ = p + ((z − p) · ˆz)ˆz,
+(17)
+where ˆz is the unit vector of z. Then, PMOD aggregates three measurements including i) the distance
+between each mapped point p′ and z, ii) the distance between p and the origin o = (0, . . . , 0)⊤, and
+iii) the unbiased standard deviation of all mapped points as:
+PMOD(P) =
+1
+|P|
+�
+P′∈P′
+�
+dist(p′, z) + α dist(p, o)
+�
++ SD
+�
+{dp′}p′∈P′�
+,
+(18)
+where P′ is a set of |P| mapped points. In (18), α is a penalty parameter for mapped points outside
+the preferred region of radius r centered at z, where r is a parameter of PMOD. When p′ is inside
+the preferred region (i.e., dist(p′, z) ≤ r), α = 1. Otherwise, α > 1, e.g., α is set to 1.5 in [28]. SD
+returns the unbiased standard deviation of input values. For each p′ ∈ P′, dp′ in (18) is the minimum
+Manhattan distance between p′ and another point q ∈ P′, i.e., dp′ =
+min
+q∈P′\{p′}
+�m
+i=1 |p′
+i − qi|.
+The smaller PMOD value is, the better quality P is in terms of i) the convergence to z, ii) the
+convergence to the origin (not the PF), and iii) the uniformity. Note that PMOD assumes the ideal
+point always does not dominate the origin. When the ideal point dominates the origin, PMOD prefers
+points far from the PF in view of above ii). Let us consider the shifted point sets in Fig. 2 again, if
+the offset is −100, PMOD is likely to prefer P7 to P3 because P7 is closer to the origin than P3.
+4.8
+IGD-CF and HV-CF
+A user-preference metric based on a composite front (UPCF) [27] is a framework for evaluating the
+quality of P. IGD-CF and HV-CF are the UPCF versions of IGD and HV, respectively. Algorithm 1
+11
+
+Algorithm 1: IGD-CF and HV-CF
+1 PCF ← Select all non-dominated points from P1 ∪ · · · ∪ PK;
+2 pc ← argminp∈PCF{dist(p, z)};
+3 Rpref ← {p ∈ Rm | dist(p, pc) < r};
+4 for i ∈ {1, . . . , K} do
+5
+IGD-CF(Pi) ← IGD(Pi ∩ Rpref) using PCF as S;
+6
+HV-CF(Pi) ← HV(Pi ∩ Rpref);
+shows how to calculate the IGD-CF and HV-CF values of K points sets P1, . . . , PK. First, let PCF
+be a set of all non-dominated points in the K point sets (line 1), where PCF was called a composite
+front in [27]. Then, the closest point pc to z is selected from PCF (line 2). A preferred region Rpref is
+defined as the set of all points in the region of a hypersphere of radius r centered at pc (line 3). Note
+that Rpref can include dominated points. If PCF = F, Rpref is equivalent to the ROIC shown in Fig.
+1(a).
+For each i ∈ {1, . . . , K}, the IGD-CF value of Pi is calculated based only on the points in Pi∩Rpref
+(line 5). In other words, points outside Rpref are removed from Pi. For example, in Fig. 1(a), points
+outside the large dotted circle are removed from a point set. IGD-CF uses PCF as an approximation
+of the IGD-reference point set S. The HV-CF value of Pi is calculated in a similar manner (line 6),
+where the previous study [27] did not give a rule of thumb to set the HV-reference point y. When the
+trimmed Pi is empty, we set its IGD-CF value to ∞ and its HV-CF value to 0 in this study.
+Li et al. [26] pointed out that IGD-CF and HV-CF cannot distinguish point sets outside the
+preferred region. This is because IGD-CF and HV-CF do not consider any point outside Rpref. In the
+example in Fig. 2, all point sets except for P3, P9, and P10 are equally bad, i.e., IGD-CF(Pi) = ∞
+and HV-CF(Pi) = 0 for i ∈ {1, 2, 4, 5, 6, 7, 8}.
+4.9
+PMDA
+The preference-based metric based on specified distances and angles (PMDA) [31] is built upon the
+concept of light beams [46]. It consists of three algorithmic steps.
+Step 1: It lets a set of points Q = {qi}m+1
+i=1 on a hyperplane passing through z while m+1 light beams
+pass from the origin (0, . . . , 0)⊤ to q1, . . . , qm+1, respectively. For i ∈ {1, . . . , m}, qi is given
+as:
+qi = z + α (ei − z),
+(19)
+where ei is the standard-basis vector for the i-th objective function, e.g., e1 = (1, 0)⊤ and
+e2 = (0, 1)⊤ for m = 2. In (19), α controls the spread of the light beams. The remaining
+qm+1 in Q is set to z.
+Step 2: All the points in Q are further shifted as:
+Q′ = βQ,
+(20)
+where β is the minimum objective value in P′ for all m objectives, i.e., β = min
+p∈P′{
+min
+i∈{1,...,m}pi}3.
+Here, P′ is a set of points in ∪K
+i=1Pi that are in a preferred region defined by m light beams,
+which pass through q1, . . . , qm but do not pass through qm+1 = z.
+Step 3: PMDA measures the distance between each point in P and its closest point in Q′ as:
+PMDA(P) =
+1
+|P|
+�
+p∈P
+�
+min
+q∈Q′{dist(p, q)} + γθp
+�
+,
+(21)
+3 [31] defined that β is the minimum objective value of P, not P′. Since β can be different for different point sets in
+this case, this version of PMDA is not reliable.
+12
+
+Algorithm 2: R-IGD and R-HV
+1 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK;
+2 for i ∈ {1, . . . , K} do
+3
+Pi ← Pi ∩ Pall;
+4
+pa ← argminp∈Pi {s(p)};
+5
+Pi ←
+�
+p ∈ Pi | |pj − pa
+j| ≤ r for j ∈ {1, . . . , m}
+�
+;
+6
+k ← argmaxj∈{1,...,m}
+� pa
+j−zj
+zw
+j −zj
+�
+;
+7
+piso ← z +
+�
+pa
+k−zk
+zw
+k −zk
+�
+(zw − z);
+8
+for p ∈ Pi do
+9
+p ← p + (piso − pa);
+10
+R-IGD(Pi) ← IGD(Pi) using a trimmed S;
+11
+R-HV(Pi) ← HV(Pi) using zw;
+where γ is a penalty value and was set to 1/π in [31]. In (21), θp is an angle between p and
+z. If p is in the preferred region defined by the m light beams passing through q1, . . . , qm,
+θp = 0. Thus, points outside the preferred region are penalized.
+A small PMDA value indicates that points in the corresponding P are close to the m + 1 points in
+Q′ and the preferred region. Thus, PMDA does not evaluate the diversity of P. Note that all elements
+of a point are implicitly assumed to be positive in [31].
+4.10
+R-IGD and R-HV
+R-metric [26] is a framework that applies general quality indicators to the performance evaluation
+of K PBEMO algorithms. R-metric assumes that the DM prefers points along a line from z to the
+worst point zw defined by the DM. As recommended in [26], we set zw = z + 2 × u, where u is a
+unit vector. We set u = (1/√m, . . . , 1/√m)⊤ in this study. The previous study [26] considered the
+R-metric versions of IGD and HV, denoted as R-IGD and R-HV.
+Algorithm 2 gives the pseudo code for calculating R-IGD and R-HV. A set of all non-dominated
+points Pall are selected from the union of K point sets (line 1). After that, the following steps are
+performed for each point set Pi. First, points dominated by any point in Pall are removed from Pi
+(line 3). Then, the best point pa is selected from Pi in terms of the ASF (line 4), where the previous
+study [26] used s in (4) as the ASF. R-metric defines a preferred region based on a hypercube of
+size 2 × r centered at pa. Points outside the preferred region are removed from Pi (line 5). For the
+example shown in Fig. 4(a), only the three points in the dotted box are considered for the R-metric
+calculation. This trimming operation can penalize a point set that does not fit the preferred region.
+Next, R-metric obtains a projection of pa on the line from z to zw by the ASF (lines 6 and 7). This
+projection is called an iso-ASF point piso. R-metric transfers all the points in Pi by the direction
+vector from piso to pa (lines 8 and 9). For the example shown in Fig. 4(b), the three points are shifted
+horizontally. This transfer operation redefines the convergence to the PF as the convergence to z along
+a line based on the DM’s preference information.
+Finally, the R-IGD and R-HV values of Pi are calculated (lines 10 and 11). More specifically, for
+R-IGD, the same trimming operation (lines 4 and 5) is first applied to the IGD-reference point set S in
+R-IGD. Thus, all points in S are inside the preferred region. Then, the IGD value of Pi is calculated
+using the trimmed S. For R-HV, zw is used as the HV-reference point y.
+4.11
+EH
+The expanding hypercube metric (EH) [29] is based on the size of a hypercube centered at z that
+contains each point and the fraction of points inside the hypercube. While the former evaluates the
+convergence of a point set P to z, the latter tries to evaluate the diversity of P.
+The pseudo code of calculating the EH for K point sets P1, . . . , PK is given in Algorithm 3. First,
+EH removes duplicated points for each point set (lines 1 and 2). In the meanwhile, it also removes
+13
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+z
+zw
+(a) The trimming operation
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+z
+zw
+(b) The transfer operation
+Figure 4: Examples of the two operations in R-metric. In this example, zw = z + 0.7 × u.
+dominated points from each point set (lines 3 and 5). Note that if a point set is empty after these
+removal operations, its EH value is set to 0 (line 19).
+Then, the following steps are performed for each point set Pi. EH calculates the size of a hypercube
+centered at z that contains each point p in Pi (lines 10 and 11). Thereafter, all elements in h are
+sorted (line 12). Note that |h| = |Pi|. The maximum size hmax in h is maintained for an adjustment
+described later (lines 13 and 14). EH calculates “the area under the trade-off curve” ai between the
+hypercube size and the fraction (lines 16 and 17). While l/|Pi| is the fraction of points in the l-th
+hypercube, “(hl − hl−1)” is the incremental size of the hypercube. Finally, the EH value of each point
+set Pi is calculated by adjusting ai using hmax (lines 18 and 20).
+A large EH value means the corresponding P has good convergence to z. Due to the operation
+for removing dominated points (line 3), EH also implicitly evaluates the convergence of P to the PF.
+Since EH does not define a preferred region, EH fails to evaluate the diversity of P in some cases. Let
+us consider a set of non-dominated unduplicated points that are close to z and distributed at intervals
+of ∆. EH is maximized when ∆ is a positive value as close to zero as possible. For the example shown
+in Fig. 2(a), EH prefers P9 to P3.
+5
+Experimental Setup
+This section introduces the settings used in our experiments including the quality indicators, the
+benchmark problems, and preference-based point sets used in our analysis.
+5.1
+Quality Indicators
+In our experiments, we empirically analyze the performance and properties of 14 quality indicators
+reviewed in Section 4.
+As a baseline, we also take the results of HV and IGD into account.
+In
+particular, the implementations of HV and R-IGD and R-HV are taken from pygmo [47] and pymoo [48],
+respectively, while the other quality indicators are implemented by Tanabe in Python. The innate
+parameters of the 14 quality indicators are set according to the recommendation in their original
+paper. For the IGD-based indicators, we uniformly generated 1 000 IGD-reference points on the PF
+of a problem. For those HV-based indicators, we set the HV-reference point y in HV and HV-CF to
+(1.1, . . . , 1.1)⊤. We set the radius r of a preferred region to 0.1 for all the quality indicators. We also
+set the radius ζ of the ROIC and ROIA to 0.1.
+5.2
+Benchmark Test Problems
+DTLZ1 [49], DTLZ2 [49], convDTLZ2 [50] are chosen to constitute the benchmark test problems, which
+have linear, nonconvex, and convex PFs, respectively. To ensure the fairness of our experiments, the
+PF of the DTLZ1 problem is normalized to [0, 1]m. As a first attempt to investigate the properties
+14
+
+Algorithm 3: EH
+1 for i ∈ {1, . . . , K} do
+2
+Pi ← {p ∈ P | ̸ ∃pdup ∈ P s.t. pdup = p};
+3 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK;
+4 for i ∈ {1, . . . , K} do
+5
+Pi ← Pi ∩ Pall;
+6 hmax ← ∅;
+7 for i ∈ {1, . . . , K} do
+8
+h ← ∅;
+9
+for p ∈ Pi do
+10
+h ← maxj∈{1,...,m}{|pj − zj|};
+11
+h ← h ∪ {h};
+12
+h ← Sort all elements in h in ascending order;
+13
+hmax ← maxh∈h{h};
+14
+hmax ← hmax ∪ {hmax};
+15
+ai ← 0;
+16
+for l ∈ {1, . . . , |Pi|} do
+17
+ai ← ai +
+l
+|Pi| × (hl − hl−1) ;
+// h0 = 0
+18 for i ∈ {1, . . . , K} do
+19
+if Pi = ∅ then EH(Pi) ← 0 ;
+20
+else EH(Pi) ← ai + (maxh∈hmax{h} − hmax
+i
+) ;
+of preference-based quality indicators, we mainly focus on the two-objective scenarios to facilitate the
+analysis and discussion about the impact of the distribution of points on the quality indicators.
