Paper Title: Electromagnetic $|G_E/G_M|$ ratios of hyperons at large timelike $q^2$.







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             LFTC-24-08/91
            
            ^1Department of Physics and OMEG Institute, Soongsil University, 
            
            Seoul 06978, Republic of Korea 
            ^2LIP, Laboratório de Instrumentação e Física 
            Experimental de Partículas, 
            
            Avenida Professor Gama Pinto, 1649-003 Lisboa, Portugal
            
            ^3Departamento de Física e Departamento de Engenharia e Ciências Nucleares, 
            
            Instituto Superior Técnico (IST),
            Universidade de Lisboa, 
            Avenida Rovisco Pais, 1049-001 Lisboa, Portugal 
            ^4Laboratório de 
            Física Teórica e Computacional – LFTC,
            Universidade Cidade de  São Paulo,  
            01506-000,  São Paulo, SP, Brazil
            0§ ABSTRACT 
            In recent years, it has become possible to
            measure not only the magnitude of the electric (G_E)
            and magnetic (G_M) form factors of spin 1/2 baryons,
            but also to measure the relative phases
             of those quantities in the timelike kinematic region.
             Aiming to interpret present |G_E/G_M| data on hyperons of the baryon octet,
             as well as to predict future data, we present model calculations of
             that ratio for large invariant 4-momentum square q^2 in the timelike region (q^2>0). 
             Without any further parameter fitting, we extend to the timelike region a covariant quark model previously developed
              to describe the kinematic spacelike region (q^2 ≤ 0)
              of the baryon octet form factors.
              The model takes into account
              both the effects of valence quarks and the excitations
            of the meson cloud which dresses the baryons.
             We report here that our results are in good agreement with very recent data from BESIII,
             within the uncertainty limits of our model. We also make 
            new predictions which may be tested in the near future,
             when the range of the present measurements is expanded
             to the 20–50 GeV^2 region.
            
             
              Electromagnetic |G_E/G_M| ratios of hyperons
            at large timelike q^2
                G. Ramalho^1, M. T. Peña^2,3,  
            K. Tsushima^4 and Myung-Ki Cheoun^1
                August 5, 2024
            =======================================================================
            
            
            
            § INTRODUCTION
            
            
            In recent years it became possible to experimentally probe the structure
            of short-lived baryons through
            e^+ e^- → B B̅ and p p̅→ B B̅ experiments in
            the timelike kinematic region above the production threshold
            q^2 ≥  4 M_B^2 (M_B is the mass of the
            baryon) <cit.>.
            
            These timelike studies complement the two last decades knowledge
            on the spacelike region[In this work,  for simplicity,  we include into the spacelike region the photon point q^2 = 0,
            due to the physical continuity between the regions q^2 < 0 and q^2=0.]
            (q^2 ≤ 0) based on results from electron scattering experiments <cit.>.
            
            
            The electromagnetic structure of the baryons is represented
            by spin (J) and parity (P)
            dependent structure form factors.
            In the timelike region these form factors are complex functions of the 
            e^+ e^- or p p̅ square center-of-mass energy 
            s= q^2 <cit.>.
            
            Of particular interest are the spin 1/2 baryons (J^P =1/2^±),
            including the nucleon and hyperons 
            characterized by two form factors only, the electric and magnetic form factors.
            
            We focus here on hyperons because the experimental information
            about the nucleon is at the moment less complete.
            Also, our model uncertainty decreases with increasing q^2,
            and the convergence of the nucleon data to the asymptotic large q^2
            behavior is expected to emerge at much larger q^2 values, due
            the dominance of light quark dynamics.
            
            Given the short lifetime of hyperons, the challenges in the determination
            of their electromagnetic properties
            in the timelike region, where their weak decays can be analyzed
            in detail <cit.>
            are less significant than in the spacelike region.
            
            
            The first experiments measured the total integrated cross section
            of the e^+ e^- → B B̅ reactions for spin 1/2 baryons,
            which determine the magnitude of the  "effective"
            form factor G defined as <cit.>
            |G (q^2)|^2 = 2 τ |G_M (q^2)|^2 + |G_E (q^2)|^2/2 τ + 1,
            
            where τ = q^2/4M_B^2.
            
            These experiments provided information only about
            a combination of the electric and magnetic form factors,
            and did not reveal their relative phases.
            
            The separate determination of |G_E| and |G_M| or of the ratio
            |G_E/G_M| for spin 1/2 baryons in the timelike region 
            can be obtained by further measuring  the differential
            cross section <cit.>, while
            the knowledge of the relative phase between the two form
            factors demands in addition the determination of polarization cross sections
            (which involve complete spin separation) <cit.>.
            Today, indeed it is possible to measure the ratio |G_E/G_M|
            and the relative angle between the form factors
            for a few baryons <cit.>.
            Most measurements to date have been done
            for ground state hyperons, but in the future excited states
            may be accessed. In a first experimental period the proton and the neutron form factors
            were extracted under some assumptions about
            the relation between G_E and G_M <cit.>.
            
            
            Although the initial measurements were performed near threshold,
            it is expected that with increased accelerator power,
            the experimental energy will go higher up to the 20–50 GeV^2
            range. This extension will allow us to test the predictions described in this work.
            
            Measurements of the effective elastic form factors
            of the Λ, Σ^0,±, Ξ^0,-
            as well as the transition form factors Σ^0 →Λ
            have been performed at BaBar <cit.>, CLEO <cit.>,
            Belle <cit.> and BESIII <cit.>.
            Measurements of the ratios  |G_E/G_M| for the Λ and Σ^+
            have also been determined recently at BESIII <cit.>.
            
            Complete experiments, that measure  |G_E|, |G_M|, 
            and the relative phase between G_E and G_M have been performed
            for the Λ and Σ^+ at BESIII <cit.>, 
            and even more measurements are planned for a near future at BESIII
            and PANDA <cit.>.
            
            
            In this work, we enlarge a previous theoretical study of the effective form factors
            of J^P= 1/2^+ hyperons <cit.> to calculate the |G_E/G_M| ratios 
            for large q^2.
            
            Calculations in this momentum transfer region are important,
            given that at very large q^2, |G_M (q^2)| dominates
            the effective form factor |G (q^2)|, allowing us to probe where that regime sets in.
            
            
            The numerical calculations are based on the
            covariant spectator quark model formalism 
            for the spacelike region <cit.>
            extended to the timelike region.
            The model has been used in the study
            of electromagnetic structure of 
            nucleon resonances <cit.>
            and other baryons
            systems <cit.>.
            
            
            
            
            In the case of the nucleon is it possible to
            compare the magnitude of G_E/G_M in the spacelike
            and timelike regions <cit.>.
            
            Timelike data on the nucleon electromagnetic
            form factors can be found
            in Refs. <cit.>
            for the proton and in Refs. <cit.> for the neutron.
            
            Theoretical studies of the nucleon electromagnetic form factors
            in the timelike region can be found in Refs. <cit.>.
            
            At the moment the measurements of the nucleon form factors in
            the timelike region are incomplete, since their relative phases
            have not been measured.
            The data of the proton and neutron effective form factors
            revealed an oscillatory dependence on the center-of-mass
            energy <cit.>.
            Different explanations have been proposed <cit.>.   
            In the present work we do not discuss this subject because
            we are focused on the asymptotic region, where
            the oscillatory component is expected to be suppressed.
            
            
            Beyond the members of the baryon octet, there are also measurements
            of the effective form factors of the Ω^-, Δ
            and Λ_c^+ <cit.>.
            Theoretical works on hyperon form factors
            in the timelike region can be found in 
            Refs. <cit.>.
            
            
            
            
            
            
            § FORMALISM
            
            
            
            Within the one-photon-exchange approximation, <cit.>
            we can write the e^+ e^- → B B̅ differential
            cross section, where B stands for
            a spin 1/2 baryon B( 1/2^±), as
                  d σ_ Born/d Ω (q^2) =
            
                  α^2 β C/4 q^2[ |G_M (q^2)|^2 (1 + cos^2 θ)
                + 1/τ |G_E (q^2)|^2 sin^2 θ],
              
            where θ is the angle between the baryon
            and the initial photon in the center-of-mass frame,
            α≃ 1/137 is the fine structure constant,
            β= √(1 - 4M_B^2/q^2) is a kinematic factor
            (the baryon speed in c=1 units) and 
            C is the Sommerfeld-Gamow factor.
            This  factor takes into account the Coulomb interaction
            and can be written as C = y/1-exp(-y)
            with y= πα/β2 M_B/√(q^2)
            for charged particles, and C=1 for 
            neutral particles <cit.>.
            For charged particles the factor C converges rapidly to C ≃ 1 
            when q^2 increases.
            
            By integrating the previous relation, we obtain
            σ_ Born (q^2) =
            4 πα^2 β C/3 q^2( 1 + 1/2 τ) |G (q^2)|^2,
            
            using Eq. (<ref>).
            We can conclude then that the effective form
            factor can be calculated directly from the integrated cross section.
            However, it does not give individual information on  G_E or G_M.
            
            
            
            To untangle information on the two form factors G_E and G_M
            one uses  the differential cross section (<ref>) by
            determining the coefficients of the angular functions
            1 + cos^2 θ and sin^2 θ <cit.>.
            
            Combining the two results, one can extract the ratio |G_E/G_M|.
            
            The measurement of the phase between the form factors
            requires in addition the measurement of cross section
            with defined polarizations of the baryons <cit.>.
            
            
            
            
            
            
            
            Finally, at threshold, where q^2= 4 M_B^2, the functions
            G_E and G_M are such that G_E = G_M,
            as a consequence of the definition
            of these Sachs from factors in terms of the kinematic independent
            Dirac and Pauli form factors. Therefore one expects that G_E/G_M ≃ 1
            for measurements 
            near threshold, and the non-zero phases between the form factors are to be sought 
            above the threshold  only (q^2 > 4 M_B^2) <cit.>.
            
            
            The main conclusion of this section is that in the case
            of a spin 1/2 baryon, to obtain 
            the absolute values of the two form factors from
            experiments of the  differential cross section
            it is necessary to measure accurately angular distributions
            of the differential cross section
            besides the effective form factor |G(q^2)|.  However, as we will see in the next section the range for non-zero imaginary parts cannot extend to indefinitely large |q^2|.
            
            
            
            
            
            
            
            
            
            
            § CALCULATION OF ELECTROMAGNETIC FORM FACTORS AT LARGE Q^2
            
            
            
            
            In this work the electromagnetic form factors G_E and G_M
            in the timelike region  are obtained from extrapolating the results
            from the region q^2=-Q^2 ≤  0 to the region q^2 > 0.
            
            
            
            
            
            For that purpose we consider the large-|q^2| relations 
            
            G_ℓ (q^2)= G_ℓ (-q^2)
            
            for ℓ=E,M. 
            These asymptotic relations are a consequence of
            two general mathematical principles: 
            unitarity as well as the Phragmén-Lindelöf theorem,
            which is valid for analytic functions <cit.>.
            
            
            We notice that this theorem implies that the imaginary parts of
            the form factors in the timelike region 
            must be negligible for large q^2,
            since the form factors are real in the spacelike region.
            
            
            
            
            
            
            
            
            
            
            Due to the character of this approximation,
            we expect our estimates here to be accurate only for large q^2,
            deviating  from the condition G_E = G_M at the threshold.
            
            Since the relations (<ref>) are valid for
             the large-|q^2| mathematical limit, and we want to make predictions
            to finite q^2, we consider finite corrections to these
            asymptotic relations.
            
            We start noticing that the elastic form factors
            are divided into two regions of  q^2: (- ∞, 0] (spacelike region)
            and  [4M_B^2, + ∞),  with a gap of 4M_B^2 length
            between the photon point (q^2=0) and
            the threshold of the baryon-antibaryon production (q^2 = 4 M_B^2).  
            
            The interval (0,4M_B^2) defines an unphysical region
            where no physical baryons can be detected <cit.>.
            
            
            
            There is then an ambiguity about the center of reflection
            for the asymptotic symmetry relation of Eq. (<ref>).
            
            To take into account the ambiguity of the location of the center of symmetry 
            we replace
            q^2 → q^2 ( 1 - 2M_B^2/q^2) in (<ref>), 
            which introduces finite corrections (-2M_B^2)
            to the limit q^2= + ∞.
            
            Therefore we use <cit.>
            G_ℓ (q^2)= G_ℓ^ SL (q^2 - 2 M_B^2)
            
            for ℓ =E,M. 
            The label SL on the r.h.s. indicates that 
            theG_ℓ^ SL (Q^2) is calculated
            in the spacelike region (Q^2 = -q^2 ≥ 0).
            
            The expression (<ref>) estimates the reflection
            symmetry point underlying (<ref>)
            exactly at the center of the unphysical interval (0,4M_B^2).
            
            
            
            
            In summary, the measured form factors in the timelike region are complex functions
            characterized by their magnitudes, |G_E|, |G_M|, and their different phases.
            
            Our model calculation is based on results obtained in the q^2 ≤ 0
            region and the application of the large-q^2 relation (<ref>).
            This way, by construction, we obtain real numbers for G_E
            and G_M that carry ± signs.
            
            In principle, for very large q^2 the comparison of our results with the
            experimental values for G_E/G_M is possible
            since in that region the relative phase can only have two possibilities,
            ΔΦ =0 (when the form factors have the same sign)
            or  ΔΦ = π (if they have different signs).
            
            
            
            
            For a control of the approximation we explored the 
            band of variation of the form factors with respect to a reasonable variation
            of the symmetry point around this central value:
            when the functions G_ℓ (q^2) are positive (negative), since
            they are expected to monotonically decrease (increase)
            with increasing q^2, we take
            the upper (lower) limit values to be G_ℓ (q^2)= G_ℓ^ SL (q^2 - 4M_B^2)
            and the lower (upper) limit  G_ℓ (q^2)= G_ℓ^ SL (q^2).
            
            This procedure provides a band of variation of the
            finite corrections to the asymptotic symmetry identity.
            
            When q^2 is very large, the variation of the results becomes
            very narrow and the three estimates (central, upper and lower limits)
            converge to the same value, providing accurate predictions
            that can be tested by experimental data. Moreover, the comparison with data
            in that region provides information of the onset of the regime of
            the asymptotic symmetry (<ref>).
            
            
            
            
            
            
            
            
            § NUMERICAL RESULTS FOR |G_E/G_M|
            
            
            
            In this section we present our results of the ratio |G_E/G_M| for large q^2 values.
            The calculation method of the form factors G_E and G_M
            for the baryon octet family in the timelike region is discussed
            in detail in Ref. <cit.> which uses the
            covariant spectator quark model formalism
            for the spacelike region <cit.>. 
            In its nutshell, we use impulse approximation for the electromagnetic
            quark current, which is justified for large Q^2, since then
            the interacting photon has sufficient resolution to interact with a single quark at a time.
            As a result we can integrate over the relative internal variables
            of the spectator quarks  and therefore we end up with the
            baryon three-quark system regarded 
            as a quark-diquark configuration.
            In the model, the SU(3) symmetry quark flavor symmetry
            is broken at the level of the baryon radial wave functions
            and the constituent quark current <cit.>,
            and gluon and quark-antiquark dressing
            are effectively taken into the constituent quark structure.
            
            
            The different baryon radial wave functions are parametrized
            by two parameters, a short range scale and a long range scale,
            consistent with the expected size of the systems
            (baryons with strange quarks are more
            compact than baryons with light quarks) <cit.>.
            The quark current is parametrized by
            vector meson dominance regulated by
            light (ρ, ω) and intermediate (ϕ)
            vector meson mass poles <cit.>.
            
            
            
            
            
            
            In particular, we use the parameterization
            of the baryon octet from Refs. <cit.>.
            To the valence quark contributions
            we add also an effective description of meson cloud
            dressing processes <cit.>,    
            which are inevitably present in the low-Q^2 region.
            
            In the large-|Q^2| region, the meson cloud contributions are suppressed
            and their effects are manifest on the rescaling of
            the bare contributions to the form factors due to
            the normalization of the hyperon wave functions <cit.>.
            
            This formalism has been used to successfully describe the spacelike electromagnetic
            structure data of baryons including the low-lying nucleon resonances <cit.>.
            The free parameters of the model are fixed by the
            data for the nucleon,
            baryon octet and baryon decuplet <cit.>.
            
            
            Our calculations of the ratio |G_E/G_M| and their comparison with the data
            are shown in Figs. <ref> and <ref>,
            that depict the results for Λ, Ξ^-,0 and  the Σ^0,±, respectively.
            In the graphs for the Λ and Σ^+ we
            include the available data for |G_E/G_M|.
            How we estimate the uncertainty of the results of our model for each form factor
            was defined in the previous section.
            
            The bands of variation for the magnitude of G_E/G_M are determined
            from the independent band intervals of variation for the G_E and G_M functions.
            
            
            
            
            
            The results and their uncertainty bands near the threshold provide only 
            qualitative estimates since our results are expected to be valid only for large q^2,
            as discussed in the previous section.
            
            The deviation from the result  |G_E/G_M|=1 at threshold
            is a consequence of the approximation used.
            
            The signs of our results for the form factors in the asymptotic
            large q^2 values are indicated on the
            right side in Figs. <ref> and <ref>. 
            
            
            
            
            The separated calculations of the form factors G_E and G_M
            for Λ, Σ^0,±  and Ξ^-,0 are presented
            in the Supplementary Material <cit.>.
            
            Note that we are not attempting to predict the zeros
            of G_E/G_M, rather, we are focused on determining its magnitude at sufficiently large 
            q^2.
            In the current formalism, the zeros can only be determined within an uncertainty band of width 
             4 M_B^2 for q^2.  
            
            
            
            We discuss now how we can take into account
            the information of the relative angle
            between the form factors in the analysis of the ratio |G_E/G_M|.
            
            In complete experiments
            the ratio  G_E/G_M is determined,
            and can be written in the form
            G_E/G_M = |G_E|/|G_M| e^i ΔΦ,
            
            where |G_ℓ| is the magnitude of
            the complex form factor G_ℓ and ΔΦ
            is the phase between G_E and G_M.
            