+Remark 6. We are aware of a previous study [29] evaluated the performance of R-NSGA-II and
+g-NSGA-II on the DTLZ problems with m ∈ {3, 5, 8, 10, 15, 20} by EH and R-HV. The previous study
+discussed the influence of the distribution of m-dimensional points on EH and R-HV using the parallel
+coordinates plot. However, the parallel coordinates plot is likely to lead to a wrong conclusion [51]. In
+fact, the results in [29] did not show the undesirable property of EH.
+5.3
+Experimental Settings
+We conduct two types of experiments.
+• One is an experiment using the 10 synthetic point sets as shown in Fig. 2. Fig. S.1 shows the
+distributions of the 10 point sets on the PF of the DTLZ1 and convDTLZ2 problems. Fig. S.1
+is similar to Fig. 2.
+• The other is an experiment using point sets found by the six PBEMO algorithms introduced
+in Section 2.4. Note that comprehensive benchmarking of the PBEMO algorithms is beyond the
+scope of this paper. Instead, we focus on an analysis of the behavior of the PBEMO algorithms.
+This contributes to the understanding of RQ2. Moreover, we also investigate how the choice
+of quality indicators influences the rankings of the PBEMO algorithms. This contributes to
+addressing RQ4.
+In particular, the source code of the PBEMO algorithms are provided by
+Li [11] while the weight vectors used in MOEA/D-NUMS are generated by using the source
+code provided by Li [15]. Each PBEMO algorithm is independently run 31 times with different
+random seeds.
+The population size µ is set to 100.
+The parameters associated with these
+PBEMO algorithms are set according to the recommendations in their original papers, except
+PBEA of which δ is set to 0.01 in this study.
+15
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+p1
+p25
+p50
+p75
+p100
+(0.1, 0.1)
+(-0.1, -0.1)
+(a) 100 points
+20
+40
+60
+80
+100
+Point IDs
+0
+20
+40
+60
+80
+100
+Rankings
+z = (0.1, 0.1)
+z = (-0.1, -0.1)
+(b) Rankings (distance)
+20
+40
+60
+80
+100
+Point IDs
+0
+20
+40
+60
+80
+100
+Rankings
+z = (0.1, 0.1)
+z = (-0.1, -0.1)
+(c) Rankings (ASF)
+-3 -2 -1 0 1 2 3
+f1
+3
+2
+1
+0
+-1
+-2
+-3
+f2
+-1
+0.5
+0
+0.5
+1
+(d) Kendall τ
+Figure 5: (a) Distribution of 100 uniformly distributed points, (b) the rankings of the 100 points by
+the distance, (c) the ranking of the 100 points by the ASF, and (d) the Kendall τ values on the DTLZ2
+problem.
+6
+Results
+This section is dedicated to addressing the four RQs raised in Section 1. First, Section 6.1 analyzes the
+relation between the distance to the reference point z and the ASF value. Then, Section 6.2 investigates
+differences in the three ROIs and the behavior of EMO algorithms. Thereafter, Section 6.3 examines
+the properties of the 14 quality indicators using the synthetic point sets shown in Figs. 2 and S.1.
+Finally, Section 6.4 analyzes the influence of quality indicators on the rankings of EMO algorithms.
+6.1
+Relation between the distance to z and the ASF value
+Fig. 5(a) shows 100 uniformly distributed points p1, . . . , p100 on the PF of the DTLZ2 problem.
+Fig. 5(a) also shows the two reference points z0.1 = (0.1, 0.1)⊤ and z−0.1 = (−0.1, −0.1)⊤. While z0.1
+is dominated by the ideal point, z−0.1 dominates the ideal point.
+As shown in Fig. 5(a), intuitively, the 50-th point p50 on the center of the PF is closest to both
+z0.1 and z−0.1. However, this intuition is incorrect. Figs. 5(b) and 5(c) show the rankings of the 100
+points by the Euclidean distance to z and the ASF in (4), respectively. A low ranking means that the
+corresponding point is close to z or obtains a small ASF value. As seen from Fig. 5(b), p50 is closest
+to z0.1. In contrast, p50 is farthest from z−0.1. The two extreme points (p1 and p100) are closest to
+z−0.1. Thus, the closest points to z0.1 and z−0.1 are different. As shown in Fig. 5(c), the rankings by
+the ASF are consistent when using either one of z0.1 and z−0.1. This is because both z0.1 and z−0.1
+are in the same direction.
+Fig. 5(d) shows the Kendall rank correlation τ value of the distance to z and the ASF value, where
+τ ∈ [−1, 1]. In Fig. 5(d), we uniformly generated z from (−3, 3)⊤ to (3, −3)⊤ at intervals of 0.01. Then,
+we calculated the τ value for each z. The τ value quantifies the consistency of the two rankings, where
+one is based on the distance to the corresponding z, and the other is based on the ASF value. Positive
+and negative τ values indicate that the two rankings are consistent and inconsistent, respectively.
+As seen from Fig. 5(d), the rankings by the distance to z and the ASF value are inconsistent when
+setting z close to the line passing through (0, 0)⊤ and (−3, −3)⊤. We can also see that the rankings
+are weakly inconsistent when setting z to other positions.
+Note that the inconsistency between the distance to z and the ASF value depends on not only
+the position of z, but also the shape of the PF. Figs. S.2 and S.3 show the results on the DTLZ1
+and convDTLZ2 problems, respectively.
+As shown in Fig.
+S.3(a), we set z to (2, 2)⊤ instead of
+(−0.1, −0.1)⊤ for the convDTLZ2 problem. As shown in Figs. S.2(b) and (c), the rankings by the
+distance to z and the ASF value on the DTLZ1 problem are always consistent regardless of the
+position of z. In contrast, as seen from Figs. S.3(b) and (c), the inconsistency of the rankings can
+be observed on the convDTLZ2 problem. While Fig. S.3(b) is similar to Fig. 5(b), Fig. S.3(d) is
+opposite from Fig. 5(d). Unlike Fig. 5(d), Fig. S.3(d) indicates that the inconsistency between the
+two rankings occurs when z is dominated by the nadir point pnadir on the convex PF.
+16
+
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(a) ROIC (z0.1)
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(b) ROIA (z0.1)
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(c) ROIP (z0.1)
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(d) ROIC (z−0.1)
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(e) ROIA (z−0.1)
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(f) ROIP (z−0.1)
+Figure 6: Distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using
+z0.1 and z−0.1.
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(a) R-NSGA-II
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(b) MOEA/D-NUMS
+0
+0.5
+1
+f1
+0
+0.5
+1
+f2
+(c) g-NSGA-II
+Figure 7: Distributions of points found by three PBEMO algorithms on the DTLZ2 problem when
+using z−0.1.
+Answers to RQ1: Our results show that the closest Pareto-optimal point to the reference point z
+does not always minimize the ASF. Although it has been believed that minimizing the ASF means
+moving closer to z, this is not always correct. We observed that the inconsistency between the
+distance to z and the ASF value depends on the position of z and the shape of the PF. Roughly
+speaking, the inconsistency can be observed when z dominates the ideal point on a problem with a
+nonconvex PF, and z is dominated by the nadir point on a problem with the convex PF.
+6.2
+Analysis of the three ROIs
+First, this section investigates the differences between the three ROIs. Similar to Fig. 1, Fig. 6 shows
+the distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using
+z0.1 = (0.1, 0.1)⊤ and z−0.1 = (−0.1, −0.1)⊤. Figs. S.6 and S.7 show the results on the DTLZ1 and
+convDTLZ2 problems, respectively. Figs. 6(a) and (b) are exactly the same as Figs. 1(a) and (b),
+respectively. Thus, the ROIC and ROIA are the same even when using either one of z0.1 and z0.5. In
+contrast, as shown in Figs. 6(d) and (e), the ROIC and ROIA are totally different when using z−0.1.
+While the ROIA is on the center of the PF, the ROIC is on either one of the two extreme points (1, 0)⊤
+and (0, 1)⊤. Since the two extreme points (1, 0)⊤ and (0, 1)⊤ are equally close to z−0.1, Fig. 1(d) shows
+two possible ROIs. This strange result is due to the inconsistency between the distance to z and the
+17
+
+ASF value reported in Section 6.1. As seen from Fig. S.7(g), a similar result can be observed on the
+convDTLZ2 problem. Results similar to those in Fig. 6(d) can be obtained by using z with a small
+Kendall τ value in Fig. 5(d).
+Fig. 6(c) significantly differs from Fig. 1(c). The extent and cardinality of the ROIP in Fig. 6(c)
+are much larger than those in Fig. 1(c). As shown in Fig. 6(f), the ROIP and the PF are identical
+when the reference point dominates the ideal point. The same is true when the reference point is
+dominated by the nadir point. In this case, preference-based multi-objective optimization is the same
+as general one. The size of the ROIP increases as z moves away from the PF. This undesirable property
+of the ROIP is similar to that of the g-dominance relation pointed out in [11]. As seen from Figs. S.6
+and S.7, this undesirable property of the ROIP can be observed on other problems. Since the DM
+does not know any information about the PF in practice, it is difficult to set a reference point that is
+neither too close nor too far from the PF.
+Next, we point out that the differences in the target ROIs caused the unexpected behavior of some
+PBEMO algorithms in [11]. Fig. 7 shows the final point sets found by R-NSGA-II, MOEA/D-NUMS,
+and g-NSGA-II on the DTLZ2 problem when using z−0.1. The results in Fig. 7 are consistent with the
+results in [11]. Figs. S.8–S.16 show the results of the six PBEMO algorithms on the three problems.
+As discussed in Section 3.1, R-NSGA-II, MOEA/D-NUMS, and g-NSGA-II aim to approximate the
+ROIC, ROIA, and ROIP, respectively. As demonstrated here, the three ROIs can also be different.
+For these reasons, the three EMO algorithms found different point sets, as shown in Fig. 7.
+The previous study [11] concluded that R-NSGA-II and g-NSGA-II failed to approximate the
+“ROI” when z is far from the PF. However, this conclusion is not very correct. Correctly speaking,
+as shown in Figs. 7(a) and (c), R-NSGA-II and g-NSGA-II failed to approximate the “ROIA” but
+succeeded in approximating the “ROIC” and “ROIP”, respectively.
+Answers to RQ2: There are two takeaways generated from the analysis in this subsection. First,
+our results showed that the three ROIs can be significantly different depending on the position of the
+reference point z and the shape of the PF. We demonstrated that the ROIA is not always a subregion
+of the PF closest to z due to the inconsistency observed in Section 6.1. In addition, we found that
+the size of the ROIP significantly depends on the position of z. Unless the DM knows the shape
+of the PF in advance, it would be better not to use the ROIP. Second, we also demonstrated that
+the differences in the three ROIs could cause the unexpected behavior of PBEMO algorithms. For
+this reason, we argue the importance to clearly define a target ROI when benchmarking PBEMO
+algorithms and performing a practical decision-making.
+18
+
+Table 2: Rankings of the 10 synthetic point sets on the DTLZ2 problem by the 16 quality indicators when using z0.5 and z−0.1.
+(a) z0.5 = (0.5, 0.5)⊤
+MASF
+MED
+IGD-C
+IGD-A
+IGD-P
+HVz
+PR
+PMOD
+IGD-CF
+HV-CF
+PMDA
+R-IGD
+R-HV
+EH
+HV
+IGD
+P1
+9
+10
+9
+9
+9
+5
+7
+7
+4
+4
+10
+6
+6
+6
+7
+9
+P2
+5
+5
+5
+5
+5
+5
+7
+3
+4
+4
+5
+4
+4
+4
+3
+4
+P3
+2
+2
+1
+1
+2
+1
+1
+1
+1
+1
+2
+1
+1
+2
+2
+2
+P4
+5
+4
+6
+6
+5
+5
+7
+5
+4
+4
+4
+5
+4
+4
+3
+4
+P5
+9
+9
+10
+10
+9
+5
+7
+9
+4
+4
+9
+7
+6
+6
+7
+9
+P6
+7
+7
+7
+7
+7
+5
+4
+4
+4
+4
+8
+8
+8
+8
+9
+7
+P7
+4
+3
+4
+4
+4
+4
+1
+2
+4
+4
+3
+8
+8
+8
+6
+3
+P8
+7
+7
+8
+8
+7
+5
+4
+8
+4
+4
+7
+8
+8
+8
+10
+8
+P9
+1
+1
+3
+3
+3
+3
+1
+6
+3
+3
+1
+2
+3
+1
+5
+6
+P10
+3
+6
+2
+2
+1
+2
+6
+10
+2
+2
+6
+3
+2
+3
+1
+1
+(b) z−0.1 = (−0.1, −0.1)⊤
+MASF
+MED
+IGD-C
+IGD-A
+IGD-P
+HVz
+PR
+PMOD
+IGD-CF
+HV-CF
+PMDA
+R-IGD
+R-HV
+EH
+HV
+IGD
+P1
+9
+2
+1
+9
+9
+8
+1
+7
+1
+1
+10
+6
+7
+6
+7
+9
+P2
+5
+4
+3
+5
+4
+4
+1
+3
+3
+3
+5
+4
+4
+4
+3
+4
+P3
+2
+6
+5
+1
+2
+2
+1
+1
+3
+3
+2
+1
+1
+2
+2
+2
+P4
+5
+4
+8
+6
+4
+4
+1
+5
+3
+3
+4
+5
+4
+4
+3
+4
+P5
+9
+1
+10
+10
+9
+7
+1
+9
+3
+3
+9
+7
+6
+6
+7
+9
+P6
+7
+8
+4
+7
+7
+9
+1
+4
+3
+3
+8
+8
+8
+8
+9
+7
+P7
+4
+10
+6
+4
+3
+6
+1
+2
+3
+3
+3
+8
+8
+8
+6
+3
+P8
+7
+9
+9
+8
+8
+9
+1
+8
+3
+3
+7
+8
+8
+8
+10
+8
+P9
+1
+7
+7
+3
+6
+3
+1
+6
+3
+3
+1
+2
+3
+1
+5
+6
+P10
+3
+3
+2
+2
+1
+1
+1
+10
+2
+2
+6
+3
+2
+3
+1
+1
+19
+
+6.3
+Analysis of quality indicators
+Table 2 shows the rankings of the 10 point sets P1 to P10 in Fig. 2 by each quality indicator when
+using z0.5 = (0.5, 0.5)⊤ and z−0.1 = (−0.1, −0.1)⊤. For the sake of reference, we show the results of
+HV and IGD. Table 2 shows which point set is preferred by each quality indicator. For example, P9
+obtains the best MASF value in the 10 point sets. Tables S.1 and S.2 show the results on the DTLZ1
+and convDTLZ2 problems, respectively. We do not intend to elaborate Tables S.1 and S.2, as they
+are similar to Table 2.
+6.3.1
+Results for z0.5 = (0.5, 0.5)⊤
+First, we discuss the results shown in Table 2(a). P3 is the best in terms of i) the convergence to the
+PF, ii) convergence to the reference point z, and iii) diversity. Thus, quality indicators should give
+P3 the highest ranking. P3 is ranked highest by 9 out of the 16 quality indicators. However, four
+quality indicators (MASF, MED, PMDA, and EH) prefer P9 with the poorest diversity to P3. This is
+because they do not take into account the diversity of points as shown in Table 1. Since HV and IGD
+do not handle the preference information, they prefer P10 that covers the whole PF. Interestingly,
+IGD-P also prefers P10 the most. This is because the IGD-reference points of IGD-P are relatively
+widely distributed around the center of PF.
+Since PR evaluates only the cardinality, PR cannot distinguish the quality of P3, P7, and P9.
+Since IGD is Pareto non-compliant, IGD prefers P7 to P9, where all the points in P7 are dominated
+by those in P9. Similarly, PMOD gives P7 the second highest ranking. Since PMOD does not take
+into account the convergence to the PF, PMOD can evaluate the quality of point sets inaccurately.
+Since IGD-CF and HV-CF cannot distinguish point sets outside their preferred regions, most point
+sets obtain the same ranking. Although this undesirable property was already pointed out in [26], this
+is the first time to demonstrate that. The same is true for HVz and PR. Since R-IGD, R-HV, and
+EH remove dominated points from point sets, they cannot distinguish the three dominated point sets
+(P6, P7, and P8).
+6.3.2
+Results for z−0.1 = (−0.1, −0.1)⊤
+Next, we discuss the results shown in Table 2(b). In this setting, P1 and P5 are the best in terms of
+all three criteria i), ii), and iii). Thus, quality indicators should give P1 or P5 the highest ranking.
+However, the rankings by four ASF-based quality indicators (MASF, IGD-A, R-IGD, and R-HV)
+are the same in Tables 2(a) and (b). This is because the point with the minimum ASF value and the
+ROIA are the same regardless of whether z0.5 or z−0.1 is used, as demonstrated in Sections 6.1 and
+6.2. The same is true for PMOD, PMDA, and EH. Thus, these quality indicators fail to evaluate the
+convergence of the point sets to the reference point.
+In contrast, the rankings by other quality indicators based on the ROIC and ROIP are different in
+Tables 2(a) and (b). As demonstrated in Section 6.2, the ROIC is on either one of the two extreme
+points when using z−0.1. For this reason, four quality indicators based on the ROIC (MED, IGD-C,
+IGD-CF, and HV-CF) prefer P1 or P5 to P3. While IGD-C and IGD-A are perfectly consistent for the
+results of z0.5, they are inconsistent for the results of z−0.1. This is due to the inconsistency revealed
+in Section 6.1.
+As discussed in Section 6.2, when z dominates the ideal point or is dominated by the nadir point,
+the ROIP is equivalent to the PF. For this reason, three quality indicators based on the ROIP (IGD-P,
+HVz, and PR) cannot handle the DM’s preference information. Thus, like HV and IGD, IGD-P, HVz,
+and PR prefer P10 the most. Since IGD-P and IGD use the same IGD-reference point set S, their
+rankings are perfectly consistent. Although the position of the HV-reference point y is different in
+HVz and HV, their rankings are almost the same. PR cannot distinguish all the 10 point sets.
+20
+
+Answers to RQ3: Our results indicated that most quality indicators have some undesirable prop-
+erties, which have not been noticed even in their corresponding papers. We demonstrated that the
+quality indicators based on the ROIA cannot evaluate the convergence of a point set to the reference
+point accurately in some cases. We also demonstrated that the quality indicators based on the ROIP
+cannot take into account the DM’s preference information. Our results imply that IGD-C may be
+the most reliable quality indicator when considering the practical ROIC. However, IGD-C is Pareto
+non-compliant.
+21
+
+Table 3: Rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality indicators when using z0.5 = (0.5, 0.5)⊤. “NUMS”stands for MOEA/D-NUMS.
+MASF
+MED
+IGD-C
+IGD-A
+IGD-P
+HVz
+PR
+PMOD
+IGD-CF
+HV-CF
+PMDA
+R-IGD
+R-HV
+EH
+HV
+IGD
+R-NSGA-II
+3
+2
+6
+6
+5
+6
+4
+4
+6
+6
+3
+5
+5
+1
+5
+5
+r-NSGA-II
+4
+3
+3
+3
+4
+4
+1
+3
+3
+4
+2
+3
+3
+3
+4
+4
+g-NSGA-II
+5
+5
+1
+1
+1
+1
+1
+1
+1
+1
+5
+2
+1
+6
+2
+2
+PBEA
+1
+6
+2
+2
+2
+2
+6
+5
+2
+2
+6
+1
+2
+5
+1
+1
+R-MEAD2
+6
+4
+5
+5
+3
+3
+5
+6
+4
+3
+4
+6
+6
+4
+3
+3
+NUMS
+2
+1
+4
+4
+6
+5
+1
+2
+5
+5
+1
+4
+4
+2
+6
+6
+22
+
+6.4
+On the rankings of PBEMO algorithms by quality indicators
+Table 3 shows the rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality
+indicators, where z = (0.5, 0.5)⊤. We calculated the rankings based on the average quality indicator
+values of the PBEMO algorithms over 31 runs. Tables S.3–S.5 show the rankings on the DTLZ1,
+DTLZ2, and convDTLZ2 problems when using various reference points. Note that we are interested
+in the influence of quality indicators on the rankings of the PBEMO algorithms rather than the
+rankings themselves.
+As shown in Table 3, the rankings of the PBEMO algorithms are different depending on the choice
+of the quality indicator. For example, R-NSGA-II performs the best in terms of EH but the worst
+in terms of five quality indicators including IGD-C, IGD-A, HVz, IGD-CF, and HV-CF. Likewise, g-
+NSGA-II is the worst performer in terms of EH but it is the best algorithm when considering the other
+nine quality indicators. As shown in Table. S.5(b), R-MEAD2 performs the best on the convDTLZ2
+problem in terms of EH. In summary, our results suggest that any PBEMO algorithm can obtain the
+best ranking depending on the choice of the quality indicator.
+These observations can be explained as the coupling relationship between innate mechanism of
+the PBEMO algorithms and the quality indicators. As demonstrated in Section 6.2, each PBEMO
+algorithm approximates its target ROI embedded by its designer. As investigated in Section 6.3, each
+quality indicator prefers a point set that approximates its target ROI well. Thus, when benchmarking
+PBEMO algorithms, it is important to clarify which type of ROI the DM wants to approximate
+and select a suitable quality indicator. For example, if the DM wants to approximate the ROIA,
+she/he should select either of IGD-A, R-IGD, and R-HV. Otherwise, the DM can overestimate or
+underestimate the performance of PBEMO algorithms.
+Answers to RQ4: Our results showed that the choice of the quality indicator significantly influ-
+ences the performance rankings of EMO algorithms. For example, as seen from Table 3, PBEA
+performs the worst in terms of PMDA but the best in terms of R-IGD. This means that any PBEMO
+algorithm can possibly be ranked as the best (or the worst) depending on the choice of the quality
+indicator. We also discussed how to conduct meaningful benchmarking of PBEMO algorithms.
+7
+Conclusion
+In this paper, we first reviewed the 3 ROIs and 14 existing quality indicators for PBEMO algorithms
+using the reference point. Different from the descriptions in their corresponding papers, we classified
+the properties of the quality indicators from the perspective of their working principle. As a result, we
+found that some quality indicators have undesirable properties. For example, PMDA and EH cannot
+evaluate the diversity of a point set. We also discussed the target ROI of each quality indicator.
+Next, we empirically analyzed the performance and properties of those 14 quality indicators to
+address 4 RQs (RQ1 to RQ4). Our findings are helpful for benchmarking PBEMO algorithms and
+decision-making in real-world problems. In any case, we argue the importance of determining a target
+ROI first of all. Our results suggested the use of the ROIC. Afterward, a researcher and the DM should
+select a PBEMO algorithm and quality indicator based on their target ROI. For example, R-NSGA-II
+aims to approximate the ROIC. In contrast, HVz is to evaluate how a point set approximates the
+ROIP. Thus, HVz is not suitable for evaluating the performance of R-NSGA-II.
+As demonstrated in the three IGD variants (IGD-C, IGD-A, and IGD-P), we believe that the
+target ROI of some quality indicators can be changed easily.
+For example, the target ROI of R-
+IGD can be changed from the ROIA to the ROIC by revising the line 4 in Algorithm 2, i.e., pc =
+argminp∈Pi{dist(p, z)}. An investigation of this concept is an avenue for future work. Note that
+the analysis conducted in this paper focused on quality indicators for PBEMO algorithms using the
+reference point. It is questionable and important to extend our analysis for other preference-based
+optimization (e.g., a value function) in future research. There is room for discussion about a systematic
+benchmarking methodology for PBEMO.
+23
+
+Acknowledgment
+Tanabe was supported by JSPS KAKENHI Grant Number 21K17824 and LEADER, MEXT, Japan.
+Li was supported by UKRI Future Leaders Fellowship (MR/S017062/1, MR/X011135/1), NSFC
+(62076056), EPSRC (2404317), Royal Society (IES/R2/212077) and Amazon Research Award.