            Once the phase is measured,
            we can extract the real part of the ratio from
            |  Re(G_E/G_M) |=
            | G_E/G_M| |cos (ΔΦ)|,
            
            which is a fraction of the measured value to |G_E/G_M|.
            This does not only show that model calculations
            of |G_E/G_M| based on real form factors 
            agree with the absolute value of the real part of G_E/G_M
            when the imaginary parts are negligible.
            
            The result (<ref>) allows also us to infer from the comparison between the
            experimental data for the r.h.s. and our model results
            for the ratio |G_E|/|G_M| (where G_E and G_M are real functions)
            if the regime where the imaginary parts of
            the form factors vanish has been attained. 
            
            The comparison of our model results for |G_E/G_M|
            with the experimental
            values for | Re(G_E/G_M)| from Refs. <cit.>
            that measure phases, are displayed in Fig. <ref>.
            
            In the graphs we notice the agreement of our results for the form factors
            (considering the model uncertainty  band) and the experimental
            data point at the largest q^2 point measured,
             q^2 ≃ 14.3 GeV^2 for the Λ
            and q^2 ≃ 8.4 GeV^2 for the Σ^+.
            
            In the graphs we ignore the uncertainties
            of the angles ΔΦ, for simplicity, and used
            their central value.
            
            The consideration of the uncertainties on ΔΦ
            only increases the error bars of the | Re(G_E/G_M)| data.
            Hopefully, future experiments will be more precise,
            which will help to determine the point were our
            model calculations provide a good description of the data
            and determine the large q^2 region where reflection symmetry sets in.
            
            
            
            
            
            
            § EFFECTIVE FORM FACTORS 
            
            
            
            In a previous work <cit.> we presented predictions
            for the effective form factors (<ref>) for
            the baryon octet members Λ, Σ^0,± and Ξ^0,-
            for q^2=10–60 GeV^2.
            At the time, only Λ, Σ^+, Σ^-, Ξ^0 and Ξ^-
            could be tested in the region of 12–15 GeV^2 by the data from CLEO <cit.>.
            
            Our estimates were on average consistent with the high q^2 data. 
            
            Since then more data for |G(q^2)| became available.
            
            In particular data for Σ^+ and Σ^- became available from BESIII <cit.>
            for the range 5.7–9.1 GeV^2.
            Surprisingly, the data are in excellent agreement with our estimates 
            (namely with the central line values, see Supplementary Material <cit.>)
            even outside the region where model validity was expected to be justified.
            
            A rough estimate of the lower limit for the application
            of our model based on the asymptotic
            relations, can be q^2 ≈ 8 M_B^2  or  q^2 ≈ 10 GeV^2.
            
            
            Recently, data from BESIII became available for
            Λ up to q^2= 21 GeV^2 <cit.>,
            and for Ξ^- up to q^2= 23 GeV^2 <cit.>.
            
            The new data are compatible with the CLEO data and extend the previous
            range covered by the experiments.
            The comparison of the new data with the model results
            from Ref. <cit.> is presented in
            Figs. <ref> and <ref> in a linear scale and in the logarithm scale for |G(q^2)|.
            
            It is clear from the figures that there is an excellent agreement between
            our predictions and the data, indicating that
            the range of the asymptotic form for the effective form factors
            may not be too far from the present day experimentally studied region.
            Future data for other hyperons may confirm or deny this tendency.
            
            
            
            
            
            
            Calculations of the effective form factors
            for all the hyperons of the baryon octet are presented in Supplementary Material <cit.>
            and compared with the world data.
            
            
            
            
            
            
            
            
            
            
            § OUTLOOK AND CONCLUSIONS 
            
            
            Experiments in new facilities (BaBar, CLEO, Belle, PANDA
            etc.) allow us to access
            the electromagnetic structure of hyperons in the elastic
            timelike region (q^2 ≥ 4 M_B^2).
            
            Till recently we had access only to
            the integrated cross section which can be used
            to determine the effective form factor |G|,
            and encode only the information
            of a certain combination of the electric and magnetic
            form factors |G_E| and |G_M|, 
            for spin 1/2 hyperons.
            
            No separated information about  |G_E| and |G_M|
            was possible till a few years ago.
            Even the separation between |G_E| and |G_M|
            for the proton and neutron was difficult
            and affected by significant errors.
            
            In the last few years it became possible
            to determine  |G_E| and |G_M| and the relative phase
            between them for Λ and Σ^+.
            
            We expect that the ratio |G_E/G_M| became
            accessible in the following years for most
            ground state hyperons (Σ^0,- and Ξ^0,-),
            as well as for charmed baryons (starting with Λ_c^+).
            
            Our calculations show a partial 
            agreement with the data in the range
            q^2 ≃ 14 GeV^2 (√(s)≃ 3.8 GeV),
            which may be an indication that our model
            calculations may be compared with future measurements
            in the range of q^2 =20–30 GeV^2
            (√(s)= 4.5–5.5 GeV)  or higher.
            
            Also the separation between the real and imaginary
            components may be tested in that range of energies.
            
            
            
            
            
              
            G.R. was supported the Basic Science Research Program 
            through the National Research Foundation of Korea (NRF)
            funded by the Ministry of Education  (Grant No. NRF–2021R1A6A1A03043957).
            M.T.P was supported by the Portuguese Science Foundation FCT
            under project CERN/FIS-PAR/0023/2021, and FCT computing project 2021.09667.
            K.T. was supported by Conselho Nacional de Desenvolvimento 
            Científico e Tecnológico (CNPq, Brazil), Processes No. 313063/2018-4 
            and No. 426150/2018-0, and FAPESP Process No. 2019/00763-0
            and No. 2023/07313-6, and his work was also part of the projects,
            Instituto Nacional de Ciência e 
            Tecnologia - Nuclear Physics and Applications 
            (INCT-FNA), Brazil, Process No. 464898/2014-5.
            The work of M.K.C. was supported by the National
            Research Foundation of Korea
            (Grant Nos. NRF–2021R1A6A1A03043957 and NRF–2020R1A2C3006177).
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            SuppM
              Supplementary material.
            
            
            
              
            
             Supplemental Material
            
            14cmWe present here calculations of the form factors
            G_E and G_M and the effective form factor G
            in the timelike region for the hyperons
            Λ, Ξ^-,0 and  the Σ^0,±.
            
            
            § FORM FACTORS G_E AND G_M
            
            
            The absolute values of the form factors G_E and G_M
            are presented in Fig. <ref>.
            The signs of G_E and G_M can be inferred from
            the labels included in the figures.
            The representation -G_E or -G_M indicate
            that the functions are negative in that region.
            The cusps indicate the zero of G_E.
            
            
            
            § EFFECTIVE FORM FACTORS G
            
            
            
            The effective form factors are presented in Fig. <ref>.
            The limits of the theoretical calculations
            are represented by the dashed lines.
            
             DataΣ^+:  Belle <cit.>,
            BESIII <cit.>,
            and CLEO <cit.>;
             DataΣ^0: BaBar <cit.>, BESIII <cit.>, 
            CLEO <cit.> and Belle <cit.>;
             DataΣ^-: BESIII <cit.>; 
             DataΛ: BaBar <cit.>, BESIII <cit.>
            and CLEO <cit.>;
             DataΞ^0: BESIII <cit.> and CLEO <cit.>;
             DataΞ^-: BESIII <cit.> and
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 Paper Title: Lanczos for lattice QCD matrix elements.



FERMILAB-PUB-24-0407-T




 Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A.

 Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A.





§ ABSTRACT     
    Recent work <cit.> found that an analysis formalism based on the Lanczos algorithm allows energy levels to be extracted from Euclidean correlation functions with faster convergence than existing methods, two-sided error bounds, and no apparent signal-to-noise problems.
    We extend this formalism to the determination of matrix elements from three-point correlation functions.
    We demonstrate similar advantages over previously available methods in both signal-to-noise and control of excited-state contamination through example applications to noiseless mock-data as well as
    calculations of (bare) forward matrix elements of the strange scalar current between both ground and excited states with the quantum numbers of the nucleon.


Lanczos for lattice QCD matrix elements
    Michael L. Wagman
    August 5, 2024
=======================================







§ INTRODUCTION



By stochastically evaluating discretized path integrals, numerical lattice QCD calculations provide a first-principles approach to studying the dynamics of the strong force <cit.>, including hadron spectroscopy and scattering amplitudes <cit.> and various aspects of hadron structure <cit.>.
Due to the stochastic nature of these calculations, statistical analysis of noisy Monte Carlo “data” is necessary.
Although there are well-established analysis methods, developing improved techniques is still an active topic of research <cit.>.
Better analysis tools can improve both statistical precision by alleviating signal-to-noise issues, as well as accuracy by providing more robust control of systematic uncertainties.

Lattice QCD calculations often involve the extraction of hadronic matrix elements from simultaneous analysis of hadronic two- and three-point correlation functions (correlators).
This analysis task underlies the calculation of many different quantities of physical interest including form factors, parton distribution functions (PDFs), and generalizations thereof like generalized parton distributions (GPDs) and transverse-momentum distribution PDFs (TMDs) <cit.>.
However, presently standard methods may produce unreliable results <cit.> due to a combination of excited-state contamination (ESC) <cit.> and exponentially decaying signal-to-noise ratios (SNR) <cit.>.
Analysis techniques which address these issues are a topic of active research <cit.>, but obtaining full control over all sources of uncertainty remains challenging for many quantities of physical interest.


Spectroscopy—the extraction of finite-volume energy levels from analysis of hadronic two-point correlation functions—is hindered by the same issues as matrix-element extractions, i.e. ESC and decaying SNR,
and methods to improve these issues can be useful in both contexts.
There has been extensive work to develop improved spectroscopy methods less susceptible to ESC and which offer bounds on systematic uncertainties—notably 
approaches based on generalized eigenvalue problems (GEVP) that provide one-sided variational bounds on energy level systematic uncertainties <cit.>, which have already been adapted to matrix-element calculations <cit.>.
Recent work has shown that a novel formalism based on the Lanczos algorithm <cit.> can provide qualitative and quantitative improvements for spectroscopy including two-sided bounds on systematic errors, as well as a potential resolution to SNR problems <cit.>.

In this work, we present a new approach to matrix element analyses using a simple extension of this Lanczos formalism.
The new method allows direct, explicit computation of matrix elements between any eigenstates resolved by the Lanczos algorithm, with only a few analysis hyperparameters associated with eigenstate identification and none specific to the operator or three-point function.
As derived below, after constructing initial and final-state Ritz coefficients P and P' from the corresponding two-point functions C(t) and C'(t), extracting matrix elements from an arbitrary three-point function amounts to matrix multiplication, taking the form

    ⟨f'|J|i|≈⟩∑_στ P^'*_f σC^3pt(σ, τ)/√(C'(0) C(0)) P_i τ,

where τ is the source-operator separation and σ is the operator-sink separation.
As explored below, this provides several important advantages over the previous state of the art, namely simplicity, direct and explicit computation of excited-state and transition matrix elements, and avoidance of inductive biases unavoidably introduced by implicit methods involving statistical modeling and fits of correlation functions.
Furthermore, we find that the new method is dramatically less susceptible to yielding deceptive results when applied to three-point correlators with large ESC.
The data required is the same as for presently standard analyses, with the important caveat that the three-point correlator must be evaluated for a connected region of Euclidean sink and operators times from zero up to some cut.

The remainder of this paper proceeds as follows.
<Ref> defines the analysis task, and reviews both the transfer matrix formalism necessary to understand the Lanczos approach as well as previously available methods used for comparison.
<Ref> derives the method.
<Ref> applies the method to a noiseless mock-data example and compares with previous approaches, demonstrating its improved convergence properties.
<Ref> discusses how the method must be adapted in the presence of statistical noise and presents a calculation of bare matrix elements of the strange scalar current for the low-lying states in the nucleon spectrum using lattice data.
<Ref> subjects both the summation method and Lanczos to adversarial attacks; the results of these experiments suggest that Lanczos estimates are qualitatively more robust against excited-state contamination.
Finally, <ref> concludes and discusses opportunities for future work.

Note that throughout this paper, repeated indices do not imply summation and all sums are written explicitly.
We also use lattice units to simplify the notation, setting the lattice spacing a=1 throughout.
In these units, physical quantities like energies and matrix elements are dimensionless, and Euclidean times t take on integer values such that they may be used interchangeably as arguments and indices.





§ BACKGROUND



We are interested in computing hadronic matrix elements of some operator J, i.e.,

    J_fi = ⟨f'|J|i|,⟩

where |i⟩ and |f'⟩ are energy eigenstates with the quantum numbers of the initial (unprimed) and final (primed) state, which may in general be different.
For example, their momenta will differ for off-forward matrix elements, i.e. when J carries some nonzero momentum.
Hadronic matrix elements are not directly calculable using numerical lattice methods, and instead are typically extracted from simultaneous analysis of two- and three-point correlators.[The Feynman-Hellmann theorem and generalizations thereof provides a distinct approach; see <cit.> for examples.]
The ones relevant to the calculation are, in Heisenberg picture,[Suppressed lattice spatial indices on each operator are assumed to be absorbed into these quantum numbers, e.g. by projection to a definite momentum.]

    C(t)    = ⟨ψ(t)  ψ(0)|,⟩
    
        C'(t)    = ⟨ψ'(t)  ψ'(0)|,⟩
    
        C^3pt(σ, τ)    = ⟨ψ'(τ+σ)   J(τ)  ψ(0) |⟩

where ψ and ψ' are interpolating operators (interpolators) with initial- and final-state quantum numbers.
Note that the arguments of C^3pt are more often defined with a different convention, with the sink time t_f = σ + τ as the first argument.
While not explicitly notated, we generally assume definitions such that all vacuum contributions are subtracted out.
These expectations may be evaluated stochastically by lattice Monte Carlo methods.





 §.§ Transfer matrices & spectral expansions


To see how these data constrain the matrix element of interest, we use the Schrodinger-picture transfer-matrix formalism <cit.> to derive their spectral expansions.
This exercise also serves to establish notation and as a review of this formalism, used throughout this work.

We begin with the assumption[This holds only approximately for many lattice actions in standard use <cit.>.] that Euclidean time evolution can be described by iterative application of the transfer matrix T = e^-H, such that 

    T |n⟩ = λ_n |n⟩≡ e^-E_n|n⟩,

where |n⟩ are unit-normalized energy eigenstates such that ⟨n|m|=⟩δ_nm, λ_n are transfer-matrix eigenvalues, and E_n = -logλ_n are the energies.
Note that states |·⟩ and operators including J and T are not directly accessible, but rather formal objects that live in the infinite-dimensional Hilbert space of states.
Applied to the vacuum state |Ω⟩, the adjoint interpolating operator ψ excites the state |ψ⟩, which may be decomposed as

    ψ|Ω⟩≡|ψ⟩
        = ∑_n ⟨n|ψ||%s⟩⟩n≡∑_n Z_n |n⟩,

where Z_n = ⟨n|ψ|$⟩ are the overlap factors. 
The sum may be assumed to be restricted to eigenstates with the quantum numbers ofψ, asZ_n=0otherwise.
The transfer matrix acts nontrivially on|ψ⟩, as

    T^t |ψ⟩ = ∑_n Z_n e^-E_n t|n⟩ .

Under such Euclidean time evolution, the amplitudes of higher-energy eigenstates decay more quickly.
Takingt →∞, the ground eigenstate|0⟩dominates.
Importantly, this amounts to application of the power-iteration algorithm <cit.>, as discussed further below.

For the initial-state correlatorC(t)from <ref>, translating to Schrodinger picture usingO(t) = T^-t O T^tgives

    C(t) 
           = ⟨Ω | T^-tψ T^t T^0 ψ T^0 | Ω|
    ⟩   = ⟨ψ | T^t |ψ|
    ⟩   = ∑_i |Z_i|^2 e^-E_i t,

using⟨Ω| T = ⟨Ω|in the second equality.
An analogous expression holds for the final-state correlator,

    C'(t) = ∑_f |Z'_f|^2 e^-E'_f t,

whereZ'_f = ⟨f'|ψ'|$⟩.
Similar manipulations produce the spectral expansion of the three-point correlator,

    C^3pt(σ, τ)
           = ⟨ψ' | T^σ J T^τ |ψ|
    ⟩   = ∑_fi Z^' *_f Z_i J_fi  e^-E'_f σ - E_i τ .

Note that these definitions assume zero temperature.[Thermal effects are handled automatically in applications of Lanczos to two-point correlators as discussed in Sec. A of Ref. <cit.>'s Supplementary Material.
The resulting isolation of thermal states automatically removes their effects from all matrix element results.
]
Comparing <ref>, we see that C and C' carry the necessary information to isolate J_fi in C^3pt.

A standard approach to extracting J_fi is to use statistical inference, i.e. simultaneously fitting the parameters Z_i, Z'_f, E_i, E'_f, and J_fi of a truncated spectral expansion to <ref>, (<ref>), and (<ref>).
The resulting estimates converge to the underlying values given sufficiently high statistics, large Euclidean times, and states included in the models.
This has been used to much success, but has certain serious disadvantages.
Specifically, the black-box nature of statistical inference and large number of hyperparameters to vary (e.g. number of states modeled, data subset included, choice of selection/averaging scheme) induce systematic uncertainties which can only be assessed with caution and experience.





 §.§ Power iteration & other explicit methods


In this work, we restrict our consideration to explicit methods that do not involve statistical modeling.
Familiar examples include effective masses[And generalizations thereof including GEVPs <cit.> and Prony's method <cit.>.] and ratios of correlation functions.
Here, we present these as examples of the power iteration algorithm, thereby motivating the use of Lanczos in <ref>.
We also review the summation method <cit.>, for later comparison with Lanczos results.

As already noted, using Euclidean time evolution to remove excited states may be thought of as applying the power iteration algorithm <cit.> to extract the ground eigenstate of the transfer matrix <cit.>.
With power iteration, increasingly high-quality approximations of the ground eigenstate, |b^(m)⟩≈|0⟩, are obtained using the recursion

    |b^(m)⟩ 
        =  T |b^(m-1)⟩/ || T |b^(m-1)⟩ || 
        =  T |b^(m-1)⟩/√(⟨ b^(m-1) | T^2 | b^(m-1)|⟩),

starting from

    |b^(0)⟩≡|ψ⟩/ |ψ| ≡|ψ⟩/√(⟨ψ|ψ|⟩) 
        = |ψ⟩/√(C(0)) .