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+

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+Circuit Complexity through phase transitions:
+consequences in quantum state preparation
+Sebasti´an Roca-Jerat,1 Teresa Sancho-Lorente,1 Juan Rom´an-Roche,1 and David Zueco1
+1Instituto de Nanociencia y Materiales de Arag´on (INMA) and Departamento de F´ısica de la Materia Condensada,
+CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain
+(Dated: January 13, 2023)
+In this paper, we analyze the circuit complexity for preparing ground states of quantum many-
+body systems. In particular, how this complexity grows as the ground state approaches a quantum
+phase transition. We discuss different definitions of complexity, namely the one following the Fubini-
+Study metric or the Nielsen complexity. We also explore different models: Ising, ZZXZ or Dicke.
+In addition, different forms of state preparation are investigated: analytic or exact diagonalization
+techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigen-
+solvers.
+We find that the divergence (or lack thereof) of the complexity near a phase transition depends on
+the non-local character of the operations used to reach the ground state. For Fubini-Study based
+complexity, we extract the universal properties and their critical exponents.
+In practical algorithms, we find that the complexity depends crucially on whether or not the system
+passes close to a quantum critical point when preparing the state. While in the adiabatic case it is
+difficult not to cross a critical point when the reference and target states are in different phases, for
+VQE the algorithm can find a way to avoid criticality.
+CONTENTS
+I. Introduction
+2
+A. Complexity overview
+2
+1. Complexity `a la Nielsen
+3
+2. Circuit Complexity from the Fubini-Study metric
+4
+3. Some remarks comparing both approaches
+5
+B. Main results and manuscript organization
+5
+II. Complexity and the geometry of states close to a quantum phase transition.
+5
+A. Complexity and its derivative when crossing a QPT
+6
+B. Finite size scaling
+6
+III. Solvable Hamiltonians
+7
+A. Quantum Ising model
+7
+1. Complexity through QPTs
+7
+2. Relation between CN and CFS
+8
+B. The Dicke model
+8
+IV. Complexity in a quantum computer, the case of ground state preparation
+10
+A. Adiabatic algorithms
+10
+1. Complexity in adiabatic algorithms
+11
+B. Circuit Complexity in VQEs
+13
+1. Local VQE ansatz
+15
+2. VQE complexity through QPTs
+15
+V. Discussion
+17
+Acknowledgments
+18
+A. Complexity associated to the VQE
+18
+B. Other paths in the adiabatic algorithm
+19
+References
+19
+arXiv:2301.04671v1  [quant-ph]  11 Jan 2023
+
+2
+I.
+INTRODUCTION
+How much does it cost to generate a target quantum state from another reference state? This is a rather general
+question that has been discussed in quantum information for obvious reasons. In quantum computation it is desirable
+to obtain the result with the minimum set of gates. This number is, roughly speaking, the computational cost and it
+is called circuit complexity (C) [1–3]. It is, let us say, the quantum analog of the concept of computational complexity
+in computer science. Importantly enough, this cost builds upon a concrete physical architecture, i.e the available
+set of gates. Therefore, C not only depends on the reference and target states but on the restrictions for reaching
+the latter. This is quite natural if one thinks of an actual quantum computer where the possible operations have
+restrictions. Note that, if any unitary were allowed, a simple rotation would achieve the goal and every quantum
+state would be easily prepared, so that (essentially) the complexity would be a trivial quantity. Therefore, also in
+analytic calculations, the path between the reference and target is restricted to a set, e.g. gaussian states [4–8] .
+Beyond quantum computation, circuit complexity is a relevant concept in quantum gravity. In particular, for its
+consequences in holography [9–11]. For those who are not experts (like us), we can say that holography describes
+quantum gravity within a region of space by looking at the boundary of that region, that is described by a non
+gravitational theory. Then, any bulk quantity in the gravitational theory is equivalent or dual to another quantity
+in the boundary of the non gravitational theory. One of the main problems of this duality is that the volume behind
+the black hole horizon keeps growing for a very long time while the entanglement at the boundary saturates at much
+shorter times. One possible solution is to conjecture that the dual of volume is not entanglement but complexity,
+via the identification Complexity = Volume. This is because we expect that volume is an extensive quantity, while
+entanglement (typically) fulfils an area law.
+Therefore, the calculations of complexity are beyond the quantum
+information community and different calculations in field theories have been discussed in the literature [12, 13].
+The notion of complexity (C) is much related to the geometry of states (or operators). It is a measure of the
+distance between two of them. Therefore, one possible choice for C is finding the geodesics in the Fubini-Study metric
+in the projected Hilbert space for the case of pure states. For mixed states different measures have been introduced
+via state purification [14] or distance measures for mixed states as the Bures distance [15]. This geometric background
+is a powerful way to understand complexity, since it allows us to know how much it will cost to prepare a state by
+solving a geodesic equation. It is true, however, that the metric, in principle, can only be obtained in some cases:
+surely in exactly solvable models. And there we know how to prepare states. Thus, it is interesting to be able to
+predict the typical behaviour in general models. Here, we move in this direction.
+In this article we are interested in a quite generic situation, i.e.
+when a critical point is crossed to reach the
+target state. In particular, we investigate what general statements about the behaviour of the circuit complexity
+we can make. We are not the first to calculate C in a quantum phase transition (QPT) [16]. Recent papers discuss
+exactly solvable models as the topological Kitaev, Bose-Hubbard and Lipkin-Meshkov-Glick ones [17–23]. Importantly
+enough, complexity has been shown to be a useful probe of topological phase transitions. Complementary to these
+calculations, in this work, we use that close to a transition point, the concept of universality emerges naturally, so
+we expect these universal properties to be inherited by complexity. If so, we can argue for its scaling laws or how
+complexity behaves regardless of model details or even on the particular chosen definition of complexity. In addition,
+we apply our theory for state preparation in quantum computers. This is a key and hard task [24]. It is within
+the QMA complexity class [25], roughly speaking the NP-complete analogue for quantum computers. Nevertheless,
+quantum computers are expected to be better than classical methods such as density functional theory [26], density
+normalization group [27], tensor networks [28], quantum Montecarlo [29] or even ML-inspired techniques [30], in some
+instances. For a recent discussion of these issues, see [31]. Heuristic quantum algorithms as adiabatic [32] or varational
+ones [33, 34] can outperform classical calculations and serve for the generation of quantum states as quantum data,
+e.g. phase classification [35]. Motivated by all of this, we discuss how useful the concept of complexity is and how
+much one can anticipate the difficulty of state preparation in variational quantum eigensolvers (VQEs) or adiabatic
+quantum algorithms (with and without shortcuts to adiabaticity). To challenge our theory we tackle both integrable
+and non-integrable models using numerical simulations and computing C.
+A.
+Complexity overview
+We find it convenient to discuss first the different notions of circuit complexity that we will use in this paper and
+the relationship between them.
+
+3
+1.
+Complexity `a la Nielsen
+The original notion of complexity is due to the works of Nielsen and collaborators [1–3]. See [13] for a recent review.
+Restricting ourselves to unitary operations, target and reference states are related as
+|ψT ⟩ = U(t, 0)|ψR⟩ = T e−i
+� t
+0 H(τ) dτ|ψR⟩ .
+(1)
+T stands for time ordering. Notice that,
+H(τ) = i(∂τU)U † .
+(2)
+A Cost function is formally defined as:
+CN := min{U}
+� t
+0
+dτ F[U, ˙U]
+(3)
+with F some functional fulfilling some basic properties as continuity, homogeneity (F[U, λ ˙U] = λF[U, ˙U] for λ ≥ 0),
+positivity and the triangular inequality 1. If, in addition to these, smoothness is assumed and the Hessian of F as a
+function of U is strictly positive, the functional is a Finsler metric. Thus, CN is nothing but the geodesics. The suffix
+N stands for Nielsen.
+Being a little more explicit, we can write that the evolution is given by
+H(τ) =
+�
+n
+Y (n)(τ)On .
+(4)
+with On some operators and Y (n)(τ) parameters. A usual functional is then given by,
+Fk(τ) ∼
+��
+n
+|Y (n)(τ)|k
+�1/k
+.
+(5)
+If we restrict ourselves to two level systems (qubits), Fk(τ) is a natural distance in SU(2n), such that d =
+� t
+0 dτ Fk(τ),
+Cf. with Eq. (3). What has been explained so far is the continuous version of complexity, that provides a lower
+bound for the number of gates needed to approximate |ψT ⟩ from |ψR⟩ [1]. The discrete version of CN can be computed
+introducing the functional (using the same notation as in the original [1]):
+F(τ) =
+�
+�
+�
+�
+′
+�
+σ
+hσ(τ)2 + p2
+′′
+�
+σ
+hσ(τ)2
+(6)
+where the Hamiltonian in this case is H(τ) = �′
+σ hσ(τ)σ + �′′
+σ hσ(τ)σ. In the first sum, σ ranges over all possible
+one- and two-body interactions, that is, over all products of either one or two qubit gates. In the second sum, instead,
+the sum is over other tensor products of Pauli matrices and the identity. The factor p > 0 penalizes three, four, ...
+-body interactions. All put together, finding the geodesics in the continuum version is a good estimate of the resources
+needed to prepare a state.
+At this point, we think it is necessary to emphasise something. If any unitary is possible, the complexity has a
+narrow utility, since its value is given by C = arccos(|⟨ψR|ψt⟩|), i.e. of the order of one (it doesn’t matter which state
+reference and destination are chosen). This can be verified by noting that the target state can always be written as
+|ψT ⟩ = cos θ|ψR⟩ + eiγ sin θ|ψ⊥
+R⟩ with ⟨ψR|ψ⊥
+R⟩ = 0. A rotation in the subspace generated by {|ψR⟩, |ψ⊥
+R⟩} does the
+job. Therefore, some restrictions on the possible unitaries or Hamiltonian (4) will be imposed. We will discuss this
+point in some depth later.
+1 Notice that due to homogeneity, w.l.o.g. we can always set t = 1.
+
+4
+2.
+Circuit Complexity from the Fubini-Study metric
+The functionals F discussed so far, see Eqs. (4) and (5), are not unique. Others can be chosen satisfying continuity,
+homogeneity, positivity and the triangular property. We want to discuss next another possibility where the distance
+between the reference and target states is given by the Fubini-Study metric. Originally proposed for Quantum Field
+Theories in [5], we prefer to study it here from a quantum information perspective. Let us time-slice the unitary (1)
+such that
+|ψT (λ)⟩ = Uλ(t, tN−1)...Uλ(t1, t0)|ψR(λ)⟩ → |ψ(λ; tn)⟩ = U(tn, tn−1)|ψ(λ; tn−1)⟩
+(7)
+We have assumed that the unitaries and so the wave functions depend on the parameters λ. Then, for sufficiently
+small time step δτ := tn − tn−1, the fidelity between two contiguous states is
+Fn,n+1 ≡ |⟨ψ(λ; tn)|ψ(λ; tn−1)⟩| = 1 − χF δτ 2 + O(δ4)
+(8)
+where χF is denoted the fidelity susceptibility [36–41], see Ref. [42] for a review. Interestingly enough, we can relate
+χF with the geometric tensor, in fact [Cf. Eq. (11)]
+χF = gµν ˙λµ ˙λν
+(9)
+with [5, 43]
+gµν = Re (Tµν) .
+(10)
+Here, Tµν is the quantum geometric tensor which is nothing but the Fubini-Study metric (FSM) on the CP n manifold,
+namely:
+Tµν = ⟨∂λµψ|P|∂λνψ⟩
+(11)
+with P = 1 − |ψ⟩⟨ψ| 2.
+Another useful way of writing the metric tensor is as follows. Using a formal Taylor expansion for the states, the
+metric tensor can be written as,
+gµν = 1
+2 (⟨∂µψ|∂νψ⟩ − ⟨∂µψ|ψ⟩⟨ψ|∂νψ⟩ + c.c.)
+(12)
+Setting now in the Hamiltonian (4), ˙λν ≡ Y (ν) then |∂ν⟩ = Oν|ψ⟩, it is straightforward to see that [6],
+gµν = 1
+2 ⟨ψ |{Oµ, Oν}| ψ⟩ − ⟨ψ) |Oµ| ψ⟩ ⟨ψ |Oν| ψ⟩ = 1
+2
+��ψ|⟨
+�
+Oµ − ⟨Oµ⟩λ , Oν − ⟨Oν⟩λ
+��� ψ⟩ ,
+(13)
+i.e. the fluctuations of the Hamiltonian operators Oν.
+Using the fact that 1−Fn,n+1 is a distance, thus satisfying the properties we imposed for the F-functional, we have
+that we can understand C as the distance defined through the Fubini-Study metric:
+CFS := min{U}
+� t
+0
+�
+gµν ˙λµ ˙λν dτ .
+(14)
+The suffix stands for Fubini-Study metric and the notion of distance is quite explicit. This is an alternative definition
+to that given by Eq. (3) that has some remarkable properties. The first one is that knowing the metric tensor the
+geodesics can be found, at least in principle, by solving the differential equation:
+d2λµ
+dτ 2 + Γµ
+νρ
+dλν
+dτ
+dλρ
+dτ = 0
+(15)
+Here, Γ are the Christoffel symbols:
+Γµ
+νρ = 1
+2gµξ (∂ρgξν + ∂νgξρ − ∂ξgνρ) .
+(16)
+The second property of CFS is that, from its relation to the fidelity between states, F, its properties close to a QPT
+can be used when discussing the complexity, C, see also Eq. (13).
+2 Notice that the imaginary part of T is nothing but the Berry phase.
+
+5
+3.
+Some remarks comparing both approaches
+The complexity according to Nielsen estimates the minimum number of gates needed to reach the target state. For
+this purpose, a metric in the space of quantum circuits or unitary transformations is defined. The optimization of
+the trajectory in this space thus minimizes the number of accessible gates. On the other hand, with the Fubini-Study
+metric, the complexity is computed by monitoring the state changes along the preparation of the target state. The
+Fubini-Study metric defines a geometry in the space of states. A key difference is that in the latter geometry a variable
+cost is assigned to specific gates as it depends on the states they act on, while in the former, each gate is assigned a
+fixed cost. On top of that, with CN there will be degenerate operations that leave the state unchanged (e.g. adding a
+global phase). Therefore and in general, it is found that the space of unitaries has a higher dimension.
+In general, different results are obtained using both approaches [6, 44]. Depending on the application, one form is
+preferred over the other. We believe that due to the equations of the geodesics given the metric, CFS is ideal for doing
+analytical calculations while, from the point of view of quantum computation and cost estimation to prepare states,
+CN will prevail. In any case, in some particular cases it has been shown that both methods give identical results, such
+as the preparation of Gaussian fundamental states [6].
+As a final remark, let us note that typically, the fidelity, no matter how close two quantum states are, drops
+exponentially with system size. This is nothing but the Anderson orthogonality catastrophe. Therefore, in numerical
+studies a variant of CFS would be to use the fidelity per site instead,
+log f ≡ 1
+N log F ,
+(17)
+in Eq. (8). On top of that, this allows to extract the extensive part for the complexity which, on the other hand, is
+what seems to matter [5].
+B.
+Main results and manuscript organization
+For the exactly solvable systems that we discuss in this work, we find that CFS ≥ CN when crossing a phase
+transition. We understand this inequality as a consequence of the fact that the unitary space is larger, see previous
+subsection I A 3. In any case, C does not diverge at the critical point, its derivative does. For CFS we can characterize
+this divergence and its critical exponents in rather general circumstances. Let us remark, again, that throughout the
+paper we focus on the extensive part of C. Two models are studied in detail, namely the one-dimensional quantum
+Ising and Dicke models.
+After this general discussion, we focus on calculating the complexity when preparing a fundamental state in a quantum
+computer. Here, obviously, we compute CN in its discrete version. We explore two algorithms in detail. First, we
+discuss the circuit complexity in adiabatic algorithms (with and without shortcuts to adiabaticity). Here, the adiabatic
+path crosses a QPT explicitly and the complexity grows around it. There is not much difference (with respect to the
+CN) in using shortcuts. Then, we discuss the circuit complexity using VQEs. These algorithms are variational and
+do not need to cross the critical point even if the reference and target are in different phases. In such a case, CN is
+not necessarily aware of the QPT. On the other hand, if the target state is close enough to a phase transition, also in
+VQEs, the complexity grows.
+The rest of the manuscript is organized as follows. In the next section, II, we discuss the relation between circuit
+complexity, in this case CFS from Eq. (14), and the geometry of quantum states that allows extracting the critical
+exponents for the derivative of C. This is our first result that emphasizes that through phase transitions CFS has
+universal properties. In section III we perform explicit calculations for CFS and CN in two solvable systems, namely
+the one dimensional XY-anisotropic and Dicke models. We extract the critical exponents. Then, in section IV, we
+perform numerical simulations where CN is computed in two types of algorithms: variational and adiabatic ones.
+Concretely we benchmark with exactly solvable models as the Ising model, and we complement our discussion with
+non-integrable Hamiltonians as the ZZXZ model. Lastly, we discuss these results and conclude the paper in V. Some
+technical issues are left for the Appendices. The code used to obtain the numerical results is available upon request.
+II.
+COMPLEXITY AND THE GEOMETRY OF STATES CLOSE TO A QUANTUM PHASE
+TRANSITION.
+In this section, we discuss general aspects for the complexity close to a QPT. To be as general as possible, we find
+it convenient to focus on CFS, Eq. (14). Within this geometric formalism, we see that, in general, the complexity is
+
+6
+finite, but not its derivative, which can diverge when crossing a QPT. We study its finite size scaling obtaining the
+corresponding critical exponents.
+A.
+Complexity and its derivative when crossing a QPT
+We have already argued in section I A 1 that if we are allowed to use any unitary, C is of the order of one. In the liter-
+ature, several unitary restrictions have been used: considering one and two qubit gates or considering gaussian states
+when moving from reference to target states. In this subsection, we consider another kind of restriction, quite natural
+when talking about a QPT. We will consider that one (and only one) parameter, say λ, of the Hamiltonian model is
+varied to pass through the QPT, keeping other variables or parameters fixed. Thus the metric is unidimensional. We
+know, that in this case, the geodesic is given by:
+gλλ ˙λ2 = cte
+(18)
+Therefore,
+C = min
+λ(τ)
+� T
+0
+√gλλ ˙λ dτ ∼ T .
+(19)
+Below, we will work some examples and we will see that T does not diverge at the QPT. However, if we compute the
+derivative instead:
+∂C
+∂λ = √gλλ .
+(20)
+It is known that some components of the metric tensor can diverge, thus diverging the derivative of C. Equation (20)
+has two consequences. The first one is that, under quite general circumstances, the derivative of C close to a QPT is
+related to the metric tensor and inherits its universal properties. The second one is that this derivative can be used
+to witness and characterize QPTs.
+B.
+Finite size scaling
+Close to a critical point correlation length diverges as,
+ξ ∼ |λ − λc|−1/a ,
+(21)
+with a a critical exponent. Similar relations occur for other thermodynamic quantities. In particular, and for what
+interests us, the metric tensor can be written as [45],
+gµν ∼ |λ − λc|∆µν/a ,
+(22)
+with ∆µν the corresponding critical exponent. Notice that, for the reasons already explained in section I A 3, from
+now on we will be interested in the intensive part of the metric tensor gµν → gµν/Ld, with d the spatial dimensions.
+Near a phase transition, finite-size scaling dictates how quantities behave after scale transformations. Very briefly,
+after a length scale transformation x′ = αx, time scales as τ ′ = αzτ, with z its critical exponent. This fixes the energy
+fluctuations ∆E∆τ ∼ 1 → ∆E′ = α−z∆E. Putting it all together, it is interesting to extract the value of the critical
+exponent ∆µν above, which controls how the metric tensor behaves, in terms of other critical exponents that dictate
+more fundamental quantities. Looking at equations (4) and (12) and (13) and writing the scaling for the derivatives
+of the Hamiltonian as ∂µ′H′ = α−∆µ∂µH we arrive to [45],
+∆µν = ∆ν + ∆µ − 2z − d .
+(23)
+Finally, merging, (21) and (22), we find that close enough to the transition, where the relevant length is given by the
+system size, L, we arrive to
+gµν ∼ L−∆µν .
+(24)
+As a consequence of all of this and using (20), when a single parameter is varied across the QPT we have the scaling:
+∂C
+∂λ ∼ L−∆λλ/2 .
+(25)
+
+7
+It is remarkable that the complexity derivative scaling is dictated by universal exponents, whenever one parameter is
+varied to cross a critical point. In particular, if ∆λλ > −2 the derivative is sub-extensive. If ∆λλ = −2 it is extensive
+and if ∆λλ < −2 is superextensive.
+III.
+SOLVABLE HAMILTONIANS
+Let us test the above ideas on a couple of solvable models: the one dimensional quantum Ising model [46] and the
+Dicke [47–49] model.
+A.
+Quantum Ising model
+The transverse field Ising model (Periodic Boundary Conditions will be assumed) is
+H = −J
+L
+�
+j=1
+σz
+j σz
+j+1 +
+L
+�
+j=1
+σx
+j .
+(26)
+Hamiltonian (26) can be solved via the Jordan-Wigner transformation [46]. This Ising model has a second order phase
+transition occurring at Jc = 1(−1) in the N → ∞ limit. For Jc > 1(Jc < −1) the Z2 symmetry is spontaneously broken
+and the g.s. is ferromagnetically (antiferromagnetically) ordered. W.l.o.g. we fix our attention in the paramagnetic-
+ferromagnetic transition occurring at Jc = 1. On top of that, the ground state can be written in terms of fermionic
+excitations (after the Jordan-Wigner transformation) as,
+|ψgs⟩ =
+�
+k>0
+�
+cos(θk/2) + ieiφ sin(θk/2) a†
+ka†
+−k
+�
+|0⟩ .
+(27)
+with k = (2m−1)π
+L
+3 and,
+tan θk =
+−J sin k
+1 + J cos k .
+(28)
+For the rest of the section the metric tensor (12) is needed. It has been computed several times already [50, 51]
+gJJ = 1
+4
+�
+k
+�∂θk
+∂h
+�2
+.
+(29)
+In the thermodynamic limit, the k-sum is an integral �
+k → N/π
+�
+and it can be computed explicitly, yielding
+gJJ =
+−π(J2 − 1) + i
+�
+J2 + 1
+� �
+log
+�
+− 2i(J+1)
+J−1
+�
+− log
+�
+2i(J+1)
+J−1
+��
+32J2(J2 − 1)
+.
+(30)
+1.
+Complexity through QPTs
+From Eq. (30) we see that gJJ diverges at J = Jc. This is the reason behind the divergence in the derivative of the
+complexity at the QPT, Cf. Eq. (20). In figure 6, we plot CFS both in the continuum and for N-finite using either
+(30) or the sum (29). In both cases, the integral (19) is computed. It is evident that the complexity does not diverge
+at the QPT, but its derivative does, inheriting this behaviour from the metric tensor, Cf. Figs. 6a and b. For the
+Ising transition, the exponent a = 1, Cf. Eq. (21). We know that ∆hh/a = 1, so the complexity derivative diverges
+as ∼ L1/2 at the Ising transition.
+3 We have used even L and periodic boundary conditions.
+
+8
+0.9
+1.0
+1.1
+J
+0.0
+0.2
+0.4
+0.6
+FS
+(a)
+L = 100
+L = 1000
+L = 2000
+0.9
+1.0
+1.1
+J
+0
+50
+100
+150
+FS
+(b)
+L = 100
+L = 1000
+L = 2000
+log(L)
+log|(
+FS)max|
+= 0.499
+(c)
+(
+FS)max data
+fit to (
+FS)max(L) = A L + B
+0.9
+1.0
+1.1
+J
+0.0
+0.1
+0.2
+0.3
+N
+(d)
+L = 100
+L = 1000
+L = 2000
+0.9
+1.0
+1.1
+J
+0.0
+1.5
+3.0
+4.5
+N
+(e)
+L = 100
+L = 1000
+L = 2000
+L
+exp ((
+N)max)
+(f)
+(
+N)max data
+fit to (
+N)max(L) = A log(L) + B
+FIG. 1. Study of the complexity for the Transverse Field Ising model. (a) Complexity for different sizes of the chain computed
+using the Fubini-Study metric. The discretization in J used is δJ = 1e−3. (b) Derivative of the Fubini-Study complexity for
+different L, δJ = 2e−3. (c) Finite size scaling of the maximum in the derivative of the Fubini-Study complexity. See that this
+maximum diverges polynomially with the size of the chain. (d) Study of the Nielsen complexity, δJ = 3e−4. (e) Derivative
+of the Nielsen complexity for different L, δJ = 3e−4. (f) Finite size scaling of the maximum in the derivative of the Nielsen
+complexity. See that this maximum diverges logarithmically.
+2.
+Relation between CN and CFS
+Formula (27) is formally equivalent to the ground state for the 1D-Kitaev model. For the latter, CN has been
+computed in [18]. If the reference, target and intermediate states have the same form (27), CN reads:
+CN =
+�
+k
+|∆θk|2
+(31)
+where ∆θk = θT
+k − θR
+k and θT
+k (θR
+k ) are the angles (28) at the target (reference) states. Following the same procedure
+as in [18] we checked that ∂JCN ∼ log N, i.e. it diverges logarithmically. This must be confronted with the divergence
+(with critical exponent 1/2) for the case of ∂JCFS. This is an important difference. While using the FS distance the
+complexity is associated with the fluctuations, cf. Eq. (13), the CN is more related to the angles difference and its
+divergence is therefore smoothed.