The resulting approximate eigenstates |b^(m)⟩ can be used to compute ground-state matrix elements of different operators.
For example, using them to extract the ground-state eigenvalue of T yields the usual effective energy

    E^eff(2m) 
           = - log⟨ b^(m) | T | b^(m)|
    ⟩   = - log⟨ψ | T^2m+1 | ψ|⟩/⟨ψ | T^2m | ψ|⟩
       = - log C(2m+1) / C(2m) ,

where the second equality fully evaluates the recursion <ref>.
Note that while this power-iteration version of E^eff is defined for only even arguments 2m, we generalize and evaluate it for all t=2m as usual.

More relevantly to this work, power-iteration states may also be used to compute ground-state matrix elements of some operator J as

    ⟨b'^(m) | J | b^(m)|   ⟩= ⟨ψ' | T^m J T^m | ψ|⟩/√(⟨ψ'|T^2m|ψ'|⟨%s|%s⟩⟩ψ|T^2m|ψ)
       =  C^3pt(m,m) /√(C'(2m)   C(2m)) ,

where |b'^(m)⟩ are from |ψ'⟩.
We thus arrive at the usual approach of constructing ratios of three-point and two-point functions to isolate the desired matrix element.
It is straightforward to insert the spectral expansions <ref> and verify that ⟨b'^(m) | J | b^(m)|=⟩ J_00 up to excited-state effects.
Thus, in analogy to the effective energy <ref>, ⟨b'^(m) | J | b^(m)|$⟩ may be thought of as an “effective matrix element” expected to plateau toJ_00asm →∞.
The new method presented in <ref> simply applies this same idea with the improved approximations of the eigenstates afforded by the Lanczos algorithm, with the notable difference that approximate eigenstates are available for excited states as well.

The ratio <ref> derived by power iteration is not the one in standard use.
For easier comparison with other works, we instead employ the standard ratio

    R(σ, τ)    = 
         C^3pt(σ, τ) / C'(σ+τ) √( C(σ) / C'(σ)  C'(σ+τ) / C(σ+τ)  C'(τ) / C(τ) )
       = J_00 + (excited states)  .

Whenψ= ψ', this reduces to⟨b^(σ) | J | b^(τ)|$⟩; additionally taking τ = σ = m reproduces <ref> exactly.
However, when ψ≠ψ' the two expressions are inequivalent.
We use this standard ratio to define a power-iteration-like effective matrix element for sink time t as

    J^PI(t) = 
            R(t/2, t/2 ),     t even
    1/2[ R(t+1/2, t-1/2) + R(t-1/2, t+1/2) ]     t odd,

similar to <ref> for even t and averaging the two equivalently contaminated points for odd t.
This quantity is what is referred to as “Power iteration” in all plots below.

Further manipulation leads to the summation method <cit.>, presently in common use, which provides an effective matrix element[This differs superficially from the typical presentation of the summation method, which prescribes fitting Σ(t_f) to extract the part linear in t_f. Linear fits to Σ(t_f) are identical to constant fits to J^eff(t_f) if their covariance matrices are computed consistently.] for sink time t_f as

    Σ_Δ_τ(t_f) 
           = ∑_τ=Δ_τ^t_f-Δ_τ R(t_f-τ, τ),
        
    
        J^eff_00,Δ_τ(t_f)    = Σ_Δ_τ(t_f+1) - Σ_Δ_τ(t_f)
        
       = J_00 + (excited states)

for each choice of Δ_τ, the summation cut.
J^eff is expected to plateau to J_00 as t_f increases and excited states decay away.
Increasing Δ_τ further removes contamination, and curve collapse is expected as Δ_τ increases.





§ LANCZOS METHOD



The previous section discussed how standard lattice analysis methods may be thought of as implementing the power iteration algorithm to resolve the ground eigenstate of the transfer matrix T.
The Lanczos algorithm improves upon power iteration <cit.> by making use of the full set of Krylov vectors ∝ T^t |ψ⟩ obtained by iterative application of T, rather than discarding all but the last.
As explored in Ref. <cit.>, Lanczos defines a procedure to manipulate two-point correlators to extract eigenvalues of T.
Here, we extend that formalism to evaluate matrix elements in the basis of transfer-matrix eigenstates.

Specifically, the method proposed here is to evaluate

    J^(m)_fi = ⟨y'_f^(m) | J | y^(m)_i|,⟩

where |y'_f^(m)⟩≈|f'⟩ and |y^(m)_i⟩≈|i⟩ are the initial- and final-state Ritz vectors after m Lanczos iterations, the Lanczos algorithm's best approximation of the corresponding eigenstates.
The steps to do so, as worked through in the subsections below, are as follows:

    
  * Apply an oblique Lanczos recursion to compute the transfer matrix in bases of Lanczos vectors with appropriate quantum numbers. Diagonalize to obtain Ritz values and the change of basis between Ritz and Lanczos vectors. (<ref>)
    
  * Compute the coefficients relating Lanczos and Krylov vectors. (<ref>)
    
  * Compute the coefficients relating Ritz and Krylov vectors. (<ref>)
    
  * Compute overlap factors to normalize the Ritz vectors. (<ref>)
    
  * Repeat the above steps on initial- and final-state two-point functions to obtain Ritz vectors with initial- and final-state quantum numbers. 
    
  * Project the three-point function onto the Ritz vectors to obtain matrix elements. (<ref>)
    
  * Identify and discard spurious states. (<ref>)

We also discuss how bounds may be used to characterize Lanczos convergence in <ref>.
Statistical noise introduces additional complications—especially, the final step above—as discussed in <ref>.

We note immediately that the oblique formalism used here formally constructs an approximation of the transfer matrix of the form

    T ≈∑_k |k̃^R⟩λ̃_k ⟨k̃^L|

with generally complex eigenvalues and distinct left and right eigenvectors.
Meanwhile, the true underlying transfer matrix is Hermitian, i.e.,

    T = ∑_k |k⟩λ_k ⟨k|

with real eigenvalues λ_k and degenerate left and right eigenvectors.
Critically, physically sensible results must respect this underlying Hermiticity.
As discussed in <ref>, when the Hermiticity of the underlying transfer matrix is manifest in the data, the Lanczos approximation of T is Hermitian as well.
However, statistical fluctuations obscure this underlying Hermiticity in noisy data.
As explored in <ref>, this results in a Hermitian subspace and a set of unphysical noise-artifact states which must be discarded.





 §.§ The oblique Lanczos algorithm



This section serves primarily as a review of Ref. <cit.>, especially Sec. D of its Supplementary Material. However, the notation and some of the definitions—notably, of the Ritz vectors—have been altered here to better accommodate the matrix element problem.

Oblique Lanczos is a generalization of the standard Lanczos algorithm which uses distinct bases of right and left Lanczos vectors, |v^R_i⟩ and ⟨v^L_i| <cit.>.
This generalization allows application to non-Hermitian operators and, in the lattice context, off-diagonal correlators with different initial and final interpolators.
As discussed in Ref. <cit.>, oblique Lanczos is also formally necessary to treat noisy correlator data even with diagonal correlators, but naive complexification of standard Lanczos provides an identical procedure when applied only to extracting the spectrum.
However, the matrix element problem requires treatment with the full oblique formalism.

We caution that the left and right vectors treated by oblique Lanczos should not be confused with initial and final states. These are distinct labels, and left and right spaces must be constructed for each of the initial and final state eigensystems.

We first present oblique Lanczos in full generality, then discuss the specific cases used in this work at the end of the subsection.
In this spirit, consider different right and left starting states, |ψ⟩ and ⟨χ|, and a non-Hermitian transfer matrix T ≠ T^†.
The oblique Lanczos process begins from the states

    |v^R_1⟩ = |ψ⟩/√(⟨χ|ψ|⟩)  and  ⟨v^L_1| = ⟨χ|/√(⟨χ|ψ|⟩)

defined such that ⟨v^L_1|v^R_1|=⟩ 1, and after m iterations constructs the right and left bases of Lanczos vectors, |v^R_j⟩ and ⟨v^L_j| with j = 1,…,m.
The resulting right and left bases are mutually orthonormal by construction, i.e.,

    ⟨v^L_i | v^R_j|=⟩δ_ij  ,

but ⟨v^R_i|v^R_j|≠⟩⟨v^L_i|v^L_j|≠⟩δ_ij in general.
In the process, oblique Lanczos necessarily also computes the elements α_j, β_j, and γ_j of the tridiagonal matrix

    T^(m)_ij = ⟨v^L_i | T | v^R_j|=⟩[ α_1 β_2           0; γ_2 α_2 β_3        ;     γ_3 α_3   ⋱    ;           ⋱   ⋱ β_m;   0         γ_m α_m;     ]_ij  .

Beginning with α_1 = ⟨v^L_1|T|v^R_1|$⟩ and definingβ_1=γ_1=0for notational convenience, the iteration proceeds via three steps:

    |r^R_i+1⟩   = (T - α_i) |v^R_i⟩ - β_i |v^R_i-1⟩, 
    |r^L_i+1⟩   = (T^† - α_i^*) |v^L_i⟩ - γ_i^* |v^L_i-1⟩,
    
       
    β_i+1γ_i+1   = ⟨r^L_i+1 | r^R_i+1|,⟩
    
       
    |v^R_i+1⟩   = 1/γ_i+1|r^R_i+1⟩, 
    |v^L_i+1⟩   = 1/β_i+1^*|r^L_i+1⟩, 
    α_i+1   = ⟨v^L_i+1 | T | v^R_i+1| ⟩.

How preciselyβ_jandγ_jare determined from the productβ_j γ_jcomputed in step two is a matter of convention; any choice satisfying[Recall that repeated indices are not summed over in this work.]
    γ_j = ⟨r^L_j | r^R_j|⟩/β_j

is correct.
It will be helpful to consider the symmetric conventionβ_j ≡γ_j ≡√( ⟨r^L_j | r^R_j|⟩)corresponding to naive complexification of the standard Lanczos algorithm.
We have in some cases observed improved numerical behavior with a different convention,β_j = | √(⟨r^L_j | r^R_j|⟩) |<cit.>.
All physical quantities computed with this formalism are invariant to the choice of oblique convention, although the Lanczos vectors are not.

The Lanczos approximation of the transfer matrix is

    T^(m)≡∑_i,j=1^m |v^R_i⟩⟨v^L_i| T |v^R_j⟩⟨v^L_j| = ∑_i,j=1^m |v^R_i⟩ T^(m)_ij⟨v^L_j|  .

For anyt ≤2 m - 1it exactly replicates the action of the transfer matrix on the starting vectors (see <ref>):

    T^t |v^R_1⟩ = [T^(m)]^t |v^R_1⟩, 
           ⟨v^L_1| T^t = ⟨v^L_1| [T^(m)]^t  .

When applied toTof rankd, Lanczos recovers the original operator exactly afterditerations:T^(d) = T.
This motivates the identification of the physical eigenvalues and eigenvectors ofTwith those of 

    T^(m) = ∑_k=0^m-1|y^R(m)_k⟩λ^(m)_k ⟨y^L(m)_k|  ,

whereλ^(m)_kare the Ritz values and|y^R(m)_k⟩and|y^L(m)_k⟩are the right and left Ritz vectors, already mentioned above.

We may relate the Ritz and Lanczos vectors by considering the eigendecomposition of the tridiagonal matrix

    T^(m)_ij = ∑_k=0^m-1ω^(m)_ikλ^(m)_k (ω^-1)^(m)_kj

whereω^(m)_ikis theith component of thekth right eigenvector ofT^(m)_ijand(ω^-1)^(m)_kjis thejth component of thekth left eigenvector.
Comparing Eqs. (<ref>) and (<ref>), we can write

    |y^R(m)_k⟩   = 𝒩^(m)_k ∑_i=1^m |v^R_i⟩ω^(m)_ik, 
    ⟨y^L(m)_k|   = 1/𝒩^(m)_k∑_j=1^m (ω^-1)^(m)_kj⟨v^L_j|  ,

where𝒩^(m)_kis an arbitrary constant included to set the normalization.
The Ritz values and right/left Ritz vectors are the best Lanczos approximation to the true eigenvalues and right/left eigenvectors, and recover them exactly in the limitm = dwheredis the rank ofT.
By construction, 

    ⟨y^L(m)_k | y^R(m)_l|=⟩δ_k l,

but⟨y^L(m)_k | y^L(m)_l|≠⟩⟨y^R(m)_k | y^R(m)_l|≠⟩δ_klin general.

Eigenvectors are defined only up to an overall (complex) constant, which must be set by convention.
We use unit-normalized right eigenvectors such that∑_i |ω_ik|^2 = 1and set the phase byω_1k = |ω_1k|.
The left eigenvector matrixω^-1is fully defined by this convention via matrix inversion ofω. 
The left eigenvectors are not, in general, unit-normalized if the right eigenvectors are.
This same observation applies for the Ritz vectors and in <ref> is used to identify and understand unphysical states which arise due to violations of Hermiticity by statistical noise.

In lattice applications, we do not have direct access to vectors and operators in the Hilbert space of states, only correlator data of the form

    F_1(t)
        ≡⟨v^L_1 | T^t |v^R_1|=⟩⟨χ | T^t |ψ|⟩/⟨χ|ψ|⟩
        =  C(t) / C(0)  .

where hereC(t)is off-diagonal for the general case.
However, as shown in Ref. <cit.>,T^(m)_ijmay be computed only in terms of these quantities using recursion relations, which iteratively construct generalized correlators evaluated between higher-order Lanczos vectors,

    F_j(t)    = ⟨v^L_j | T^t | v^R_j|,⟩
    
        G_j(t)    = ⟨v^L_j | T^t | v^R_j-1|,⟩
    
        H_j(t)    = ⟨v^L_j-1 | T^t |v^R_j| ⟩ .

These recursions may be derived by inserting <ref> into the above expression, which gives



    F_j+1(t) 
    = ⟨v^L_j+1 | T^t | v^R_j+1|=⟩1/β_j+1γ_j+1[   ⟨v^L_j | T^t+2 - 2 α_j T^t+1 + α_j^2 T^t | v^R_j|+⟩β_j γ_j ⟨v^L_j-1 | T^t | v^R_j-1|
    ⟩   - γ_j ⟨v^L_j-1 | T^t+1 - α_j T^t | v^R_j|-⟩β_j ⟨v^L_j | T^t+1 - α_j T^t | v^R_j-1|⟩]
    
    
    = 1/β_j+1γ_j+1[
               F_j(t+2) - 2 α_j F_j(t+1) 
        + α_j^2 F_j(t) + β_j γ_j F_j-1(t)
        
       - γ_j H_j(t+1) + α_j γ_j H_j(t)
        - β_j G_j(t+1) + α_j β_j G_j(t)
        ],

    G_j+1(t) = ⟨v^L_j+1| T^t |v^R_j⟩ 
    = 1/β_j+1[
        ⟨v^L_j | T^t+1 - α_j T^t | v^R_j|-⟩γ_j ⟨v^L_j-1 | T^t |v^R_j|⟩]
    = 1/β_j+1[
        F_j(t+1) - α_j F_j(t) - γ_j H_j(t)
    ],

    H_j+1(t) = ⟨v^L_j| T^t |v^R_j+1⟩ 
    = 1/γ_j+1[
        ⟨v^L_j | T^t+1 - α_j T^t | v^R_j|-⟩β_j ⟨v^L_j | T^t |v^R_j-1|⟩]
    = 1/γ_j+1[
        F_j(t+1) - α_j F_j(t) - β_j G_j(t)
    ],

with F_0(t) = G_1(t) = H_1(t) = 0 defined for notational convenience.
Similarly, one may derive

    β_j+1γ_j+1 = ⟨r^L_j+1 | r^R_j+1|=⟩⟨v^L_j | T^2 | v^R_j|-⟩α_j^2 - β_j γ_j
    = F_j(2) - α_j^2 - β_j γ_j  .


Using these expressions, starting from the (normalized) correlator dataF_1(t), each iteration proceeds by computing firstγ_j+1andβ_j+1, thenG_j+1(t)andH_j+1(t), thenF_j+1(t), and finally

    α_j+1 = ⟨v^L_j+1 | T | v^R_j+1|=⟩ F_j+1(1)   .

Each step incorporates two more elements of the original correlator, i.e., theα_j,β_j, andγ_jare functions ofF_1(t)fromt=0,…,2j-1.
Thus, the generalized correlators grow shorter with each iteration:F_j(t),G_j(t), andH_j(t)are defined fort=0,…,N_t-2(j-1).
The recursion must terminate after allN_telements of the original correlator have been incorporated, producingα_j,β_j, andγ_jforj=1 …N_t/2afterm=N_t/2steps.

The matrix element problem requires only a subcase of this formalism.
In particular, we use only diagonal two-point correlators withχ= ψand hereafter formally assume 

    |v^R_1⟩ = |v^L_1⟩ = |ψ⟩/√(⟨ψ|ψ|⟩),

such that

    ⟨v^L_1|T^t|v^R_1|=⟩⟨v^R_1|T^t|v^R_1|=⟩⟨v^L_1|T^t|v^L_1|=⟩⟨ψ|T^t|ψ|⟩/⟨ψ|ψ|⟩  .

As already discussed at the top of this section, the underlying transfer matrix is Hermitian, i.e.T = T^†.
For any symmetric convention|β_j| ≡|γ_j|, applying this to the formalism gives|v^R_j⟩ = |v^L_j⟩for allj, and the oblique Lanczos process reduces to standard Lanczos (identically, ifβ_j ≡γ_j).
In this case, all left (L) and right (R) quantities become identical and the distinction may be dropped.[If |β_j| ≠ |γ_j|, then |v^R_j⟩≠|v^L_j⟩ even when T = T^† and |v^R_1⟩ = |v^L_1⟩. However, the L and R versions of any physical quantity will still coincide, as they are necessarily convention-independent.]
However, when statistical noise obscures the Hermiticity ofT, the distinction|v^R_j⟩ ≠|v^L_j⟩forj>1remains important as discussed in <ref>.





 §.§ Convergence & bounds


Explorations of the convergence properties of the Lanczos algorithm have produced several distinct classes of bounds on the approximation errors in its results.
This subsection reviews two, one of which can be used in practice to assess convergence and the other of which is useful to understand the convergence properties of the method.
Versus Ref. <cit.>, the definitions here have been adapted to include factors of𝒩^(m)_kand accommodate complexγ_jandβ_j.

Formally restricting to HermitianT = T^†, the Lanczos formalism allows computation of a rigorous two-sided bound on the distance between a given Ritz valueλ^(m)_kand the nearest true eigenvalueλ<cit.>.
Specifically, as derived for the oblique formalism in Ref. <cit.>, in the special case of HermitianT = T^†,

    min_λ| λ^(m)_k - λ|^2 ≤ B^R/L(m)_k

holds simultaneously for both

    B^R(m)_k ≡ R^R(m)/⟨y^R(m)_k | y^R(m)_k|⟩ ,   
        B^L(m)_k ≡ R^L(m)/⟨y^L(m)_k | y^L(m)_k|⟩,

defined in terms of the residuals[Recall that the first index of ω^(m)_ik and second index of (ω^-1)^(m)_ki correspond to Lanczos vectors and run from 1 to m.]
    R^R(m)_k    = |γ_m+1|^2 |ω^(m)_mk|^2 |𝒩^(m)_k|^2 ⟨v^R_m+1|v^R_m+1|,⟩
    
        R^L(m)_k    = |β_m+1|^2 |(ω^-1)^(m)_km|^2 1/|𝒩^(m)_k|^2⟨v^L_m+1|v^L_m+1| ⟩ .

Regrouping terms admits a convenient simplification,

    B^R(m)_k    = |γ_m+1|^2 
            |ω^(m)_mk|^2 
            V^R(m)_k,
        
    
        B^L(m)_k    = |β_m+1|^2 
            |(ω^-1)^(m)_km|^2
            V^L(m)_k   ,

where

    V^R(m)_k 
           ≡⟨v^R_m+1|v^R_m+1|%s/%s⟩ |𝒩^(m)_k|^2 ⟨y^R(m)_k | y^R(m)_k|⟩
       = ⟨v^R_m+1|v^R_m+1|⟩/∑_ijω^(m)*_ik⟨v^R_i | v^R_j|ω⟩^(m)_jk,
        
    
        V^L(m)_k    ≡⟨v^L_m+1|v^L_m+1|%s/%s⟩ 1/|𝒩^(m)_k|^2 ⟨y^L(m)_k | y^L(m)_k|⟩
       = ⟨v^L_m+1|v^L_m+1|⟩/∑_ij (ω^-1)(m)_ki⟨v^L_i | v^L_j|(⟩ω^-1)^(m)*_kj  .

The second equation in each of the above inserts the Ritz vector definition <ref>.
The cancellation of factors of𝒩^(m)_kreflect independence of Ritz vector normalization convention.
The⟨v^R_i|v^R_j|$⟩ and ⟨v^L_i|v^L_j|$⟩ factors may be computed as discussed in <ref>, and factors of⟨y^R/L(m)_k | y^R/L(m)_k|$⟩ and 𝒩^(m)_k as in <ref> and <ref>.
These bounds are directly computable whenever Lanczos is applied and can therefore be used to monitor convergence in practice. 

Separately, Ritz values and vectors converge to true eigenvalues and eigenvectors with a rate governed by Kaniel-Paige-Saad (KPS) convergence theory <cit.> even for infinite-dimensional systems.
In the infinite-statistics limit of the case of interest (|v^L_1⟩ = |v^R_1⟩ and T = T^†), differences between Ritz values and transfer-matrix eigenvalues satisfy the KPS bound[Note that T_m-n-1(Γ_n) appears here and below in place of T_m-n(γ̂_n) in Ref. <cit.>; the Chebyshev arguments are identical while the order differs by 1 because the largest eigenvalue is labeled λ_0 here as opposed to λ_1 in  Ref. <cit.>. ]

    0 ≤λ_n^(m) - λ_n ≤ (λ_n - λ_∞) [  K_n^(m)tanϕ_n/ T_m-n-1(Γ_n)]^2,

where the T_k(x) are Chebyshev polynomials of the first kind defined by T_k(cos x) = cos(k x),

    Γ_n    ≡ 1 + 2(λ_n - λ_n+1)/λ_n+1 - λ_∞ = 2 e^E_n+1-E_n - 1,

and

    K_n^(m)≡∏_l=1^n-1λ_l^(m) - λ_∞/λ_l^(m) - λ_l,  n > 0,

with K_0^(m)≡ 1 and λ_∞ = 0 for infinite-dimensional T.
For the ground state this simplifies to

    λ_0^(m) - λ_0/λ_0   ≤[ tanarccos z_0/T_m-1(2 e^δ - 1)]^2

where z_n ≡⟨n|v_1|$⟩ andδ≡E_1 - E_0.
For largek,T_k(x) ≈1/2(x + √(x^2 - 1))^k, and this further simplifies to

    λ_0^(m) - λ_0/λ_0 ≲4 (1 - z_0^2)/z_0^2× e^-2 (m-1) δ   δ≫ 1 
    
        e^-4(m-1)√(δ)   δ≪ 1 
         .

As discussed in Ref. <cit.>, near the continuum limit whereδ≪√(δ) ≪1thee^-4 m √(δ) ∼e^-2 t √(δ)convergence of Lanczos is exponentially faster than thee^-t δconvergence of the power-iteration method and standard effective mass.

An analogous KPS bound applies to the overlapsY_n^(m) ≡< n | y_n^(m) >between Ritz vectors and transfer-matrix eigenvectors.
Defining the angleϕ_n^(m) ≡arccosY_n^(m)between these vectors, the KPS bound
 ontanϕ_n^(m) = tanarccosY_n^(m)is given by <cit.>
    tanϕ_n^(m)≤K_n^(m)/T_m-n-1(Γ_n)tanarccos z_n.

For the ground state this simplifies to

    tanarccos Y_n^(m)≤1/T_m-1(2 e^δ - 1)tanarccos z_0,

which can be expanded similarly as

    tanarccos Y_n^(m)≲2 √(1 - z_0^2)/z_0× e^-(m-1) δ   δ≫ 1 
    
        e^-2(m-1)√(δ)   δ≪ 1 
         .

This demonstrates that|Y_n^(m)|^2converges to 1, which indicates that|y_n^(m)⟩is identical to|n⟩, with the same exponential rate that the Ritz values converge to transfer-matrix eigenvalues.




 §.§ Krylov polynomials


The right and left Lanczos vectors|v^R_j⟩and|v^L_j⟩are related to the right and left Krylov vectors

    |k^R_t⟩≡ T^t |v^R_1⟩    |k^L_t⟩≡ (T^†)^t |v^L_1⟩

by the Krylov coefficientsK^R_jtandK^L_jtas

    |v^R_j⟩ = ∑_t=0^j-1 K^R_jt|k^R_t⟩  |v^L_j⟩ = ∑_t=0^j-1 K^L*_jt|k^L_t⟩  .

Equivalently, these coefficients may be thought of as the coefficients of polynomials inT.
These polynomials are Hilbert-space operatorsK^R/L_jwhich excite the Lanczos vectors from the starting vectors|v^R_1⟩and|v^L_1⟩as

    |v^R_j⟩   = K^R_j |v^R_1⟩ = ∑_t=0^j-1 K^R_jt T^t |v^R_1⟩, 
    |v^L_j⟩   = K^L†_j |v^L_1⟩ = ∑_t=0^j-1 K^L*_jt (T^†)^t |v^L_1⟩ .

It is convenient to consider these objects when relating quantities defined in terms of Lanczos vectors to correlation functions, including especially Ritz vectors.

The Krylov coefficients can be computed using a simple recursion.
Beginning with

    K^R_1t = K^L_1t = [ 1,   0 ⋯]_t,
    
    
        K^R_2t   = [ -α_1 /γ_2,  1/γ_2,   0 ⋯]_t, 
    
        K^L_2t   = [ -α_1 /β_2 ,  1/β_2,   0 ⋯]_t,

the coefficients are obtained for each subsequentjfrom

    K^R_j+1,t   = 1/γ_j+1[ K^R_j,t-1 - α_j K^R_jt - β_j K^R_j-1,t], 
    
        K^L_j+1,t   = 1/β_j+1[ K^L_j,t-1 - α_j K^L_jt - γ_j K^L_j-1,t],

usingK^R_j,-1 = K^L_j,-1 = 0for notational convenience.
Note thatK^R/L_jt = 0for allt > j-1.

Once computed,K^R_jtandK^L_jtprovide a convenient means to compute the Lanczos-vector matrix elements that appear in the eigenvalue bound <ref>:

    ⟨v^R_i | v^R_j|   ⟩= ∑_s=0^i-1∑_t=0^j-1⟨v^R_1 | (T^†)^s K^R*_is K^R_jt T^t | v^R_1|
    ⟩   = ∑_st K^R*_is K^R_jtC(s+t)/√(C(0)),
    
    ⟨v^L_i | v^L_j|   ⟩= ∑_s=0^i-1∑_t=0^j-1⟨v^L_1 | T^s K^L_is K^L*_jt (T^†)^t | v^L_1|
    ⟩   = ∑_st K^L_is K^L*_jtC(s+t)/√(C(0)),

restricting to HermitianT = T^†in the second line of each equation.

With the symmetric conventionβ_j ≡γ_jthe definitions <ref> are identical andK^R_jt = K^L_jtalways.[The right and left Lanczos vectors |v^R_j⟩ and |v^L_j⟩ are then excited by K_j = ∑_t K_jt T^t and its conjugate K^†_j, respectively.]
In this case,⟨v^R_i | v^R_j|=⟩ ⟨v^L_i | v^L_j|^⟩*as computed by <ref>, given realC(t).
For other conventions, these quantities may differ nontrivially.





 §.§ Ritz rotators


The tridiagonal matrix eigenvectorsω^(m)and right/left Krylov coefficientsK^R/Lmay be combined to compute the Ritz coefficients

    P^R(m)_kt   ≡𝒩^(m)_k ∑_i=1^m ω^(m)_ik K^R_it, 
    
        P^L(m)_kt   ≡1/𝒩^(m)_k∑_i=1^m (ω^-1)^(m)_ki K^L_it,

which directly relate the Ritz and Krylov vectors as

    |y^R(m)_k⟩   = ∑_t P^R(m)_kt|k^R_t⟩, 
    |y^L(m)_k⟩   = ∑_t P^L(m)*_kt|k^L_t⟩  .

The Ritz coefficients are independent of obliqueβ,γconvention.
They may equivalently be thought of as the coefficients of a polynomial in the transfer matrixT.
These operators are the right/left Ritz rotatorsP^R/L(m), which excite the Ritz vectors from the starting ones as

    |y^R(m)_k⟩   = P^R(m)_k |v^R_1⟩≡∑_t P^R(m)_kt T^t |v^R_1⟩ , 
    |y^L(m)_k⟩   = P^L(m)†_k |v^L_1⟩≡∑_t P^L(m)*_kt [T^†]^t |v^L_1⟩  .

These objects allow straightforward relation of quantities defined in terms of Ritz vectors with expressions in terms of correlation functions.
Note that the Ritz rotatorsP^R/L(m)_kare not proper projection operators.
However, true Ritz projectors may be constructed usingT^(m)in place ofTwith the same coefficients; see <ref>.

The normalization and phase of the Ritz vectors—as encoded by𝒩^(m)_k—is in principle a matter of convention, but in this application unit normalization

    ⟨y^R(m)_k|y^R(m)_k|=⟩ 1

is required so that|y^R(m)_k⟩ = |y^L(m)_k⟩may hold for physical states; this cannot occur if their normalizations differ.
Furthermore, as discussed further in <ref>, extractions of matrix elements from off-diagonal three-point functions depend on this convention; a physical choice is necessary.
Determining𝒩^(m)_kis most straightforwardly accomplished by computing the overlap factors, as shown in the next section.

The Ritz coefficientsP^R/L(m)_ktafford an alternative means of computing the⟨y^R(m)_k | y^R(m)_k|$⟩ and ⟨y^L(m)_k | y^L(m)_k|$⟩ factors in <ref>.
Inserting <ref> gives

    ⟨y^R(m)_k | y^R(m)_l |   ⟩= |𝒩^(m)_k|^2 ∑_st⟨v^R_1 | (T^†)^s P^R(m)*_ks P^R(m)_lt T^t | v^R_1|
    ⟩   = |𝒩^(m)_k|^2 ∑_st P^R(m)*_ksC(s+t)/C(0) P^R(m)_lt,
    
    ⟨y^L(m)_k | y^L(m)_l |   ⟩= 1/|𝒩^(m)_k|^2∑_st⟨v^L_1 | T^s P^L(m)*_ks P^L(m)_lt (T^†)^t | v^L_1|
    ⟩   = 1/|𝒩^(m)_k|^2∑_st P^L(m)*_ksC(s+t)/C(0) P^L(m)_lt ,

invokingT = T^†in the second equality of each.
This expression provides a useful consistency check.
In the noiseless case, whenP^R/L(m)_ktare properly normalized, <ref> should yield⟨y^R(m)_k | y^R(m)_l|=⟩ ⟨y^L(m)_k | y^L(m)_l|=⟩ δ_kl.
In the noisy case of <ref>, similar holds for the subset of physical states.





 §.§ Overlap factors


The overlap factors may be obtained directly from the eigenvectorsω^(m)_ikof the tridiagonal matrix as

    [Z^R(m)_k]^* = ⟨ψ | y^R(m)_k|   ⟩= 𝒩^(m)_k ∑_i=1^m |ψ| ⟨v^L_1 | v^R_i|ω⟩^(m)_ik
       = 𝒩^(m)_k |ψ| ω^(m)_1k,
        
    
        Z^L(m)_k = ⟨y^L(m)_k | ψ|   ⟩= 1/𝒩^(m)_k∑_j=1^m (ω^-1)^(m)_kj⟨v^L_j | v^R_1||⟩ψ|
        
       = |ψ|/𝒩^(m)_k (ω^-1)^(m)_k1    ,

using the definition <ref>.
The specific choices ofLversusRin these definitions are motivated further in <ref>, where in the noisy case they provide useful intuition.
However, the two definitions coincide for all physical states with degenerate left and right eigenvectors, and other equally correct ones are possible.

The overlap factors provide a convenient means of determining𝒩^(m)_kto enforce unit normalization for the Ritz vectors.
Specifically, note thatZ^L(m)_k = Z^R(m)_konly if|y^R(m)_k⟩ = |y^L(m)_k⟩, which in turn requires compatible normalizations.
Demanding that this holds gives 

    |𝒩^(m)_k|^2 =  (ω^-1)^(m)*_k1/ω^(m)_1k.

WhenT^(m)is Hermitian as in the noiseless case,(ω^-1)^(m)_ki = ω^(m)*_ikso that|𝒩^(m)_k|^2 = 1automatically.
However, in the noisy case explored in <ref>, it must be set manually; for unphysical noise-artifact states this will not be possible, as(ω^-1)^(m)*_k1 / ω^(m)_1kwill be complex in general when|y^R(m)_k⟩ ∝ |y^L(m)_k⟩.

As mentioned previously, some convention is also required to set the phase of the Ritz vectors (and thus of𝒩^(m)_k).
The standard convention for the phase of the true eigenstates is typically set to give real overlap factors, so we adopt this for the Ritz vectors as well.
From <ref> we see that the conventionω^(m)_1k = |ω^(m)_1k|is equivalent to choosingZ^R(m)_kreal if𝒩^(m)_kis.
In the noiseless case whereω^(m)is unitary, this convention is inherited byZ^L(m)_k.
However, in the noisy case, unphysical states with complexZ^L(m)_karise as discussed in <ref>.

Properly normalized Ritz coefficients allow a different but equivalent computation,

    Z^L(m)_k
           = ∑_t=0^m-1 P^L(m)_kt⟨v^L_1|T^t| v^R_1||⟩ψ|
        = ∑_t P^L(m)_kt C(t) / |ψ| ,
    
    
        [Z^R(m)_k]^*
           = ∑_t=0^m-1 |ψ| ⟨v^L_1|T^t| v^R_1|P⟩^R(m)_kt
        = ∑_t P^R(m)_kt C(t) / |ψ|  ,

inserting the Ritz rotators <ref> into the definitions of <ref>.
This allows derivation of several nontrivial identities and can be useful for cross-checks.
For example, in the noiseless case, complex values indicateω^(m)_1k = |ω^(m)_1k|has not been enforced correctly; in the noisy case, this may signal to appearance of unphysical states as discussed below in <ref>.





 §.§ Matrix elements


With the right/left Ritz coefficientsP^R/L(m)_ktcomputed and normalized, we can derive an expression to directly compute matrix elements from three-point functions.
Using <ref>, the derivation proceeds as 

    J_fi^(m)   = ⟨y'^L (m)_f | J | y^R(m)_i|
    ⟩   = ∑_σ,τ=0^m-1P'^L(m)*_fσ⟨v^' L_1| T^σ J T^τ |v^R_1|P⟩^R(m)_i τ
       = ∑_στP'^L(m)*_f σ⟨ψ'| T^σ J T^τ |ψ|⟩/ |ψ'| |ψ| 
        P^R(m)_i τ
       = ∑_στP'^L(m)*_f σ C^3pt(σ, τ) /√( C'(0) C(0) )
        P^R(m)_i τ,

where the primedP'^L(m)_f σare computed from the final-state two-point functionC'(t), and the unprimedP^R(m)_i τfrom the initial-stateC(t).
This expression is the main result of this work.
Note that for any state of physical interest <ref> will reduce to <ref>—this is all states in the noiseless case where the data is manifestly Hermitian, but only a subset in the presence of noise as discussed in <ref>.
The generalization to matrix elements of products of currents or other temporally nonlocal operators follows immediately by replacingJin Eq. (<ref>) with the corresponding composite or nonlocal operator.