+B.
+The Dicke model
+The Hamiltonian for the ground state sector of the N-spin Dicke model can be written in terms of total spin
+operators of spin S = N/2 as [52]
+H = ωca†a + ωsSz +
+λ
+√
+2S
+�
+a† + a
+�
+(S+ + S−) ,
+(32)
+
+9
+where the spin and ladder operators obey the canonical commutation relations [Sz, S±] = ±S±, [S+, S−] = 2Sz. This
+model can be solved in the thermodynamic limit, S → ∞, with a Holstein-Primakoff transformation on the spins
+S+ →
+√
+2Sb†
+�
+1 − b†b
+2S ,
+(33)
+S− →
+√
+2S
+�
+1 − b†b
+2S b ,
+(34)
+Sz → b†b − S ,
+(35)
+(36)
+yielding
+H = ωca†a + ω†
+sa + λ
+�
+a† + a
+�
+�
+b†
+�
+1 − b†b
+2S +
+�
+1 − b†b
+2S b
+�
+− ωcS .
+(37)
+In the normal phase of the Dicke model we can obtain an effective Hamiltonian for S → ∞ by neglecting terms
+with 2S in the denominator in the Hamiltonian of Eq. (37), resulting in a completely symmetric model of coupled
+harmonic oscillators, one corresponding to the physical oscillator and the other corresponding to the spins within the
+Holstein-Primakoff transformation
+H = ωca†a + ω†
+sa + λ
+�
+a† + a
+� �
+b† + b
+�
+− ωcS .
+(38)
+In the superradiant phase, the bosonic modes must be displaced to accommodate the macroscopic occupations that
+the spins and field develop in this phase. Once the displacements are introduced, terms with powers of 2S in the
+denominator are again neglected in the thermodynamic limit, yielding
+H = ωc¯a†¯a + ωs
+2µ(1 + µ)¯b†¯b + ωs(1 − µ)(3 + µ)
+8µ(1 + µ)
+�¯b† + ¯b
+�2 + λµ
+�
+2
+1 + µ
+�
+¯a† + ¯a
+� �¯b† + ¯b
+�
+,
+(39)
+where µ = ωzΩ/
+�
+4λ2�
+and ¯a,¯b are the displaced bosonic operators [53]. We omit the expressions of the displacement
+as they are irrelevant in the following. Both the normal and superradiant effective Hamiltonians can be diagonalized
+in the real space basis, where they present a gaussian profile given by
+g(x, y) =
+�ϵ+ϵ−
+π2
+�1/4
+e− (R,AR)
+2
+,
+(40)
+where R = (x, y), x and y are the real-space coordinates associated to the modes a(¯a) and b(¯b), A = U −1MU with
+U a unitary matrix, M = diag [ϵ−, ϵ+] and ϵ± are the eigenmodes of the system [54]. The overlap of two different
+ground states is given by
+⟨g|g′⟩ = 2 [det M det M ′]1/4
+[det (M + M ′)]1/2 .
+(41)
+This allows us to compute the components of the quantum metric tensor for the Dicke model exactly in the thermody-
+namic limit. We combine this with finite size results from exact diagonalization of Hamiltonian (32). The results are
+shown in Fig. 2. Just like we showed for the case of the Ising model, there is no divergence in CFS, the only signature
+of the phase transition is a non-analiticity that is only noticeable in the N → ∞ case. This non-analiticity, or its
+precursor in the case of finite N is best revealed as a divergence in the derivative of the complexity, which is naturally
+the square root of the metric tensor. Here we are considering the complexity along a λ-path and the divergence is
+revealed in ∂CFS = √gλλ. We perform a finite-size scaling analysis of the metric tensor by fitting the maximal values
+(∂CFS)max(N) and critical parameters at said maxima λmax(N) to their respective scaling laws
+|(∂CFS)max(N) − B| = C · N δ ,
+(42)
+|λmax(N) − λc| = A · N −ν .
+(43)
+The resulting critical exponents ν = 0.655(22) ≊ 2/3 and δ = 0.6711(15) ≊ 2/3 are in agreement with values reported
+in the literature [55].
+
+10
+0.4
+0.6
+0
+1
+2
+3
+4
+FS(0
+)
+(a)
+N
+50
+100
+150
+200
+N
+0.4
+0.6
+100
+101
+102
+103
+FS
+(b)
+log|
+max(N)
+c|
+=  0.655(22)
+(c)
+max(N) data
+fit to |
+max(N)
+c| = A N
+log(N)
+log|(
+FS)max(N)
+B|
+=  0.6711(15)
+(d)
+(
+FS)max(N) data
+fit to |(
+FS)max(N)
+B| = C N
+FIG. 2. Fubini-Study complexity (a) and its derivative with respect to λ (b) across the phase transition of the Dicke model
+as a function of the system size (numerical results) and in the thermodynamic limit (analytical results). Plots on the right
+showcase the fits of λmax(N) (c) and (∂CFS)max(N) (d) (extracted from center plot) to their respective finite size scaling laws.
+All results are at resonance ωc = ωs = 1 and with a discretization of dλ = 10−3. Numerical results were obtained with a cutoff
+for bosonic excitations of Nexc = 30 .
+IV.
+COMPLEXITY IN A QUANTUM COMPUTER, THE CASE OF GROUND STATE PREPARATION
+In this section, we compute CN when preparing ground states in a quantum computer. We study both adiabatic
+algorithms and variational quantum eigensolvers (VQEs). Two versions of the former algorithms are discussed: with
+and without shortcuts to adiabaticity.
+In both cases, the initial state is the “trivial zero” |0⟩ ≡ |00 · · · 0⟩4. Some gates are applied to prepare the ground
+state of a given Hamiltonian. Here, we are especially interested when this initial state (that can be understood as the
+ground state in the paramagnetic phase) is in a different phase than the final one. In addition, we discuss whether or
+not a QPT is crossed during the algorithm. Finally, notice that in quantum computing applications it seems natural
+to compute CN and, in particular, its discrete version (the number of gates needed), Cf. Sec. I A 1. Thus, through
+this section, we compute CN.
+A.
+Adiabatic algorithms
+A systematic way of finding the ground state of a given Hamiltonian is by adiabatic passage or annealing. Let us
+consider the time-dependent Hamiltonian:
+H(t) = (1 − λ(t))H0 + λ(t)HT .
+(44)
+Here, H0 has a trivial ground state (easy to prepare), and HT is the hamiltonian from which we want to obtain its
+ground state. Consider that the time-dependent function λ(t) runs from λ(t = 0) = 0 to λ(t = tf) = 1, where tf is
+the final time of the algorithm. At t = 0 the state is prepared in the ground state of H0. If ˙λ is sufficiently small
+compared to the instantaneous gap, by means of the adiabatic theorem the final state is the ground state of HT
+[32]. On the other hand, the adiabatic condition alerts us that as the gap closes, for example in continuous phase
+transitions, the execution time, i.e. the circuit depth, scales with the inverse of this gap, thus also C.
+Importantly enough the adiabatic condition can be relaxed by introducing counter-diabatic terms.
+Generally
+speaking, instead of H(τ) (whose ground states are |ψ(tn)⟩) what is evolved is the “modified” Hamiltonian [56, 57]:
+H′(τ) = H(τ) + HCD(τ)
+(45)
+4 In fact, in almost all algorithms the initial state seems to be |00 · · · 0⟩.
+
+11
+The last term ensures that the time evolution exactly matches the instantaneous ground state of H(τ) no matter how
+fast the evolution is. This is known in the literature as shortcuts to adiabaticity and HCD is called counter-diabatic
+Hamiltonian. There are different ways of writing HCD. In its original form we can write:
+HCD(τ) = i ˙λµ � ⟨m|∂µH|n⟩
+En − Em
+|m⟩⟨n| + h.c.
+(46)
+with ∂µH ≡ ∂H/∂λµ. We emphasize that at times 0 and t, |ψR⟩ and |ψT ⟩ are ground states of H(0) and H(t)
+respectively. Explicitly |ψT ⟩ = T e−
+� t
+0 H′(τ) dτ|ψR⟩. Here τ means time, cf. with Eq (1). To connect this evolution
+with the previous sections, we note that the fidelity susceptibility can be written in terms of the HCD(τ) fluctuations
+[Cf. Eq. (9) and (13)]:
+χF = ⟨(H(τ) + HCD(τ))2⟩ − ⟨(H(τ) + HCD(τ))⟩2 = ⟨HCD(τ)2⟩ = ˙λµ ˙λν gµν .
+(47)
+In practice HCD is difficult to find. Therefore, a systematic although approximate way of writing is convenient.
+Following [58] it can be rewritten as,
+HCD(τ) = ˙λµAµ
+(48)
+Here, A is the adiabatic gauge potential that can be approximated as:
+A(ℓ)
+µ
+= i
+ℓ
+�
+k=1
+αk [H, [H, . . . [H
+�
+��
+�
+2k−1
+, ∂µH]]]
+(49)
+where (l) is the “degree of approximation”. On top of that, the {αk} are found variationally by minimising the action
+[59]:
+Sℓ = Tr
+�
+G2
+ℓ
+�
+,
+Gℓ = ˙λµ �
+∂µH − i
+�
+H, A(ℓ)
+µ
+��
+(50)
+In many cases of interest, in the adiabatic protocol, H(τ) is expected to be a local Hamiltonian, in particular a
+two body one. Notice that due to nested commutators, the higher the order (l), the longer the range of interaction.
+Following the functional (6) three, four, or higher order body interactions will be highly penalised. Thus, in what
+follows, we will restrict ourselves to l = 1 that introduces two body interactions at most. This, in turn, provides a
+systematic way of preparing, via trotterization, quantum states.
+1.
+Complexity in adiabatic algorithms
+As has been previously discussed, in order to compute the complexity as defined by Nielsen [Cf. Sec.(I A 1)], we
+only need to express our unitary operation as the time evolution of some Hamiltonian. In the present case, it is
+straightforward, with and without shortcuts, as the Hamiltonian (45) is given explicitly. We study the Ising model in
+transverse field and the ZZXZ model. For both, we use the function λ(t) = sin2 � π
+2 sin2 � πt
+2T
+��
+to drive the evolution
+from the initial Hamiltonian, H0, to the target one, HT.
+For the Ising model, we start from H0 = hx
+�
+i σx
+i , leaving the transverse field fixed and switching on the spin-spin
+interaction until we reach Hint = J �
+i σz
+i σz
+i+1. The counter-diabatic Hamiltonian follows from equations (46), (49)
+and (50) with l = 1 yielding HCD(t) = ˙λ(t)α(t) �
+i(σy
+i σz
+i+1 + σz
+i σy
+i+1), where α(t) is the variational parameter in Eqs.
+(49) and (50). The full-time-dependent Hamiltonian reads
+H′(t) = H0 + λ(t)Hint + HCD(t) .
+(51)
+Since we need to implement our unitary evolution via Trotter decomposition, we have to split the total time T in
+T/δT steps, where δT is the time discretization employed. The smaller δT, the more precise the implementation
+will be but more gates will be needed, increasing the complexity of the operation. In the simulations presented here
+δT = min(0.1, T/30).
+Therefore, following the prescription given for the Nielsen complexity, CN, given in (6), it is straightforward to obtain5:
+CN =
+� T
+0
+dt
+� T
+δT
+�1/2 �
+N + (N − 1)λ(t)2J2 + 2(N − 1) ˙λ2(t)α2(t)
+�1/2
+(52)
+5 For the adiabatic evolution without shortcuts, the expression would be the same but with α = 0.
+
+12
+10
+1
+100
+101
+102
+T
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+|J| = 0.25
+|J| = 0.75
+|J| = 1.00
+|J| = 1.25
+|J| = 1.75
+0.00
+0.25
+0.50
+0.75
+1.00
+t/T
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+|E1
+E0|
+(b)
+0.25
+1.00
+1.75
+|J|
+50
+100
+150
+200
+250
+/L
+(c)
+L = 6
+L = 8
+L = 10
+L = 12
+L = 14
+FIG. 3. Complexity study for the Transverse Field Ising model using the adiabatic algorithm. (a) Evolution of the fidelity
+obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing time lengths of the full
+algorithm and different target Js for L = 12. At shorter times the shortcuts provide better results, being identical to the
+simple case (without shortcuts) for the longest times. (b) Evolution of the gap between the ground state and the first excited
+state during the algorithm for the same values of J as in (a). The gap closes with an increasing value of |J|, explaining why
+longer times are needed for the larger |J| to obtain the same fidelity. (c) Complexity per spin computed for different sizes
+with shortcuts (solid lines) and without shortcuts (dashed). As the gap closes, more gates are needed to achieve the fidelity
+threshold (0.9 in this case) but we do not find relevant differences between applying shortcuts or not in the final result for the
+complexity.