The extracted value is in general sensitive to the choice of Ritz vector normalization asJ_fi^(m) ∝𝒩^(m)_i / 𝒩^'(m)_f.
In the special case of diagonal three-point correlators whereψ' = ψ, these factors cancel and the value is convention-independent.
However, convention dependence remains in the off-diagonal case, emphasizing the importance of enforcing unit normalization as noted above to obtain physically interpretable results.

It is natural to ask why <ref> should be preferred over⟨y'^R (m)_f | J | y^R(m)_i|$⟩ or ⟨y'^L (m)_f | J | y^R(m)_i|$⟩.
The definition employed is privileged in that, for the other two options, the equivalents of the last equality in <ref> must invokeT=T^†.
However, the distinction is irrelevant in practice: for physical states the right and left Ritz vectors are degenerate, in which case all three definitions produce identical results.

The Lanczos matrix-element extraction has stricter data requirements than standard analysis methods.
Specifically, evaluatingJ^(m)_firequiresC^3pt(σ, τ)for all0 ≤σ≤m-1and0 ≤τ≤m-1, which includes data for all sink times 0 ≤t_f ≤2(m-1)formLanczos iterations.
This is no concern for quark-line disconnected and gluonic operators, where three-point data is naturally available for all0 ≤τ< N_tand0 ≤t_f < N_t, or whereverC(t_f)is available.
However, this clashes with the typical strategy used when computing connected contributions with sequential source methods.
The typical mode of inverting through the sink requires a separate calculation for eacht_fdesired, each obtaining allτat fixedt_f.
Thus, to avoid computation,C^3ptis often computed sparsely int_f, typically avoiding smallt_fand skipping points in the evaluated range.
For standard analysis methods, this strategy gives better confidence in control over excited-state effects given a fixed budget.
However, it is incompatible with a Lanczos analysis, which is thus unlikely to be applicable to existing connected three-point datasets.
While inconvenient, these data requirements are not necessarily a disadvantage.
The small-t_fpoints typically discarded have good SNR, and Lanczos may usefully incorporate them without the possibility of worsening ESC.

At most, the estimate <ref> incorporates only1/4of the computable lattice three-point function, corresponding to slightly less than half of the useful data where operator ordering∼ψ' J ψis satisfied.
The useful data correspond to allt_f, τin0 …N_t-1satisfyingt_f ≥τ, but the sums in <ref> overτandσ= t_f-τrun only toN_t/2-1at maximalm=N_t/2, excluding allσ, τ≥N_t/2.
While it would be desirable to incorporate all data available in the noisy case, the formalism does not allow it; however, we note that all excluded points lie in the regiont_f ≥N_t/2, where thermal effects are significant or dominant.
Remarkably, as seen in theN_t/2-dimensional example of <ref>, the subset of three-point data incorporated is sufficient to solve for all matrix elements exactly in a finite-dimensional setting.





§ MANIFEST HERMITICITY & NOISELESS EXAMPLE


For our problems of interest, the transfer matrixTis Hermitian, with real eigenvalues and degenerate left and right eigenvectors.
As emphasized throughout <ref>, physical interpretability requires that this also holds for the Lanczos approximation of the transfer matrix, at least for states of physical interest.
As explored in this section, in the absence of statistical noise Lanczos produces a fully Hermitian eigensystem.
We demonstrate that not only do Lanczos matrix-element estimates converge, they do so much more rapidly than estimates with previously available approaches.

It is straightforward to see that Lanczos respects Hermiticity when it is manifest in the correlator data.
In this case,T = T^†may be applied in the formalism of <ref> without introducing any inconsistencies.
Taking the symmetric convention[As discussed in <ref>, all physical quantities are independent of the choice of oblique convention. The reductions discussed in this paragraph apply for any symmetric convention |β_j| ≡ |γ_j|, which produce identical results with manifestly Hermitian data.]β_j ≡γ_j, oblique Lanczos reduces identically to standard Lanczos, which produces degenerate right/left Lanczos vectors|v^R_j⟩ = |v^L_j⟩ = |v_j⟩by construction.
It follows immediately that

    T^(m)_ij = ∑_ij|v_i⟩⟨v_i | T | v_j|⟨%s|⟩v_j

is Hermitian, thus the Ritz valuesλ^(m)_kare real and the right/left Ritz vectors are degenerate|y^R(m)_k⟩ = |y^L(m)_k⟩ ≡|y^(m)_k⟩.
All left and right quantities coincide, and theL/Rdistinction may be dropped.



To verify these statements and demonstrate the Lanczos method, we apply it to a finite-dimensional, exactly Hermitian mock-data example.
The simulated problem is the most general one that can be treated with the procedure defined in <ref>: an off-diagonal three-point function and corresponding pair of diagonal initial- and final-state two-point functions.
The two-point functions are defined as

    E_i    = 0.1 (i+1) ,   
                     E'_f    = [ E_f^2 + 0.1^2 ]^1/2, 
    
        Z_i    = 1/√(2 E_i) ,
                     Z'_f    = 1/√(2 E_f), 
    
        C(t)    = ∑_i=0^N_t/2-1 Z_i^2 e^-E_i t, 
                 C'(t)    = ∑_f=0^N_t/2-1 Z_f^' 2 e^-E'_f t,

while the three-point function is defined as 

    J_fi   = √(4 E'_0 E_0/4 E'_f E_i)J̃_fi, 
    J̃_00   = J_00 = 1, 
    J̃_fi   ∼𝒩(0,1),    (f+i) > 0,
    
    
        C^3pt(σ, τ)    = ∑_i,f=0^N_t/2-1 Z'_f Z_i J_fi e^-E'_f σ -E_i τ,

whereM̃_fi ∼𝒩(0,1)indicates that those values have been drawn from a unit-width normal distribution centered at zero.[The precise values of J_fi used are provided in a supplemental data file.]
Initial- and final-state effective energies and the standard ratio <ref> are shown in <ref>.
The energiesE_iandE'_fare chosen to resemble an off-forward two-point function, with the final-state spectrum a boosted version of the initial one.
The overlap factors are flat up to the single-particle relativistic normalization of states, simulating the case of severe excited-state contamination.[In fact, if left untruncated, this choice of overlaps is unphysically severe: the infinite sum C(0) = ∑_k |Z_k|^2 ∼∑_k 1/k does not converge, whereas C(0) is always finite in practice.]
The excited-state and transition matrix elements have mixed signs, with the only structure in their magnitudes from the imposed single-particle relativistic normalization.
The value of the ground-state matrix element is fixed to 1; this is much larger than the typical magnitude ofJ_fiwithf+i > 0, so thatC^3ptis ground-state dominated.

We proceed following the steps laid out at the top of <ref> separately to each of the initial- and final-state correlators,C(t)andC'(t).
It is straightforward to numerically verify the claims above.
Note that all statements made in this section should be taken to apply for exact arithmetic.[The Lanczos algorithm is notoriously susceptible to numerical instabilities due to round-off error at finite precision. <Ref> discusses where precisely high-precision arithmetic is required.]
With any|β_j| = |γ_j|, the tridiagonal matricesT^(m)_ijare real and symmetric, and thus have real Ritz valuesλ^(m)_kand unitary eigenvector matricesω^(m).
The right/left Krylov coefficients are real and identical,K^R_jt = K^L_jt, such thatK^R_j = K^L †_j, implying|v^R_j⟩ = |v^L_j⟩.
Consistently, evaluating <ref> confirms⟨v^R_i|v^R_j|=⟩ ⟨v^L_i|v^L_j|=⟩ δ_ij.
The right and left Ritz coefficientsP^R/L(m)are also identical, such that the Ritz rotators are Hermitian as necessary for|y^R(m)_j⟩ = |y^L(m)_j⟩.
Similar statements apply for primed final-state quantities.
TheL/Rdistinction is thus dropped in the discussion of results that follows.


<Ref> shows the spectra of Ritz values extracted for different numbers of Lanczos iterationsm.
One additional Ritz value is produced after each iteration, and the spectra are recovered increasingly accurately asmincreases.
This accuracy is reflected in the decreasing size of the eigenvalue bounds <ref>, represented by the error bars in <ref>; <ref> simplifies when|v^R_j⟩ = |v^L_j⟩, so these may be computed without further effort.
Because each spectrum has onlyN_t/2states, Lanczos recovers all energies exactly at the maximalm=N_t/2iterations.




After obtaining the initial- and final-state Ritz coefficientsP'^(m)_ftandP^(m)_it, the matrix elements may be computed using <ref>, which reduces to

    J^(m)_fi = ∑_στ (P')^(m)*_f σC^3pt(σ, τ)/√(C'(0) C(0)) P^(m)_i τ

in the noiseless case, recovering the form of <ref> for all statesf,i.
The resulting estimates of the ground-state matrix elementJ_00are shown in <ref>, alongside effective matrix elements computed with power iteration (<ref>) and the summation method (<ref>).
The Lanczos estimate converges rapidly to the true value, reproducing it exactly at maximalm=N_t/2where Lanczos solves the system.
The advantages in comparison to the other methods are apparent, neither of which converge near to the true value before the full Euclidean time range available is exhausted.
As a more qualitative advantage, the Lanczos estimate does not approach the true value smoothly, advertising that results are unstable until convergence is achieved.
The benefit is made apparent by considering the summation curve, which appears to be asymptoting—but to an incorrect value.
This disceptiblity is investigated more directly in <ref>.


At maximalm = N_t/2, applied to thisN_t/2-dimensional example, Lanczos extracts not only the trueJ_00exactly but all(N_t/2)^2matrix elements:J^(N_t/2)_fi = J_fi.
As illustrated in <ref>, the convergence is more rapid for lower-lying states.
Comparing with <ref>, this may be associated to a combination of the faster convergence of lower-lying eigenvalues, as expected given the relationship between eigenvalue and eigenvector convergence discussed in <ref>, and misidentification of higher-lying states with true ones.



Finally, <Ref> shows the initial- and final-state overlap factors computed using <ref>.
As with the other quantities estimated, theZ^(m)_krapidly converge to their true values, with lower-lying states converging more quickly, and recovers the trueZ_kexactly atm = N_t/2 = 12.





§ NOISE & THE STRANGE SCALAR CURRENT


The previous section explored application of the Lanczos matrix-element procedure to a noiseless example.
However, as emphasized throughout the preceding discussion, important differences arise when introducing statistical noise.
Without noise, the underlying Hermiticity of the transfer matrix is manifest in the correlator data, but noise obscures this Hermiticity, resulting in unphysical states which must be identified and discarded.
This section explores these issues and techniques to treat them in an application to noisy lattice data.





 §.§ Problem statement & data


Specifically, we apply the Lanczos method to extract forward matrix elements of the strange scalar current

    J(τ) = ∑_y⃗s(y⃗,τ)   s(y⃗,τ)

in the nucleon and its first few excited states.
We employ a single ensemble ofN_cfg = 1381configurations generated by the JLab/LANL/MIT/WM groups <cit.>, using the tadpole-improved Lüscher-Weisz gauge action <cit.> andN_f = 2+1flavors of clover fermions <cit.> defined with stout smeared <cit.> links on a48^3 ×96lattice volume. Action parameters are tuned such thata ≈0.091 fmandM_π≈170 MeV<cit.>. 
The data for the example are a diagonal three-point function with zero external momentum (i.e.p⃗ = p⃗' = 0) and the single corresponding nucleon two-point function projected to zero momentum, all generated in the course of the studies published in Refs. <cit.>.
Details are largely as in those references, but reproduced here for completeness.
Ref. <cit.> used data generated independently on configurations from the same ensemble.

Each nucleon two-point function measurement is computed as

    C(t; x_0) = ∑_x⃗[ Γ⟨χ(x⃗, t+t_0) χ(x_0) |⟩],

wherex_0 = (x⃗_0, t_0)is the source position and with the trace over implicit Dirac indices, including those of the spin projector

    Γ = P_+ (1 + γ_x γ_y)    with    P_+ = 1/2 (1+γ_t)   .

The interpolator employed is

    χ(x) = ϵ^abc [u_b^S(x)^T C γ_5 d_c^S(x)] u_a^S(x),

whereCis the charge conjugation matrix, andu^S(x)andd^S(x)are up- and down-quark fields smeared using gauge-invariant Gaussian smearing to radius 4.5 with the smearing kernel defined using spatially stout-smeared link fields <cit.>.
This is evaluated at 1024 different source positionsx_0on each configuration, arranged in two interleaved4^3 ×8grids with an overall random offset.
Averaging over all source positions yields the per-configuration measurements ofC(t)used in this analysis.
<Ref> shows the effective mass computed from it.

Crucially, we invoke the underlying Hermiticity ofTto discard the measured imaginary part of the two-point correlatorC(t), which is real in expectation.
While the procedure is well-defined for complex correlators, the clear separability of physical from noise-artifact states discussed below appears to arise as a result of manually enforcing Hermiticity at this level.

The three-point function is computed as

    C^3pt(σ, τ; x_0) = ∑_x⃗[ Γ⟨χ(x⃗, t_f+t_0) J(τ+t_0) χ(x_0)|,⟩]

wheret_f = σ+τ.
Integrating over quark fields results in a quark-line disconnected diagram.
The strange quark loops are evaluated stochastically <cit.> using one shot ofZ_4noise per configuration, computing the spin-color trace exactly and diluting in spacetime using hierarchical probing <cit.> with a basis of 512 Hadamard vectors.
These are convolved with the grids of two-point functions and vacuum subtracted to produce the three-point function.
As with the two-point correlator, we discard the measured imaginary part.
<Ref> shows the standard ratio computed from these data.




Note that the methodology assumes unit-normalized eigenvectors and the corresponding definitions

    C(t)    = ∑_k |Z_k|^2 e^-E_k t, 
    
        C^3pt(σ, τ)    = ∑_fi Z^' *_f Z_i J_fi e^-E'_f σ - E_i τ  .

This absorbs kinematic factors and relativistic normalizations into the definitions ofZandJ.
In general, the matrix elements and overlaps extracted must be rescaled by appropriate factors to isolate the quantities of interest; this is no different than when using methods based on correlator ratios.
However, in this example, all such normalization and kinematic factors cancel other than a factor of√(2)absorbed into the overlap factors (for single-particle states).
The matrix elements extracted here thus correspond directly to the physically normalized ones, up to adjustments required if any of the resolved states is a multi-particle one.
However, we emphasize that the quantities extracted are bare.
Accounting for renormalization and operator mixing to obtain a physical quantity would requires secondary calculations unrelated to the subject of this work.
In this case, treating mixing is particularly important: the strange scalar current mixes with the light one, whose matrix element is much larger.





 §.§ Lanczos with noise


Applying Lanczos to noisy correlator data yields spurious noise-artifact states which must be discarded to obtain physically meaningful results.
Ref. <cit.> introduced one strategy to identify them based on the Cullum-Willoughby (CW) test <cit.>; we employ it here as well.
In addition, consideration of the Lanczos approximation of the transfer matrix eigensystem provides a different and complementary view of this issue.

Aftermiterations, it can be shown from <ref> that Lanczos reproduces the incorporated correlator data (i.e., fort ≤2m-1) exactly:[We thank Anthony Grebe for this important insight.]
    C(t)    = ⟨ψ | [T^(m)]^t |ψ|
    ⟩   = ∑_k ⟨ψ | y^R(m)_k|(⟩λ^(m)_k)^t ⟨y^R(m)_k | ψ|
    ⟩   = ∑_k Z^R(m)*_k Z^L(m)_k  e^-E^(m)_k t ,

whereE^(m)_k = -logλ^(m)_k.
For noisyC(t), this requires contributions from states with complex eigenvalues that oscillate int, as well as generally complex overlap productsZ^R(m)*_k Z^L(m)_k, which may only occur with distinct left and right Ritz vectors.
Due to the enforced reality ofC(t), these states necessarily contribute in pairs with complex-conjugate Ritz values and overlap products.
However, we find that we are able to identify a Hermitian subspace of statesHwith real Ritz values and degenerate left and right Ritz vectors

    |y^R(m)_k⟩ = |y^L(m)_k⟩ = |y^(m)_k⟩  ∀  k ∈ H ,

such that, for that part of the approximation ofT,

    T^(m)_H ≡∑_k ∈ H|y^(m)_k⟩λ^(m)_k ⟨y^(m)_k|

is manifestly Hermitian.
These states are physically interpretable, while the others are clearly associated with (or at least contaminated by) noise.
This separation may provide a mechanistic explanation for how Lanczos avoids exponential SNR degradation, as discussed in the conclusion.
Besides insight, in practice this observation also provides a hyperparameter-free prescription for state filtering as detailed below.
While additional filtering with the CW test remains necessary, the requirement for tuning is reduced.

The remainder of this subsection presents this filtering prescription as we proceed through the analysis, as well as general observations.
We employ bootstrap resampling to study the effects of statistical fluctuations and to estimate uncertainties in the next subsection.
Each bootstrap ensemble is constructed by randomly drawingN_cfg = 1381configurations with replacement from the original ensemble.
On the correlator for each bootstrap ensemble, we independently apply the steps laid out in <ref> to compute all the various quantities therein.
Unlike in the noiseless example, high-precision arithmetic is necessary in only a few places, none of which are computationally expensive; see <Ref> for details.

Running the oblique Lanczos recursion of <ref> produces inN_t/2=48iterations the elementsα_j,β_j,γ_jof the tridiagonal matricesT^(m).
Allα_jand productsβ_j γ_jare real becauseC(t)is, but may be negative; whichβ_j γ_j < 0varies per bootstrap.
With the symmetric conventionβ_j ≡γ_j ≡√(β_j γ_j), negative fluctuations produce pure imaginaryβ_jandγ_j.

DiagonalizingT^(m)for eachmyields Ritz valuesλ^(m)_kand eigenvector matricesω^(m).
The majority of Ritz values extracted are complex.[Diagnosed as Im λ / |λ| > 10^-8 numerically. This convention is used for all similar statements in this section.]
The corresponding states may be excluded from the Hermitian subsetHimmediately.
The number of real and complex Ritz values at fixedmvaries per bootstrap draw; for allm ≥3Lanczos iterations, there are a minimum of 3 real Ritz values in each.

Unit normalization cannot be simultaneously enforced for states outside the Hermitian subspace while maintaining our definitions, providing a useful means of identifying them.
Defined as they appear in the eigendecomposition, the conventions of the left eigenvectors are fully determined by those of the right.
Attempting to compute|𝒩^(m)_k|^2using <ref> for such states thus yields complex-valued(ω^-1)^(m)*_k1 / ω^(m)_1k, with the apparent contradiction because⟨ψ|y^R|$⟩ and ⟨ψ|y^L|$⟩ cannot be made equal if|y^L⟩ ≠|y^R⟩.
We thus identify the Hermitian subspaceHas those stateskfor which(ω^-1)^(m)*_k1 / ω^(m)_1kis real and positive (and with realλ^(m)_k).
Manual normalization is necessary for states inHafter the firstmwhereβ_j γ_j < 0, for which|𝒩^(m)_k|^2 ≠1in general.[This corresponds to the iteration where the standard Lanczos process would terminate or refresh, and oblique Lanczos is required to proceed. Before this, oblique and standard Lanczos coincide.]

We diagnose the remaining spurious states using the CW test <cit.> as in Ref. <cit.>.
To do so, we defineT̃^(m)asT^(m)with the first row and first column removed and diagonalize it to obtain them-1CW valuesλ̃^(m)_l.
Ritz values of spurious states will have a matching CW valueλ̃^(m)_l; non-spurious states will not.
Thus, we keep all states inHwhich satisfy

    Δ^CW(m)_k > ϵ^CW(m),

whereϵ^CW(m)is some threshold value, and

    Δ^CW(m)_k = min_λ̃∈{λ̃^(m)_l}_ℝ | λ^(m)_k - λ̃ |,

restricting the minimum only the real CW values; when there are none, we accept all states.
As noted in Ref. <cit.>, results are sensitive to the cutϵ^CW(m)and thus the procedure to choose it introduces the primary source of hyperparameter dependence in these methods.
In this work, we use a simple heuristic choice versus the more extensive analysis employed in Ref. <cit.>, taking 

    ϵ^CW(m) = 
        max_k [Δ^CW(m)_k] - min_k [Δ^CW(m)_k]
        /
            a |H'| + bm
        ,

where|H|is the number of states inH.
We adopt the relatively permissivea = 10andb = 1and rely on outlier-robust estimators to compensate for mistuning, as discussed below.
The surviving subset are identified as the physical ones.

Without restricting to the Hermitian subspace, the CW test alone admits100%of complex-eigenvalue states.
After restricting to real-eigenvalue states, the normalizability condition and CW test are highly redundant.
Of the real eigenvalues, normalizability removesO(1%)of states admitted by CW alone, while CW removesO(10%)of states admitted by normalizability alone.
Of the surviving states, none haveλ^(m)_k > 1(corresponding to negativeE^(m)_k, which may be a thermal state).
Due to statistical fluctuations,O(2%)haveλ^(m)_k < 0(corresponding to imaginaryE^(m)_k); notably, the CW test removesO(75%)of such states admitted by normalizability.
Note that these statistics depend on the choice of CW cut.




 §.§ Results


With state filtering complete, we may proceed to computing observables and estimating their statistical uncertainties.
We note that for all states that survive filtering, the left and right Ritz projector coefficients are equal up to round-off error, as expected for states from the Hermitian subspace.
It follows that theLandRdefinitions of all observables will coincide for these states, so we may drop the distinction for the results in this section.
For matrix elements in particular, <ref> is recovered for all statesf,i.

Different numbers of states survive filtering in each different bootstrap ensemble.
To avoid dealing with the complications of error quantification with data missingness, we present results for only three states, the maximum number that survives in every ensemble.
However, we note that≈85%of ensembles have at least four states,[This is sufficient to calculate some quantities for this state with some reasonable but ad-hoc definitions; see <ref>.] and≈7%have at least five, with the precise fraction depending onm.

Uncertainty quantification requires associating states between different bootstrap ensembles.
There is no unique or correct prescription for doing so, so this represents another primary source of hyperparameter dependence.
In this analysis, we make the simple choice of associating the states by sorting on their Ritz valuesλ^(m)_kand taking the ground state as the one with the largestλ^(m)_k, the first excited state the one with the second largestλ^(m)_k, etc.

Inspection of bootstrap distributions indicates frequent misassociations by this procedure.
Rather than tuning our filtering and association schemes, we compensate using outlier-robust estimators to compute central values and uncertainties from the bootstrap samples.
We use the by-now de-facto community standard as implemented in the  software package <cit.>: we take the median rather than the mean, an estimator based on the width of the1σconfidence interval rather than the standard deviation,[
Specifically, we use
σ = max(Δ^+, Δ^-) where 

    ŷ   = Median_b[y_b] 
    Δ^-    = ŷ - Percentile_b[y_b, 100 s] 
    Δ^+    = Percentile_b[y_b, 100 (1-s)] - ŷ

with b indexing bootstrap samples and s = ∫_-∞^-11/√(2 π) e^-x^2/2≈ 0.16
is the CDF of the unit normal distribution evaluated at -1.] and the usual Pearson correlation matrix to construct covariance matrices.
We note immediately that, across all results presented here, this gives central values that fluctuate less than their errors and measured correlations would suggest.
Separately, we observe that the median over bootstraps appears to have a regulating effect versus simply carrying out the analysis on the central value (i.e. meanC(t)over bootstraps).
It will be important to build up a statistical toolkit better suited for treating Lanczos results sensitive to the appearance of spurious eigenvalues.

We now present the results of this analysis, beginning with quantities computed fromC(t)only.
<Ref> shows the energies of the three lowest-lying states as extracted by Lanczos.
The results are similar to those seen in Ref. <cit.>: Lanczos energy estimates exhibit no exponential decay in SNR, in contrast to the effective energy, where useful signal is available only up tot ≈30.
The ground state is resolved with excellent signal. Noise increases moving up the spectrum.
<Ref> shows results for the nearly-resolved third excited state.

With the precision available, Lanczos does not resolve several known intermediate states in the spectrum.
WithM_π≈0.078andM_N ≈0.42, theN(1)π(-1)andNππmulti-particle states both lie nearE ≈0.6, between the ground and first excited state found by Lanzcos.
This is to be expected: at finite precision, Lanczos is known to miss eigenvectors (here, states) with small overlap with the initial vector (here,|ψ⟩) <cit.>, and these states are known to have very small overlaps with the single-hadron interpolating operators used here <cit.>.
Their absence in the results points immediately to several topics requiring further study: the dynamics that determine which states are extracted by Lanczos, and how badly such missed intermediate states contaminate Lanczos matrix-element estimates.


As with effective energies and matrix elements, Lanczos estimates at differentmcarry independent information that can be combined to obtain a more precise estimate.
To demonstrate, for all estimates in this section, we fit a constant to allm ≥8to exclude the less-noisy points at earlym.
This simple but arbitrary choice serves the point of demonstrating that estimates at differentmcarry independent information, but it will be important to find more principled convergence diagnostics for real applications.
Uncertainties on fitted values are estimated by linear propagation from the data covariance matrix to avoid the complications of nested bootstrapping or underestimation from sharing a covariance matrix between different bootstraps.
The reduction in uncertainty versus the average uncertainty of the data gives a notion of the amount of independent measurements in the41points included in each fit.
For the energiesE_k, the fitted values are≈9,4,4times more precise than the data fork=0,1,2, respectively.
Reduction by a factor√(41) ≈6.4corresponds to the expected reduction for 41 statistically independent points, but
fluctuations about this value are expected due to noise.
Systematic deviations from this value arising from correlations between data points are not observed.

 Paper Title: Real-time chiral dynamics at finite temperature from quantum simulation.


Methodological Foundations of Modern Causal Inference in Social Science Research
    Guanghui Pan[Department of Sociology, University of Oxford. 41 Park End Street, Oxford, the United Kingdom, OX1 1HH. Email: guanghui.pan@sociology.ox.ac.ukguanghui.pan@sociology.ox.ac.uk. Manuscript in progress, do not cite or redistribute without permission.]
    August 5, 2024
========================================================================================================================================================================================================================================================================





§ INTRODUCTION

In recent years, the exploration of quantum simulations for field theories <cit.> has garnered significant attention across many subfields of physics <cit.>. This growing interest is largely driven by the advancements in quantum computing technology, which promises to revolutionize our ability to tackle complex and computationally intensive problems. Traditional classical computers, despite their power, often fall short when dealing with the intricate dynamics and properties of quantum systems due to the exponential growth of computational resources required. On the other hand, quantum simulations hold the potential for breakthrough discoveries in areas such as high-energy physics, material science, and chemistry, aiding in understanding phase transitions <cit.>, solving quantum many-body problems <cit.>, and discovering new materials with unique properties <cit.>.

To explore these potential developments, we focus on the Schwinger model <cit.>, quantum electrodynamics (QED) in (1+1) dimensions. 
The Schwinger model provides an excellent testing ground for studying various phenomena, including confinement, dynamical mass generation, and anomaly-induced symmetry breaking. It offers the simplicity of fewer degrees of freedom while still encapsulating the rich physics of gauge theories, making it an ideal benchmark model for quantum simulations on both quantum and classical hardware <cit.>.


In this study, we explore the real-time dynamics of the chiral magnetic effect (CME) at finite temperature within the massive Schwinger model. The CME refers to the generation of an electric current along a magnetic field in systems where there is an imbalance between left-handed and right-handed fermions <cit.>. This imbalance is typically produced by the presence of a chiral chemical potential. To investigate the CME within the context of the Schwinger model, we employ a quench protocol as outlined in our previous study <cit.>. By introducing a chiral chemical potential through this quench process, we drive the system out of equilibrium and analyze the resulting vector currents and how they evolve over time. The Hamiltonian is adjusted to include the time-dependent chiral chemical potential, enabling the investigation of the CME within a quantum computing framework.

The interest in the study of the temperature dependence of the real-time CME stems from the following consideration. The CME is intrinsically a non-equilibrium, real-time phenomenon. However, when the external fields and the chiral chemical potential vary slowly with time, the CME response is driven by the fermion zero modes, and should thus be temperature-independent <cit.>. This independence of CME current on the temperature has been established in analytical computations <cit.>. However, under a rapid quench, the CME current receives contributions from excited fermion modes <cit.>, and can thus develop a dependence on the temperature. This temperature dependence  has not been evaluated before from first principles. In this paper, we will evaluate it using the methods of quantum simulation in massive Schwinger model at finite temperature.  

We employ the quantum imaginary time evolution (QITE) algorithm to study the thermal states and utilize the Suzuki-Trotter decomposition for real-time evolution. The QITE algorithm is essential for efficient quantum simulation on quantum circuits for several reasons, and various related algorithms have been developed <cit.>.
It enables the simulation of thermal states, crucial for studying phenomena like phase transitions at finite temperatures <cit.>. Since imaginary time evolution involves non-unitary operations, which cannot be directly implemented on quantum hardware, QITE approximates these operations with unitary ones, making it feasible on quantum circuits. Additionally, traditional classical algorithms struggle to prepare thermal states of many-body systems, while QITE allows for efficient exploration of these state spaces. Its versatility makes it applicable to a wide range of quantum systems and models, bridging theoretical models and practical implementations on quantum hardware. By iteratively applying small imaginary time steps and recalibrating the system, the QITE ensures the system gradually reaches the desired state where subsequent measurements are performed to extract physical observables.

Our motivation follows the successful quantum simulations performed on the (1+1)-dimensional Nambu–Jona-Lasinio (NJL) model <cit.>, where quantum algorithms provided insights into the chiral phase transition at finite temperature and chemical potential. Extending this line of research, our previous work on the quantum computational study of field theories has demonstrated the feasibility and efficacy of these simulations in delivering significant theoretical insights. This work on the Schwinger model bridges the study of simple models accessible by current quantum hardware and the more complex theories underpinning the standard model of particle physics. By exploring the unique properties and dynamics of the Schwinger model, we aim to provide insights into chiral dynamics in low-dimensional quantum field theories.

The rest of this article is organized as follows. In Sec. <ref>, we present the model and the quench protocol that we study in this work. In Sec. <ref>, we describe our algorithm of the quantum imaginary time evolution to implement the thermal states and subsequent expectation evaluation to extract physical observables at finite temperature. In Sec. <ref>, we show the results from our quantum simulations, where we discuss the real-time chiral dynamics of the current at finite temperatures, the phase diagram of the Schwinger model, and the temperature dependence of the electric charge (axial current). Finally, the article is concluded in Sec. <ref>. 




§ THE MASSIVE SCHWINGER MODEL WITH THE TOPOLOGICAL TERM
 


 §.§ The Schwinger Hamiltonian

The action of the massive Schwinger model <cit.> with θ term in (1+1)-dimensional Minkowski space is

    S = ∫ d^2x[-1/4 F^μν F_μν + gθ/4πϵ^μνF_μν + ψ̅(iD-m)ψ],

with D=γ^μ(∂_μ + i gA_μ).
Here, A_μ is the U(1) gauge potential, E=Ȧ_1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass, and γ^μ are two-dimensional γ-matrices satisfying the Clifford algebra. When g≠0, the θ-term induces the classical background electric field, which breaks the parity symmetry. The gauge field A_μ and the coupling constant g have mass dimensions 0 and 1, respectively.
By a chiral transformation, ψ→ e^iγ^5θψ and ψ̅→ψ̅e^iγ^5θ, the action is transformed to 

    S = ∫ d^2x[-1/4 F^μν F_μν  + ψ̅(iγ^μ D_μ-me^iγ^5θ)ψ].

The action is invariant under this transformation only up to the boundary term.
It is evident from eq. (<ref>) that the massive theory with a positive mass (m>0) at θ=π is equivalent to the theory at θ=0 but with a negative mass (-m).  

The Hamiltonian in the temporal gauge A_0=0 is given by

    H=∫ dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1 - me^iγ^5θ)ψ]  ,

where the space-time coordinate is labeled by x^μ=(t,z). We denote the Pauli matrices as X, Y, and Z, and use the following convention for the Dirac matrices: γ^0 = Z, γ^1 = i Y, γ^5=γ^0 γ^1 = X.
In (1+1) dimensions, the axial charge density Q_5(x)≡ψ̅(x)γ^5γ^0ψ(x) 
and the vector current density J(x)≡ψ̅(x)γ^1ψ(x) are related 
by Q_5(x) = -J(x). Moreover the vector charge density Q(x)≡ψ̅(x)γ^0ψ(x) and the 
axial current density J_5(x)≡ψ̅(x)γ^5γ^1ψ(x) are related by Q(x) = J_5(x).



 §.§ Real-time chiral dynamics at finite temperature


As we have mentioned in the introduction, the chiral magnetic effect refers to the generation of an electric current along a magnetic field in a system with an imbalance in the number of left-handed and right-handed fermions, induced by the presence of a chiral chemical potential. In the Schwinger model, when the chiral quench is applied, the system is driven out of equilibrium, creating a non-zero chiral chemical potential μ_5. This chiral chemical potential represents an asymmetry between left-handed and right-handed fermions.

Here, we consider the μ_5-quench <cit.> protocol: The system Hamiltonian with θ=0 is prepared at time t<0. Then, starting at t=0, the Hamiltonian rotates the θ angle according to θ=-2μ_5t corresponding to constant chiral chemical potential (i.e. θ̇ = -2μ_5.). The Hamiltonian after the quench is modified as

    H'= dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1-γ^1θ̇/2 - me^iγ^5θ)ψ]  .

The real-time evolution of an operator O can be evaluated by the time-ordered integral 

    O(t)=𝒯[e^i∫_0^tdt'H'(t')]O𝒯[e^-i∫_0^tdt'H'(t')],

and the thermal expectation value of O evaluated at time t at finite temperature T is

    ⟨ O(t)⟩_β=[e^-β HO(t)]/ [e^-β H],

where H is the unquenched Hamiltonian in eq. (<ref>) at θ=0 and β=1/T is the inverse of the temperature of the system.

Notably, this evolving θ = -2μ_5 t after the quench at t = 0 indicates a dynamical chiral chemical potential, driving the system into a non-equilibrium state where left-handed and right-handed fermions are imbalanced. In the presence of this chiral chemical potential, a vector current J (which is typically zero before the quench) is induced. The vector current's time evolution at a finite β, influenced by the quench, aligns with the characteristic behavior observed in the CME at a finite temperature.





 §.§ The lattice Hamiltonian on the qubits


To discretize our Hamiltonian, we use staggered fermions <cit.> on a lattice such that

    ψ(x≡ na) = 1/√(a)(
        [          χ_2n;        χ_2n+1 ]) ,

where a is the finite lattice spacing. Then the lattice Hamiltonian corresponding to eq. (<ref>) is <cit.> 

    H    = a g^2/2∑_n=1^N-1 L_n^2
     -i/2a∑_n=1^N-1[χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]
    + mcosθ∑_n=1^N (-1)^n χ^†_nχ_n
       + imsinθ/2∑_n=1^N-1 (-1)^n [χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]-θ̇/4a∑_n=1^N-1[χ^†_n+1χ_n+χ^†_nχ_n+1] ,

where L_n is the electric field operator satisfying the Gauss' law constraint

    L_n-L_n-1 =  χ_n^†χ_n-1-(-1)^n/2.

Using the Gauss' law constraint, we eliminate the link fields U_n by the gauge transformation,

    χ_n→ g_nχ_n,  χ^†_n→χ^†_n g_n^†,  
        U_n → g_n+1U_n g_n^†,

with

    g_1=1,    g_n = ∏_i=1^n-1U^†_i.


For the purpose of quantum simulation, we rewrite the lattice Hamiltonian in the spin representation using the Jordan–Wigner transformation <cit.>:

    χ_n=X_n-iY_n/2∏_i=1^n-1(-i Z_i).

The full Hamiltonian in the qubit representation considered in this study is

    H=   ∑_n=1^N-1(1/4a-m/4(-1)^nsinθ)[X_n X_n+1+Y_n Y_n+1]
         +mcosθ/2∑_n=1^N(-1)^n Z_n
       +a g^2/2∑_n=1^N-1L^2_n-θ̇/8a∑_n=1^N-1[X_n Y_n+1-Y_n X_n+1].