+Figure 3 summarises our results for the Ising model with adiabatic algorithms. The transverse field was fixed to
+hx = 1 and different values of J were studied. In Figure 3a we plot the fidelity between the final state obtained
+adiabatically and the target state. As expected, the longer the time the better. We also confirm that at lower times
+higher fidelities are achieved thanks to the counter-diabatic term. Figure 3b shows the gap evolution within the
+adiabatic algorithm, giving insights about why as |J| is greater, it takes more time to achieve a high fidelity: the
+gap becomes smaller. Finally, the last panel 3c shows the actual Nielsen complexity values. Reflecting the fidelity
+behaviour, the complexity jumps around the transition as the gap is closing. As the adiabatic theorem states, crossing
+a QPT is hard for this kind of algorithms and complexity serves the purpose of quantifying such difficulty.
+In order to check if this holds in other models, we also study the so-called antiferromagnetic ZZXZ model:
+HT = J
+�
+i
+σz
+i σz
+i+1 + hx
+�
+i
+σx
+i + hz
+�
+i
+σz
+i .
+(53)
+Due to the combination of longitudinal and transverse fields, this is a non-integrable model. It is ideal, then, to
+explore the phenomenology of complexity beyond the exactly solvable models considered so far. In Fig. 4 we draw
+the phase diagram of the model at zero temperature as a function of the fields applied to the spins and the exchange
+constant [60]. The critical line separates paramagnetic and antiferromagnetic phases. For our particular purposes,
+keeping the same initial Hamiltonian, H0, we set the transverse field, hx = 1 and the target longitudinal field to
+hz = 0.75. We thus study the quantum phase transition appearing when moving to different target values of J. This
+path is shown as the red line in Fig. 4, where the final point marks the maximum value simulated for the target
+J. Therefore, the transverse field is going to be fixed while we turn on both the longitudinal field and the magnetic
+interaction. The counter-diabatic term can be computed in the same fashion as before, getting the same result as in
+[58]. The time-dependent Hamiltonian reads
+H′(t) = H0 + λ(t)
+�
+i
+�
+Jσz
+i σz
+i+1 + hzσz
+i
+�
++ HCD(t)
+(54)
+and the complexity acquires the following expression
+CN =
+� T
+0
+dt
+� T
+δT
+�1/2 �
+N
+�
+1 + h2
+zλ2(t)
+�
++ (N − 1)��(t)2J2 + ˙λ2(t)
+�
+Nα2(t) + 2(N − 1)
+�
+β2(t) + γ2(t)
+���1/2
+.
+(55)
+
+13
+0.0
+0.5
+1.0
+1.5
+hx / J
+0.0
+1.0
+2.0
+hz / J
+AFM
+PM
+FIG. 4. Phase diagram of the ZZXZ model for zero temperature. The black dotted line signals the critical region between
+phases for different ratios of the fields (hx, hz) to the magnitude of the exchange interaction (J). The coloured lines depict the
+path followed for the adiabatic algorithm (red) and the values computed in the VQE (blue) [Cf. Sec. IV B].
+In figure 5 we show the results obtained for the different values of J and the chain sizes, N. The behaviour is equiva-
+lent to the previous model except that for sufficiently large values of J, the gap decreases sharply, closing completely
+(see figure 5b), causing the counter-diabatic terms to cause more error than the simple evolution itself, as we can see
+in panel (a) of the same figure. This is a consequence of the fact that our expression for the counter-diabatic term is
+not exact, but a first-order approximation of a general expression [cf Eq. (49)]. The smaller the gap, the more careful
+we will have to be with the design of the CD term.
+Putting all together, we can conclude that, due to the gap closing at the QPT the CN with adiabatic algorithms
+diverges with system size. The inclusion of shortcuts does not provide any significant advantage in terms of complexity
+reduction. This is because we have constrained these shortcuts to be as local as possible, in our case l = 1 in (50),
+introducing two body interactions at much. It is expected that by introducing long-range terms in (46) the complexity
+decreases as the system approaches to the QPT. This can be compared to the previous section II, where there was no
+restriction to local operations, so both CF and CN remained finite despite crossing the QPT. Other paths investigated
+in this work are sent to App. (B).
+B.
+Circuit Complexity in VQEs
+VQEs, introduced in [34], use the fact that any quantum state can be written in terms of a unitary operation as
+|φ(⃗θ)⟩ = U(⃗θ)|0⟩ ,
+(56)
+where U(⃗θ) is a parameterized unitary that transforms the initial state into the desired wave function |φ(⃗θ)⟩. This
+unitary can be implemented in a quantum circuit as a set of quantum gates. The expectation value of the Hamiltonian
+where we encode our problem (H) results
+⟨H⟩ = ⟨0|U †(⃗θ)HU(⃗θ)|0⟩ ≥ E0 .
+(57)
+The optimization process consists on minimizing the average energy of the parameterized state:
+EVQE = min
+θ
+⟨0|U(⃗θ)†HU(⃗θ)|0⟩ ≥ E0 .
+(58)
+
+14
+10
+1
+100
+101
+102
+T
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+J = 0.25
+J = 0.75
+J = 1.00
+J = 2.00
+J = 5.00
+0.00
+0.25
+0.50
+0.75
+1.00
+t/T
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+|E1
+E0|
+(b)
+10
+2
+10
+1
+100
+J
+102
+103
+/L
+(c)
+L = 6
+L = 8
+L = 10
+L = 12
+L = 14
+FIG. 5. Complexity study for the ZZXZ model using the adiabatic algorithm. The phenomenology is essentially the same as
+for the TFI model. (a) Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed
+lines) for increasing time lengths of the full algorithm and L = 12. In this case we see that, for sufficiently large values of J, no
+applying shortcuts works better than applying them. This is explained by the gap closing much more abruptly than in the TFI
+model, as can be seen in (b). (c) Complexity per qubit computed for different sizes with shortcuts (solid lines) and without
+shorcuts (dashed). As the gap closes, more gates are needed to achieve the fidelity threshold (0.9 in this case).
+The algorithm can be divided into three different stages. First, we need to choose the trial wave function (see
+Eq.(56)). Choosing the unitary U(⃗θ) is equivalent to constructing the quantum circuit that transforms the initial
+state into the parameterized wave function.
+The circuit used to achieve |φ(⃗θ)⟩ is called the ansatz and can be
+represented as,
+q0 :
+U(⃗θ)
+q1 :
+q2 :
+q3 :
+q4 :
+(59)
+Choosing an appropriate ansatz is crucial for the optimization process. This choice depends completely on the
+model we are simulating and the set of gates available. We will dig into our choice of unitary below. The next step
+is constructing the Hamiltonian of the problem. Since this Hamiltonian is going to be evaluated later, Eq. (57), it
+must be written in terms of Pauli strings {I, σx, σy, σz}⊗N. Pauli operators are related to spin observables, which
+are suitable for direct measurement in quantum devices [61]. With the Hamiltonian and the wave function defined,
+we can measure the energy of the state, which is the cost function. To compute this cost function, the expectation
+values of the Pauli observables are measured determining the value of the energy. Since the technique uses quantum
+and classical processors, VQEs are cast as hybrid algorithms. Our results are numerical and our Python code simply
+computes the product of the matrices U(⃗θ)†HU(⃗θ) previously defined and then projects onto the zero state obtaining
+⟨0|U(⃗θ)†HU(⃗θ)|0⟩. We will not discuss its measurement overhead. Here, we are interested in the circuit complexity
+for reaching the desired ground state.
+The final step is to minimize this cost function through the variation of the parameters θ in the wave function. At
+the end of each iteration we obtain the value of the energy (58). Then, a classical optimizer determines the best
+direction of variation of the parameter vector ⃗θ to minimize this value. We use as many iterations as needed until we
+converge to a final solution for the coordinates of the parameter vector. Ideally, this solution is the absolute minimum
+in the space of parameters. Still, obtaining this minimum is not an easy task. The optimizer can get trapped in
+local minima which will imply serious limitations in the minimization process. This problem and others have been
+previously discussed in the literature [61, 62] and are out of scope for this work.
+
+15
+Summarizing, we assume a given ansatz, the set of available gates in U(⃗θ) in (56) and the hybrid algorithm finds the
+optimal solution. CN counts the number of gates, and once the VQE circuit is chosen, it can be done systematically.
+1.
+Local VQE ansatz
+We focus on a fixed geometry that is suitable for one-dimensional systems with single and two-qubit gates, besides
+the two-qubit gates act only on contiguous qubits. This ansatz can be interpreted as a Trotter approximation of
+continuous evolution by a local 1D Hamiltonian [63]. In this case, we can separate the terms of the Hamiltonian that
+act on even and odd links and obtain two sets, each made of mutually commuting gates. In particular, the circuit is
+given by
+q0 :
+RY (θ[0])
+•
+•
+RZ (−π/2)
+q1 :
+RY (θ[1])
+RZ (π/2)
+RZ (−π/2)
+RY (θ[5])
+•
+•
+RZ (−π/2)
+q2 :
+RY (θ[2])
+•
+•
+RZ (−π/2)
+RY (θ[6])
+RZ (π/2)
+RZ (−π/2)
+q3 :
+RY (θ[3])
+RZ (π/2)
+RZ (−π/2)
+RY (θ[7])
+•
+•
+RZ (−π/2)
+q4 :
+RY (θ[4])
+RZ (π/2)
+RZ (−π/2)
+(60)
+i.e.
+it consists of fundamental blocks (or layers) (separated by dashed lines above).
+Each layer is made out of
+single-qubit rotations Ry(θ) and control-Z gates (CZ). At the end of the circuit, we add a final column of rotations
+(Ry).
+For computing CN we rewrite the CZ gates in terms of Pauli operators, count the gates and use equation (6). This
+is a routine process that we send to Appendix A. Here, we just give the final result:
+CN =
+d
+�
+j=1
+�
+�
+�
+�
+2(L−1)
+�
+i
+�
+θj
+i
+2
+�2
++ 3(L − 1)
+�π
+4
+�2
+.
+(61)
+2.
+VQE complexity through QPTs
+As before, we focus on Ising and ZZXZ models, Eqs. (26) and (53). In figure 6 we summarize our results for the
+Ising Hamiltonian. In panel a) we plot the complexity using the local VQE ansatz for obtaining the ground state at
+a given J. We see that CN grows when the ground state approaches the QPT, that in this case is given by Jc ∼= 1 6.
+In fact, close enough to the transition the VQE cannot reach an acceptable ground state for a maximum depth of 8
+(in our simulations). This can be checked in panel b) where the fidelity between the state obtained within the VQE
+algorithm and the exact ground state falls below 0.8 in the gray region of panel b). Therefore, all indications are that,
+also with VQE, complexity increases as QPT is approached. With what has been said so far this should not be a
+surprise. Perhaps, the most remarkable thing here is that the complexity is only high near the transition. When the
+target state is far from the critical point the complexity drops, even though the latter and the reference state may be
+in different phases. This is due to the fact that, contrary to the adiabatic algorithm, the VQE does not necessarily
+need to visit states in the transition region to go from |ψR⟩ to |ψT ⟩, it can circumvent criticality and go directly from
+one phase to another. This is easy to understand in the Ising model, because in the paramagnetic phase the ground
+state is approximately given by |+, ..., +⟩ (|+⟩ = 1/
+√
+2(|0⟩ + |1⟩)), Cf. Eq. (26). This is easy to prepare: it can be
+obtained with single qubit rotations from the reference state |ψR⟩ = |0, ..., 0⟩.
+Since we are dealing with finite simulations, deep in the ferromagnetic phase, the Z2 symmetry is not broken so
+the ground state manifold found by exact diagonalization is spanned by the states 1
+2 (|0, ..., 0⟩ ± |1, ..., 1⟩). The VQE
+6 We say Jc ∼
+= 1 since our simulations are done in finite systems. Jc = 1 in the thermodynamic limit.