Moreover, the local vector and axial charge densities are, respectively <cit.>, 

    Q_n ≡    ψ̅γ^0ψ = Z_n+(-1)^n/2a,
    
        Q_5,n≡    ψ̅γ^5γ^0ψ = X_nY_n+1-Y_nX_n+1/4a .

For later convenience, we introduce the average vector and axial charge operators that sum over all lattice sites,

    Q≡a/N∑_n=1^NQ_n,     
        Q_5≡a/N-1∑_n=1^N-1Q_5,n.

Notably, Q commutes with the Hamiltonian while Q_5 does not. With the boundary condition L_0=0, the Gauss' law constraint, eq. (<ref>), leads to the solution L_n =  a∑_j=1^n Q_j.

Notably, one can project the Hamiltonian to different charge sectors to work with states with specified quantum numbers. In the case of the Schwinger model, the projection operator P to the charge neutral sector (Q=0) can be written as P_Q=0 = ∑_n|n⟩⟨n| where n are all the bitstrings with even number of 1s and 0s. On the qubits, P_Q=0 can be always written as the sum of Pauli-Z operators such that H_Q=0=P_Q=0 H P_Q=0^† is the projected Hamiltonian to the Q=0 sector[For example, at N=4, the projection operator is

    P_Q=0 = P_Q=0^† = 0.375 (1 + Z_1 Z_2 Z_3 Z_4) - 0.125 ( Z_3 Z_4 + Z_2 Z_4 + Z_2 Z_3 + Z_1 Z_4 + Z_1 Z_3 + Z_1 Z_2 )
]. Here the coefficients are determined in such a way that P_Q=0^2=P_Q=0. For the numerical simulation results presented in this paper, we always work in the Q=0 sector. This is because 
the Schwinger model spectrum does not have charged states due to confinement at all scales.
Therefore, when we say Hamiltonian, in what follows, we always mean the charge neutral Hamiltonian H=H_Q=0. 



§ QUANTUM SIMULATION OF QED_2 AT FINITE TEMPERATURE



 §.§ Quantum imaginary time evolution

Quantum imaginary time evolution (QITE) is an efficient way to investigate eigenstates and thermal states on a quantum computer <cit.>. Here, we briefly review the QITE algorithm used in this work for preparing the thermal state and extracting the finite temperature observable.

In QITE, one attempts to approximate the action of the imaginary time evolution
operator on a state | Ψ⟩ with a parameterized unitary
operation:

    c(Δβ)^-1/2 e^-ΔβĤ | Ψ⟩≈ e^-i ΔβÂ(a⃗)| Ψ⟩,

where Δβ is a small time step and c(Δβ)^-1/2 is a normalization coefficient approximated
by 1-2Δβ⟨Ψ | Ĥ | Ψ⟩. The real-time evolution operator Â(a⃗)
is given by a linear combination of N_μ Pauli string:

    Â(a⃗) = ∑_μ^N_μ a_μP̂_μ.

Here, P̂_μ=∏_lσ_μ_l,l is a Pauli string and the subscript μ of a_μ labels the various Pauli strings, and their coefficients a_μ are solved from the linear equation ( S+ S^T)  a= b, where the matrix S and vector b are defined by

    S_μν   =⟨Ψ(β)|P̂_ν^†P̂_μ|Ψ(β)⟩ ,
    
    b_μ   =-i/√(c(Δβ))⟨Ψ(β)| (ĤP̂_μ+P̂_μ^†Ĥ)|Ψ(β)⟩ .

The QITE procedure is repeated for a specified
number of time steps to reach a target total evolution time. Using this iterative approach, we are able to evolve an initial quantum state |Ψ_k(0)⟩ under the unitary operator e^-iΔβ A to any imaginary time β by trotterization:

    |Ψ_k(0)⟩→ |Ψ_k(β)⟩ +𝒪(Δβ) .

Note the sum of |Ψ_k(β)⟩ over all basis states approximates e^-βĤ/(e^-βĤ) at finite temperature T=1/β in the sense that it gives the thermal expectation of an observable Ô:

    1/Z∑_k⟨Ψ_k(β/2)|Ô|Ψ_k(β/2)⟩≈⟨Ô⟩_β=1/Z (e^-βĤÔ),

where Z= (e^-βĤ)=∑_k⟨Ψ_k(β/2)||Ψ_k(β/2)⟩ is the canonical partition function, which is also the sum of the normalizations over all the quantum states.





 §.§ Real-time evolution at finite temperature


Studying the dynamics of the quantum system at finite temperatures is usually complicated. In this work, we are interested in the real-time evolution of the thermal mixed state at finite temperatures of the Schwinger Hamiltonian. Using another evolution Hamiltonian H' that is not equal to the Hamiltonian H of the thermal system, we define a quench process by the real-time evolution of the operator. See eq. (<ref>) for our quench protocol.

To implement this protocol on the quantum circuit, we first use the QITE algorithm to obtain the collection of the |Ψ_k(β/2)⟩ states to represent the thermal state at finite temperature T=1/β. Then, we simply evolve each |Ψ_k(β/2)⟩ in real time as usual:

    |Ψ_k(t,β/2)⟩=𝒯[e^-i∫_0^tdt'Ĥ'(t')]|Ψ_k(0,β/2)⟩.

The final thermal expectation at real-time t is

    ⟨Ô(t)⟩_β=1/Z∑_k⟨Ψ_k(t,β/2)|Ô|Ψ_k(t,β/2)⟩.

Compared with eq. (<ref>), we see that thermal expectation evaluated on the quantum circuit is associated with the states instead of observables. Nevertheless, the two methods are entirely equivalent, which can be seen in our comparison between the quantum simulation and exact diagonalization results. To perform the real-time evolution, we trotterize the time into small time steps using second-order Suzuki-Trotter product formula <cit.>, which gives good convergence.




§ NUMERICAL RESULTS




 §.§ Thermal state preparation

To study the real-time evolution of physical properties at finite temperatures, we must first prepare the initial thermal state at the desired temperatures. In this work, we use the QITE algorithm developed using Qiskit <cit.> in a proceeding work <cit.> for the thermal medium preparation. Specifically, the unitary evolution is implemented using PauliEvolutionGate library with SuzukiTrotter method, and the thermal expectation is evaluated with the Sampler class on the statevector quantum simulators. In particular, we use the QITE to prepare thermal states with four qubits from β/a=0 (aT=∞) to β/a=4.0 (aT=0.25) for the three masses ma=0.25, 0.5, 4.0, and at coupling g=a^-1. A universal trotter step size Δβ = 0.01 is used.



To measure the validity of the quantum simulation, we use the fidelity between two density matrices σ and ρ:

    F(σ,ρ)=(√(√(σ)ρ√(σ)))^2.

Here, we take σ=e^-β H/Z to be the exact density matrix calculated via exact diagonalization and ρ=∑_k|Ψ_k(β/2)⟩⟨Ψ_k(β/2)|/Z for the thermal mixed state prepared by the QITE algorithm. In Fig. <ref>(left), we show the fidelity of these two density matrices as a function of the QITE imaginary evolution parameter β (i.e., inverse temperature). 
For all three cases of different masses, the fidelity is between 0.999 and 1.0, very close to 1, suggesting that the thermal state prepared by QITE is a good practical approximation of the corresponding thermal equilibrium state. In particular, we see a comparatively slight decrease of fidelity in the early β (high temperature) region for the heaviest mass ma=4.0, which is due to the insufficiently small step size Δβ = 0.01 used in the simulation for the heavy mass.

In Fig. <ref>(right), we show the purity of the mixed quantum state ρ at different temperatures and masses. Purity is defined by [ρ^2] and is an index used to determine how much a quantum state is mixed, with values equal 1 indicating a pure state, and values approaching 0 signifying a mixed state. For every mass, we expect that the purity to approach 1 as the temperature approaches 0 (equivalently, as β approaches infinity), since the state is pure at zero temperature (i.e., the ground state). For smaller mass values, such as ma = 0.25, the curve shows that the purity decreases more slowly with temperature, indicating smaller mass states transition to mixed states more gently as the temperature rises. In constrast, as the mass ma increases, the purity drops more rapidly. Despite these differences, all curves will eventually approach a lower purity limit as the temperature continues to increase, showing the natural tendency of quantum systems to shift towards mixed states under a finite temperature. 




 §.§ Real-time evolution of the charge and the current at finite temperature


Now that the thermal state is prepared using the QITE protocol, we study real-time evolution of the total electric and axial charge densities for the system at a selected temperature with both light and heavy masses. Our simulations are conducted using the exact statevector quantum simulator provided by Qiskit <cit.> and compared to the T = 0 limit obtained from exact diagonalization. Specifically, we perform our simulation using eqs. (<ref>) and (<ref>), following the quench protocol provided in Sec. <ref> at various chiral chemical potentials.


The real-time evolution is compared against the zero-temperature limit using exact diagonalization, serving as a benchmark for the results obtained from the quantum simulator at finite temperatures. We first note that the results at T=0 are consistent with the results in <cit.>, which performed the study under the periodic boundary condition at g=0 while we work with the open boundary condition at g≠0. We also observe that ⟨ Q⟩ =0 at all finite temperatures and it does not evolve over time, which is expected. The quantum simulation uses four qubits and applies the QITE algorithm for thermal state preparation. These methods allow for accurate modeling of thermal states and real-time evolution, offering a robust framework for understanding the underlying physics. The second order Suzuki-Trotter product formula <cit.> is used for the real-time evolution, with step sizes dt varying based on the masses: 0.1 for light masses and 0.01 for heavy masses. 



Fig. <ref> shows the real-time evolution of the thermal average of the axial charge ⟨ Q_5(t) ⟩_β for various temperatures, aT = 10.0,1.0,0.25. The mass takes values of ma=0.25, 1.0, 4.0 to examine how light and heavy masses affect the evolution of the vector and axial charge densities, showing how it evolves under different temperatures. The results highlight the role of a chiral chemical potential μ_5 and include both exact solutions and approximate results using quantum simulation. To confirm consistency with theoretical expectations, we show that the total vector charge is conserved and does not depend on time. 

In studying the chiral dynamics in the Schwinger model, our results share several important similarities with our previous results of nonlinear chiral magnetic waves (CMWs) <cit.>, particularly in observing how the oscillation of axial charges is influenced by mass. Here, chiral magnetic waves are collective excitations in a (1+1)-dimensional chiral medium under the influence of a magnetic field. These waves are a direct consequence of the interplay between axial (chiral) and vector (electric) currents in the presence of magnetic fields. Notably, the tendency that heavier mass results in faster oscillations of the axial charge is common <cit.>.
In both studies, it is observed that the axial charge density oscillates more rapidly with increasing mass. In the nonlinear CMWs, the nonlinearity arises because, in massive Schwinger model, the frequency of axial charge oscillations becomes significantly higher than that of the vector charge. This results in the axial charge oscillating rapidly while being nearly confined within static electric dipoles, a phenomenon referred to as the “thumper” solution  <cit.>. The interplay between the excited state (causing rapid oscillations) and the ground state (with almost static behavior) creates a non-linear dynamic in the system.




In Fig. <ref>, we present a comparison between our simulation results and exact diagonalization, observing good agreement when sufficiently small time steps are used. This figure illustrates how accurately our simulation can predict the system's behavior, validating our approach. Furthermore, the rapid oscillations observed in the simulation at a larger mass (ma = 4) also verified our expectation. These oscillations are attributed to the interplay between different energy eigenstates, and their amplitude increases as the temperature rises. The consistency between the simulation in Fig. <ref> underscores the accuracy of our model in capturing the dynamic properties of the system at varying conditions.

To study the dynamical interplay between finite temperature and real-time evolutions, we also present the density plots for ⟨ Q_5(t) ⟩_β in Fig. <ref>. The axial charge oscillates as a function of time t at different temperatures T, with increasing frequency for the heavier mass. Notably, the thermal state is first obtained using the QITE algorithm at finite β/a=1/(aT) values in the range of [0, 4], and then we obtain the density plot versus aT to the range of [∞, 0.25] with interpolation to T=0 temperature. The number of modes for the time oscillations is directly related to the mass in the Schwinger Hamiltonian. 



With the goal of extrapolating our results in the continuum, we perform the exact diagonalization with N=10 qubits in Fig. <ref>. Importantly, we have direct access to the finite temperature thermal state with exact matrix exponentiation using eq. (<ref>), and our exact results are in terms of aT and t. Here, we show the density plot of the thermal expectation value of ⟨ Q_5(t)⟩_T (with temperature in units of aT in the range of [0, 4]). Comparing to results simulated on the quantum simulator shown in Fig. <ref>, we observe the same behavior for both ma=1.0 and 4.0 cases, despite having many more qubits. Quantitatively, we can also observe good agreement between the quantum and classical simulations by counting the oscillation modes and the peak intensity values in the density plots.



§ CONCLUSION

This paper investigates the real-time dynamics of the chiral magnetic effect (CME) at finite temperature in the context of the massive Schwinger model, quantum electrodynamics (QED) in (1+1) dimensions. The study utilizes quantum simulation techniques, combining the quantum imaginary time evolution (QITE) algorithm for thermal state preparation and trotterization for real-time evolution, to examine the thermal properties of the system and the transitions induced by a chiral chemical potential via a quench process.

Our results illustrate the dynamic interplay between the fermion mass and the real-time dynamics of the axial charge. At zero temperature, the corresponding dynamics has been investigated in our previous studies   <cit.>. 
The new result in this paper is the temperature dependence of the real-time CME response. Specifically, at temperatures small compared to the fermion mass, the CME response is similar to the one found previously at zero temperature. However, at larger temperatures, the oscillations of the axial charge induced by the CME are dampened by thermal effects. This can be understood as a result of the contribution from excited fermion modes to the real-time CME response, that may depend on the temperature <cit.>. 

In addition, at large fermion mass the axial charge oscillations become very non-linear. The origin of this  non-linearity can be explained in the following way: in bosonized description of the massive Schwinger model, fermion mass induces non-linear interactions among the bosons, so the chiral magnetic waves become highly non-linear <cit.>.

Finally, our study of CME dynamics suggests that quantum simulations can be used to study other real-time phenomena at finite temperatures in gauge theories that form the basis of the Standard Model. 




§ ACKNOWLEDGEMENT

We thank Meijian Li for helpful comment on the manuscript. We are also grateful to Andrea Palermo and 
Ismail Zahed for useful discussions. This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704 (KI, DK, FZ), the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Grants Nos. DE-FG88ER41450 (DK), DE-SC0012704 (DK) and DE-SC0011090 (FZ), and the  National Science Foundation under grant No. PHY-1945471 (ZK). WQ is supported by the European Research Council under project ERC-2018-ADG-835105 YoctoLHC; by the Spanish Research State Agency under project PID2020-119632GB-I00; by Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019-2022), by European Union ERDF, and by the Marie Sklodowska-Curie Actions Postdoctoral Fellowships under Grant No. 101109293.


JHEP




 Paper Title: Gravitational form factors of pion from top-down holographic QCD.






[]daisuke@rcnp.osaka-u.ac.jp
Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai, 319-1195, Japan
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]iwanaka@rcnp.osaka-u.ac.jp
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]tanaka@hken.phys.nagoya-u.ac.jp
Department of Physics, Nagoya University, Nagoya 464-8602, Japan

§ ABSTRACT 

The gravitational form factors (GFFs) of pions has been calculated from a top-down holographic QCD approach with momentum transfer dependence for the first time. It is important because the GFFs of hadrons have information on the internal stress distribution that may provide insight into the mechanism of how QCD forms hadrons. The forward limit value of this GFFs, i.e. the D-term, was also obtained. Furthermore, in this approach, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 



Gravitational form factors of pion from top-down holographic QCD
    Mitsuru Tanaka
    
================================================================



Introduction.—
How forces act inside hadrons to confine quarks and gluons forming hadrons has not been solved. 
Addressing this question is inseparable from developing an insight into the non-perturbative properties of quantum chromodynamics (QCD), such as confinement and spontaneous chiral symmetry breaking, which is a crucial problem in modern physics. 

Recently, the stress inside hadrons has attracted attention (see a recent review <cit.>) due to the fact that experimentally it has been measured. The stress distribution inside nucleons, extracted from the matrix elements of the nucleon energy-momentum tensor (EMT), is obtained from deeply virtual Compton scattering <cit.>. 
The form factors that characterize the matrix elements are called gravitational form factors (GFFs), which are calculated theoretically from their coupling with gravity. 

The forces inside nucleons were analyzed experimentally in Ref. <cit.> for the quark part (2018) and in Ref. <cit.> for the gluon part (2022). Experimental results show that around the center of the nucleon, the force pushes the quarks outwards, while around the surface it reverses into a confining force, indicating an extremely anisotropic behavior. As a result, the pressure near the center of the nucleon may reach the highest value currently observed in nature. The pion GFFs have also been analyzed experimentally in Ref. <cit.>. The GFFs of nucleons <cit.> and pions <cit.> have also been measured by lattice QCD analysis, a first-principles numerical simulation of QCD. 

In this letter, we are the first to calculate the momentum transfer dependence of the pion GFFs from a top-down holographic QCD approach. We use the Sakai-Sugimoto model, a top-down holographic QCD model based on the D4-D8 brane construction <cit.>. With this approach, the spectroscopy <cit.>, decay widths and electromagnetic form factors <cit.> of mesons, baryons and glueballs have been investigated with strong coupling and large N_c limit. Recently, using the same approach, nucleon GFFs have been formulated and D-terms have been estimated <cit.>. The GFFs of pions <cit.> and other hadrons <cit.> have also been calculated using bottom-up holographic QCD, which has no direct connection with QCD through string theory. 

It is interesting that stress distributions and GFFs of pions<cit.>, which are the focus of this letter, show anomalous behavior. The stress distribution inside hadrons is very sensitive to interactions in the system. The forward limit of the GFFs giving the stress distribution is called the D-term, which takes the value -1 for spin-zero particles in the free Klein-Gordon theory. On the other hand, even an infinitesimally small interaction causes a significant deviation from -1 <cit.>. 