+
+16
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+J
+1.0
+2.0
+3.0
+4.0
+/L
+(a)
+1
+2
+3
+4
+5
+6
+7
+8
+Layers
+0.5
+0.75
+1.0 (b)
+J = 0.90
+J = 0.92
+J = 0.94
+J = 0.96
+J = 0.98
+J = 1.00
+J = 1.02
+J = 1.04
+J = 1.06
+J = 1.08
+J = 1.10
+J = 1.30
+J = 1.50
+J = 1.70
+FIG. 6. Transverse Field Ising model with bias, ϵ = 0.001 and size N = 12. (a) Complexity per size as a function of J. The
+grey zone indicates that the VQE does not converge for points inside that region in a reasonable number of layers to the fidelity
+threshold (0.9). (b) Fidelity obtained for different numbers of layers for points inside the grey box in (a) and in its vicinity.
+For those points whose fidelity is above the threshold (0.9) it has only been plotted the best result for clarity’s sake.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+J
+1.0
+2.0
+3.0
+4.0
+/L
+(a)
+1
+2
+3
+4
+5
+Layers
+0.5
+0.75
+1.0 (b)
+J = 0.90
+J = 0.92
+J = 0.94
+J = 0.96
+J = 0.98
+J = 1.00
+J = 1.02
+J = 1.04
+J = 1.06
+J = 1.08
+J = 1.10
+J = 1.30
+J = 1.50
+J = 1.70
+J = 1.90
+J = 2.10
+J = 2.30
+J = 2.50
+FIG. 7. ZZXZ Ising model for size N = 12. (a) Complexity per size as a function of J. The grey zone, as in the TFI model,
+indicates that the algorithm fails to achieve fidelity over 0.9 for points within that region. (b) The fidelity behaviour with the
+depth of the ansatz shows that, again, once the QPT is crossed the algorithm cannot reach fidelities over 0.9. In contrast to
+the TFI model, here we don’t recover high fidelity once we are fully in the antiferromagnetic phase, reaching a maximum value
+of 0.5 for the highest values of J.
+reaches instead one of the fully polarized states, either |0, ..., 0⟩ or |1, ..., 1⟩, given that they are degenerate with the
+symmetric ground state. Our convergence criterion is based on reaching a fidelity of 0.9 between the state generated
+by the VQE and the result of exact diagonalization. Because of the discrepancy in the ground states obtained by
+both methods, in the ferromagnetic phase the fidelity is capped at 0.5 and the convergence criterion is never satisfied.
+Driven by the physics of actual QPTs in the thermodynamic limit, where the symmetry is (spontaneously) broken,
+we decide to add a small bias, ϵ � σz
+i in (26). In doing so, the VQE should a priori be able to reach full convergence.
+This is indeed the case as can be seen in Fig. 6. Additionally, convergence is reached in very few layers, equivalently
+to what is observed in the PM phase. This low complexity can be explained by noticing that the symmetry broken
+ferromagnetic ground state is either the reference state or can be obtained from it by means of single-qubit rotations.
+We now consider the ZZXZ model, Hamiltonian (53)7. Here, we are not going to explicitly break the symmetry
+7 The parameters employed in the simulations are depicted as the blue line in Fig. 4, namely hx = 1, hz = 0.75 and J ∈ (0., 2.5]
+
+17
+0
+1
+2
+J
+-0.5
+0.0
+0.5
+1.0
+Magnetization
+(a)
+Total
+Even sites
+Odd sites
+0
+1
+2
+J
+0.5
+0.75
+1.0 (b)
+Single state
+Subspace
+0
+1
+2
+J
+0.994
+1.0
+Energy accuracy
+(c)
+FIG. 8. VQE state characterization in the ZZXZ model for N = 12. (a) Magnetization of the spin chain as a function of J
+obtained from the states generated by VQE. The solid black line represents the total magnetization per site that the spin chain
+should have (obtained via exact diagonalization) whereas the dashed black line sets the magnetization per site in even/odd
+sites. (b) Evolution of the best fidelity obtained as a function of J. In blue it is computed the fidelity as the overlap between
+the state generated by the VQE and the exact ground state; in red it is computed as the projection onto the subspace generated
+by the ground state and the first excited state. (c) Energy accuracy obtained for the same configurations displayed in the other
+panels computed as 1 − Erel, being Erel the relative error between the energy obtained from VQE and the exact value.
+in order to discuss the scenario in which the symmetric ground state is sought.
+In the ZZXZ model, the QPT
+separates paramagnetic (PM) and antiferromagnetic (AFM) phases. In the PM phase, the behavior is analogous to
+the Ising model, Cf. Figs. 6 and 7. Deep in the AFM phase, the ground state manifold is spanned by the states
+|ψAFM⟩ ∼=
+1
+√
+2(|1, 0, 1, 0, ...⟩ ± |0, 1, 0, 1, ...⟩). Following the previous discussion, the VQE does not reach the symmetric
+ground state. Therefore, we see that CN grows as it approaches the phase transition (with our parameters Jc ∼= 1,
+see Fig. 4) but does not decrease afterwards. At some point near criticality, the VQE cannot produce a ground state
+with a fidelity larger than 0.9, see panel b), similar to the Ising model case. Here, however, the state remains difficult
+for the VQE after the near-transition region is surpassed. This is further confirmed in figure 8. There, we can see
+that although the total magnetization is well reproduced by the VQE (also the energy, in panel c), once we enter
+the antiferromagnetic phase the VQE generates either |1, 0, 1, 0, ...⟩ or |0, 1, 0, 1, ...⟩, as can be seen by computing the
+magnetization per site, which should be close to 1/2 in the exact ground state. However, the VQE gives 0 (1) for the
+even (odd) sites. To conclude our characterization, we see that all this is consistent with obtaining a F = 0.5, as well
+as a F ∼= 1 if we compare the state generated by the VQE with the projection onto the subspace generated by the
+ground state and the first excited state.
+V.
+DISCUSSION
+Knowing in advance how much a computation will cost, even if only approximately, is of great help. Unfortunately,
+this estimation can pose a great challenge. Computer science has traditionally categorized problems into different
+complexity classes, allowing one to know whether a given problem is tractable on a classical computer.
+For a
+quantum computer, we can ask a similar question to know if the task we want to tackle is going to be feasible with
+the architecture we have at hand. For this purpose, the concept of circuit complexity was invented. Again, knowing
+the complexity of each task in any architecture seems too general to be able to give a concrete answer. On the other
+hand, we can shed some light on generic situations where some kind of general statement can be made. This is the
+idea that motivated us to write this manuscript. We have studied the situation in which a critical region is crossed in
+the process of preparing a state.
+Our work has shown that, regardless of the type of complexity one chooses, and for diverse models, it appears
+that complexity grows if the algorithm visits states near a phase transition. We have further proven that this is a
+characteristic trait of typical algorithms for state preparation such as VQE and adiabatic evolution. The degree of
+divergence does depend on the definition of complexity used and on the allowed gates. In the case of local ans¨atze
+or evolutions, C tends to diverge as the system size grows. Importantly, we have shown that VQEs, to the extent
+that they can go “directly” from the reference to the target state, can potentially avoid the divergence in complexity
+
+18
+even if the reference and target states lie in different classes. Whether this is possible depends on the model, as it
+is determined by the degree of entanglement of the target and reference states. In the case of adiabatic algorithms,
+keeping the complexity down seems to be a matter of allowing non-local gates in the evolution, to fully exploit
+shortcuts to adiabaticity. This is supported analytically in Sec. III. Here, the Ising critical point is traversed along
+a restricted path of states of the form (27). Despite this restriction, these states are sufficiently non-local for CN to
+remain finite.
+The impact of our work on the preparation of states in a quantum machine seems straightforward. What our
+results mean in the field of holography is another matter. Unfortunately, we do not have the knowledge to anticipate
+anything, but it would be interesting to think in this direction. Other ideas not discussed here would be the use of
+other types of complexity such as Krylov [22, 64–66] or mixed states and their behavior in thermal phase transitions.
+We leave this for future work.
+Note Added in Proof.- While we were finishing writing this manuscript, the paper [67], which discusses the impor-
+tance of local and non-local gates in the computation of complexity, appeared in the arXiv.
+ACKNOWLEDGMENTS
+The authors thank Fernando Luis for his helpful comments and insights during the preparation of this manuscript.
+The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS),
+the Spanish Government Grants PID2020-115221GB-C41/AEI/10.13039/501100011033 and TED2021-131447B-C21
+funded by MCIN/AEI/10.13039/501100011033 and the EU “NextGenerationEU”/PRTR, the Gobierno de Arag´on
+(Grant E09-17R Q-MAD) and the CSIC Quantum Technologies Platform PTI-001. This work has been financially
+supported by the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through
+the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery,
+Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda”. J
+R-R acknowledges support from the Ministry of Universities of the Spanish Government through the grant FPU2020-
+07231.
+Appendix A: Complexity associated to the VQE
+To compute F we must express our ansatz as a unitary of the form U = T e−i
+� T
+0 H(τ) dτ where H is written in terms
+of Pauli matrices {σx, σy, σz} and tensor products of these matrices. To do so, recall that the local VQE ansatz only
+contains one and two qubit gates (between nearest neighbors). To construct the effective Hamiltonian, notice that
+Ry(θi) = e−i θi
+2 σy .
+(A1)
+Now, the C-Z gate, can be decomposed
+q0 :
+•
+=
+q0 :
+•
+•
+RZ (−π/2)
+q1 :
+•
+q1 :
+RZ (π/2)
+RZ (−π/2)
+Therefore
+C-Z = e−i π
+4 (σ0
+zσ1
+z−σ0
+z−σ1
+z) = e−i π
+4 σ0
+zσ1
+zei π
+4 σ0
+zei π
+4 σ1
+z .
+(A2)
+If we substitute in the representation of a layer of the ansatz, we find that each one of the building blocks marked
+with a dashed line in the main text is represented by a unitary of the form
+U = e−i �
+j
+θj
+2 σj
+ye−i π
+4 (σ0
+zσ1
+z−�
+j σj
+z) ≈ e−i(�
+j
+θj
+2 σj
+y+ π
+4 σ0
+zσ1
+z− π
+4
+�
+j σj
+z) ,
+(A3)
+Finally,
+H =
+�
+j
+θj
+2 σj
+y + π
+4 σ0
+zσ1
+z − π
+4
+�
+j
+σj
+z .
+(A4)
+
+19
+More generally, each layer of the ansatz can be written as an operator of the type
+H = Heven + Hodd ,
+(A5)
+where
+Heven = 1
+t
+��
+i
+θi
+2 σi
+y − π
+4
+L−1
+�
+i=0
+σi
+z + π
+4
+�
+i=even
+σi
+zσi+1
+z
+�
+,
+(A6)
+Hodd = 1
+t
+�L−2
+�
+i=0
+θi+L
+2
+σi
+y − π
+4
+L
+�
+i=1
+σi
+z + π
+4
+�
+i=odd
+σi
+zσi+1
+z
+�
+.
+(A7)
+Now we use a Trotter decomposition to compute the complexity of this circuit. We have fixed the total evolution
+time to 1 and each layer is considered a Trotter step. This way, t = T/#steps = 1/d, where d is the number of layers
+of the circuit. Now, using Eq. (6) we find
+F(U) =
+�
+�
+�
+�
+2(L−1)
+�
+i
+�
+dθi
+2
+�2
++ 3(L − 1)
+�
+dπ
+4
+�2
+.
+(A8)
+Here, L − 1 corresponds to the number of C-Zs in the layer, with L is the number of qubits. Now, the complexity is
+nothing but the integral of this functional across the number of layers in the circuit
+CN =
+� 1
+0
+F(U)dt ≈
+d
+�
+j=1
+F(U)1
+d ,
+(A9)
+which leads to Eq. (61) in the main text.
+Appendix B: Other paths in the adiabatic algorithm
+In Sec. IV A we show an adiabatic evolution for the Transverse Field Ising model where we let the field fixed as we
+increase the interaction between the neighbouring spins. However, we could have let the interaction fixed and switched
+on the transverse field, going from a classical Ising model to the TFI. In Fig. 9 we show this possible adiabatic path.
+The behaviour of the gap between the ground state and the first excited state is qualitatively different, to the point
+of even closing. This results in a much worse performance for small values of the field.
+Similarly, the gap behaviour also causes a big impact in the ZZXZ model. In Fig. 10 we show that for odd number
+of spins in the chain we get a higher complexity as the gap presents a dip at intermediate times which makes necessary
+longer times to achieve the fidelity threshold.
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+|J|
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+103
+104
+105
+C/L
+(c)
+L = 5
+L = 7
+L = 9
+L = 11
+L = 13
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