However, the D-term of the Nambu-Goldstone (NG) bosons in the strongly coupled system becomes -1 like the free particles, in the soft pion limit because of the restriction by chiral symmetry <cit.>. 
This fact may provide some insight into the role of chiral symmetry breaking in hadron formation since the D-term carries information about the forces inside hadrons. 
These insights are expected to be validated in more detail via analysis of hadron GFFs from future hadron experiments <cit.>. 




Formulation.—
In this section, we provide the method to derive the GFFs of pions in the top-down holographic QCD <cit.>. 
The method for calculating the GFFs of nucleons with spin 1/2 has been formulated in Ref. <cit.> and will be applied in the following. 


The matrix elements of EMT of a spin-0 particle are expressed as

    <π^a(p_2)|T̂^μν(x)|π^b(p_1)> 
       =δ^ab[2P^μ P^ν A(t)+1/2(Δ^μΔ^ν-η^μνΔ^2)D(t)]e^ix·Δ,

where a,b are the indices for isospin and the kinematic variables are defined as P^μ=(p^μ_1+p^μ_2)/2, Δ^μ=p^μ_2-p^μ_1 and t=Δ^2 with Lorentz indices μ,ν <cit.>. 
In this paper, we employ the mostly plus metric η^μν= diag(-1,1,1,1) and the normalization condition as <p⃗_⃗2⃗|p⃗_⃗1⃗>=2E(2π)^3δ^(3)(p⃗_⃗1⃗-p⃗_⃗2⃗),    E=√(p⃗_⃗1⃗^2+m^2).


The method for calculating the matrix elements of the EMT by holographic QCD is as follows. 
Within the framework of holographic QCD, perturbations to the metric caused by matter fields in the bulk propagate to the boundary, allowing us to extract the expectation values of the EMT in the boundary theory from these perturbations. The top-down holographic QCD used in this study, the Sakai-Sugimoto model, consists of N_f D8 branes embedded as a probe in black 4-brane background <cit.>, a holographic dual of N_c D4 branes. This background is a 10-dimensional curved spacetime of type IIA supergravity, derived from 11-dimensional supergravity related to M-theory on doubly Wick rotated AdS_7 black hole × S^4, 

    ds^2=   r^2/L^2[f(r)dτ^2-dx^2_0+dx_1^2+dx^2_2+dx^2_3+dx^2_11] 
       +L^2/r^2dr^2/f(r)+L^2/4dΩ^2_4 , 
    
        f(r)=    1-R^6/r^6,    R = L^2M_KK/3,    L^3=8π g_s N_c l^3_s,    
    
        R_11=   
        λ/2π N_c M_KK,    λ N_c=L^6M_KK/32π g_s l^5_s=216π^3κ,

where M_KK is the Kaluza-Klein mass in the τ direction, g_s string coupling, l_s string length, λ 't Hooft coupling, and R_11 the radius of the circle compactified in the x_11 direction.

The expectation values of EMT on the boundary, <T_μν>, can be extracted from the boundary values of the metric fluctuations in the bulk in the background spacetime (<ref>) propagated to the boundary as follows, 

    ∫_x_11,τδ g_μν∼2κ^2_7 L^5/6<T_μν>/r^4+...,

where κ^2_7 is related to the 11-dimensional gravitational constant as κ^2_11= Vol(S^4)κ^2_7=8π^2/3(L/2)^4κ^2_7.
Note that, to obtain the above, we need to apply the holographic renormalization, developed by Ref. <cit.>. 


The metric fluctuations δ g_MN of the bulk obey the following linearized Einstein equations:

    ℋ̅_MN=   ℋ_MN-g_MN/2ℋ^P_P=-2κ^2_7𝒯_MN, 
    ℋ_MN≡   ∇^2δ g_MN+∇_M∇_Nδ g^ P_P 
       -∇^P(∇_Mδ g_NP+∇_Nδ g_MP)-12/L^2δ g_MN,

where M, N and P are the Lorenz indices of the 7-dimensional space-time after integration over S^4, as in M,N,P=0,1,2,3,τ,r,11, and the source 𝒯_MN is the EMT of the matter fields which here considered only the pion field. 
We solve Eq. (<ref>) under the axial gauge, δ g_Mr = 0. 

In order to examine the EMT 𝒯_MN in the bulk, we embed the D8 branes into the 11-dimensional space, considering the action as

    S_D8= 
       -C/2π R_11g_s∫ dτ dx_11d^4xdrdΩ_4(δ(τ)+δ(τ-π/M_KK)) 
       ×√(-G_(11))/√(G_ττ)1/4G^MNG^PQ tr[F_MPF_NQ]+...,

with C=(64π^6l_s^5)^-1 and the metric tensor G_MN in Eq. (<ref>). We define the EMT as

    𝒯_MN=-2/√(-G_(11))δ S_D8/δ G^MN_(11) Vol(S^4) ,

which leads to the following expression:



    𝒯_MN(Δ⃗,r) ≡ 2π R_11∫ dτ^'𝒯_MN(τ^',k⃗,r), 
       𝒯_μν (Δ⃗,r)=L^5/r^5π^2Cr^2 L^2/3g_s √(f(r))) tr[F_μρF_ν^ρ+(1+z^2)^4/3M_KK^2F_μ zF_ν z-η_μν/4(F_ρσF^ρσ+2(1+z^2)^4/3M^2_KKF_ρ zF^ρ_z)], 
       𝒯_11,11(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[-1/4F_ρσF^ρσ-(1+z^2)^4/3M^2_KK/2F_ρ zF^ρ z], 
       𝒯_μ z(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[F_μρF_z^ρ],    𝒯_zz(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[1/2F_zρF_z^ρ-1/4(1+z^2)^4/3M^2_KKF_ρσF^ρσ],


where the radial coordinate r  (R≤ r<∞) is related to z  (-∞<z<∞) as z=±√(r^6/R^6-1).
In this study, we consider the case where the pion field is the only matter field as the source of the gravitational waves and decompose A_M(x^ν,z) as follows

    A_μ(x^ν,z)=0,     A_z(x^ν,z)=π(x^μ)ϕ_0(z), 
       ϕ_0(z)=c_0(1+z^2)^-1,

where c_0^-2=2M^2_KKκ∫ dz(1+z^2)ϕ_0(z)^2 is the normalization constant of the eigenfunction ϕ_0(z) <cit.>. Since the pion field is an external line, it can be replaced by a plane wave as π(x^ν)∼ e^ip_ν x^ν with incoming momentum p_ν.

Since we are interested in the EMT of QCD at the boundary such as to satisfy Δ^μ<T_μν>=0, we consider the metric fluctuations that are transverse Δ^μδ g_μν∼0 near the boundary, which can be parametrized as δ g_ij∼δ g^ TT_ij+(δ_ij-Δ_iΔ_j/Δ⃗^2)/5δ g^α_α with the transverse-traceless (TT) part δ g^ TT_ij, i,j=1,2,3 and α=μ,τ,11. Under the axial gauge, the Einstein equation requires the six-dimensional traceless condition δ g^α_α∼0 in the asymptotic AdS region (r→∞) labeled by (x^μ,τ,x^11). 
Therefore, we consider TT modes in the metric fluctuation, which is nothing but glueball spectra.
Among the 14 TT modes named in Ref. <cit.> (see also Ref. <cit.>), only T_4 and S_4 contribute to the EMT. Actually, substituting the pion field (<ref>) into equation (<ref>), we confirm that δ g^ other_μν falls faster than T_4 and S_4 mode at r→∞. 
By decomposing the metric fluctuations into each mode, δ g_μν=δ g^ T(2)_μν+δ g^ T(0)_μν+δ g^ S(0)_μν+δ g^ other_μν, the Einstein equations reduce to 

    r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)∂_r)-L^2Δ^2/r^2]L^2/r^2δ g^ T_μν=-2κ^2_7𝒯^ T_μν, 
       r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^5-2R^6)(r^6+2R^6))) 
       -L^2Δ^2/r^2]L^2/r^2δ g^ S(0)_μν|_r→∞=-2κ^2_7𝒯^ S(0)_μν,

where the source of each mode is decomposed as 𝒯_μν=𝒯^ T_μν+𝒯^ S(0)_μν+𝒯^ other_μν (𝒯^ T_μν≡𝒯^ T(2)_μν+𝒯^ T(0)_μν), and the two T_4-modes are combined as δ g^ T_μν=δ g^ T(2)_μν+δ g^ T(0)_μν, since they obey the same equation. 

Following the same manner as in Ref. <cit.>, each part of EMT is written as 


    2κ^2_7𝒯^ T_μν=2κ^2_7𝒯_μν-2κ^2_7𝒯^ S_μν-2κ^2_7𝒯^ other_μν, 
       2κ^2_7𝒯^ S_μν=r^6+2R^6/4(r^6-R^6)(η_μν-Δ_μΔ_ν/Δ^2)[1/L^4r^3∂_r(r(r^6-R^6)∂_r(a+b)-3R^6(a+b))-Δ^2b], 
       2κ^2_7𝒯^ other_μν=1/L^4r^3∂_r[(η_μν-Δ_μΔ_ν/Δ^2)r(r^6-R^6)∂_r a-3R^6η_μνb],  
       ∂_r a=-3R^6/(5r^6-2R^6)b+9r^11/(5r^6-2R^6)R^62κ^2_7𝒯_zz,    b=-iΔ^μ/Δ^2r^6√(r^6-R^6)/R^9 2κ^2_7 𝒯_μ z,


with the ansatz, 

    δ g^ other_μν(Δ⃗,r)=r^2/L^2Δ_μΔ_ν/Δ^2a(r,Δ), 
       δ g^ other_11,11(Δ⃗,r)=r^2/L^2b(r,Δ), 
       δ g^ other_ττ(Δ⃗,r)=r^2f(r)/L^2(-b(r,Δ)).



We determine the perturbation to the metric caused by the pion field as the bulk EMT source using Green's function method as follows, 

    δ g^ T/S_μν(Δ⃗,r)∼r^2/L^2∫ dr^'√(-G_(7)(r^'))G^ T/S(Δ⃗,r,r^') 
       ×L^2/r^'^22κ^2_7𝒯^ T/S_μν(Δ⃗,r^'),

where, for S_4 mode, we can write as this form in the r→∞ limit, and ∼ implies this point. 
The Green's function G^ T/S is written as 

    G^ T/S(Δ⃗,r,r^')=∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2,

by expanding in the complete system {Ψ^ T/S_n(r)} which obeys the eigenvalue equation, 

    1/r^3L^4∂_r((r^7-rR^6)∂_rΨ^ T_n)=-(m^ T_n)^2Ψ^ T_n, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))Ψ^ S_n) 
       =-(m^ S_n)^2Ψ^ S_n,

where m^ T/S_n is n'th glueball mass of T/S mode and Ψ^ T/S_n(r) satisfy the completeness conditions,

    r^3/L^3∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')=δ(r-r^'), 
       ∫^∞_R drr^3/L^3Ψ^ T/S_n(r)Ψ^ T/S_m(r)=δ_mn.








Results.—
Substituting the pion field into the EMT of the bulk, we read off the matrix elements of the boundary EMT from Eq. (<ref>) with the dictionary Eq. (<ref>) as follows, 

    A(t)=6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ t𝒯_μν(Δ⃗,r^'), 
       D(t)= 
       -6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s(𝒯_μν+𝒯^ S_μν+𝒯^ other_μν) 
       +6/L^7∫^∞_Rdr^'∑^∞_n=1α^ S_nΨ^ S_n(r^')/(m^ S_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s𝒯^ S_μν, 
       α^ T/S=(m^ T/S_n)^2L^4/6∫^∞_R dr r^3Ψ^ T/S_n(r),
 
where P^μν_ t and P^μν_ s are defined as P^μν_ t=(3P^μ P^ν/P^2-η^μν)/P^2 and P^μν_ s=(P^μ P^ν/P^2-η^μν)/Δ^2. 

To calculate infinite sums, we expand as 

    6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T/S_nΨ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2=∑^∞_k=0F_k^ T/S(r^')(-Δ⃗^2)^k, 
       F^ T/S_k(r^')=∫_R^∞ drr^3/L^3∑_n=0^∞Ψ_n^ T/S(r)Ψ_n^ T/S(r^')/(m_n)^2k.

From the eigenvalue equations (<ref>), F^ T/S_k(r) satisfies the following equations:

    1/r^3L^4∂_r((r^7-rR^6)∂_rF^ T_k)=-F^ T_k-1, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))F^ S_k) 
       =-F^ S_k-1,

where we can prove F^ T/S_0=1 by using Eq. (<ref>). 
We determine F^ T/S_k(r) by solving it as the recurrence formula with boundary conditions on ∂_r F^ T/S_k(r)|_r→ R=1,  F^ T/S_k(r)|_r→∞=0. 

We show the results of GFFs that truncate the expansion at k=10 using Eq. (<ref>) in FIG. <ref> (also including results that truncate at k=8,9), where we set κ =0.00745 and M_KK =949  MeV which are determined by ρ meson mass 776   MeV and the pion decay constant f_π=92.4  MeV <cit.>. 
In the current calculation, D(t) begins to deviate from around t∼0.5   GeV^2 while A(t) converges well in the range 0<t<0.8   GeV^2 on the expansion up to k=10 with momentum transfer. The current approach begins to deviate from QCD in the region where t is close to the M_KK, so further efforts are unnecessary. 

In the top-down holographic QCD approach, the amplitude of D(t) dumps more rapidly than that of A(t), which is qualitatively consistent with the results of lattice QCD <cit.>. 
These differences are due to the contribution of S_4 modes to D(t), which is not observed in the bottom-up holographic QCD framework <cit.>. 
Also in this approach, as in previous studies <cit.>, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 

As you can see in FIG. <ref>, our result shows D(0)=-1. 
Note that this is a result of our calculations with loop effects in the large Nc limit, unlike the fact A(0)=1, given by the general constraints. 
This is consistent with the low-energy theorem obtained in the soft pion limit <cit.>. The result is as expected since in our present work we consider the Sakai-Sugimoto model, which is the holographic dual of QCD in the chiral limit. Several methods have been proposed on how to introduce quark masses into this model <cit.>, thus analysis beyond the chiral limit is a future work. 










Summary and outlook.—
We calculated the GFFs of pions from the top-down holographic QCD. 
We solve the Einstein equations in bulk theory with pion fields as matter fields and derived the GFFs using the dictionary (<ref>).
Our results are qualitatively compatible with those from lattice QCD <cit.> and successfully reproduce the feature where the behaviors of A(t) and D(t) are different, which is distinct from bottom-up holographic QCD <cit.>.
Additionally, we obtained the D-term as D(0)=-1 in the chiral limit. 

Finally, this study opens up several avenues for future research. First, it is possible to generalize to other hadrons, such as nucleons and vector mesons with non-zero spin. Also, it is very interesting to investigate beyond the chiral limit, because the connection between the Gell-Mann–Oakes–Renner relation and the stability of the stress distribution has been discussed in Ref. <cit.>. It is also important to investigate the behavior of the GFFs and the stress distribution inside hadrons under finite temperatures, densities, and magnetic fields, which change the nature of the interaction in the system. 




Acknowledgements.—
We are grateful to the authors of Ref. <cit.> for generously supplying the data tables utilized in FIG. <ref>.
We also thank Atsushi Hosaka, Kei Suzuki and Shigeki Sugimoto for useful discussions. This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Nos. JP24K17054, JP24KJ1620). 





*










 Paper Title: Universal Euler-Cartan Circuits for Quantum Field Theories.



 

ananda.roy@physics.rutgers.edu
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

Division of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

§ ABSTRACT 
Quantum computers can efficiently solve problems which are widely believed to lie beyond the reach of classical computers. In the near-term, hybrid quantum-classical algorithms, which efficiently embed quantum hardware in classical frameworks, are crucial in bridging the vast divide in the performance of the purely-quantum algorithms and their classical counterparts. Here, a hybrid quantum-classical algorithm is presented for the computation of non-perturbative characteristics of quantum field theories. The presented algorithm relies on a universal parametrized quantum circuit ansatz based on Euler and Cartan's decompositions of single and two-qubit operators. It is benchmarked by computing the energy spectra of lattice realizations of quantum field theories with both short and long range interactions. Low depth circuits are provided for false vacua as well as highly excited states corresponding to mesonic and baryonic excitations occurring in the analyzed models. The described algorithm opens a hitherto-unexplored avenue for the investigation of mass-ratios, scattering amplitudes and false-vacuum decays in quantum field theories.




Universal Euler-Cartan Circuits for Quantum Field Theories
    David Rogerson
    August 5, 2024
==========================================================






  
*Introduction.
Quantum field theory (QFT), which reconciles quantum mechanics and the special theory of relativity, is fundamental to the description of a vast array of physical phenomena. The latter range from confinement and asymptotic freedom in theories of elementary particles to high-temperature superconductivity and topological order in condensed matter systems. The investigation of non-perturbative characteristics of generic QFTs remains an outstanding challenge with ab-initio computations likely to stay beyond the reach of the most powerful classical computers. Quantum computation <cit.> provides an alternate paradigm for the analysis of QFTs with quantum algorithms boasting exponential speedup over classical ones for the computation of scattering amplitudes in massive scalar <cit.> and fermionic <cit.> theories. 

Despite their potential, it is a daunting challenge to implement quantum algorithms on state-of-the-art quantum simulators and solve QFT problems of widespread interest. This is because the current quantum simulators contain only a few hundred qubits with modest error rates. In the current noisy quantum era, hybrid quantum-classical algorithms, which combine the power of a quantum processor with that of a classical one, are crucial in bridging the enormous divide between purely-quantum algorithms and their classical counterparts. Notable examples include the variational quantum eigensolver <cit.> and its modifications <cit.> as well as the quantum approximate optimization algorithm <cit.>, which have been successful in analyzing problems in quantum chemistry <cit.> and combinatorial optimization <cit.>. 

There are two main elements of a hybrid algorithm that govern its efficacy: i) a parametrized quantum circuit ansatz built with a predetermined pool of unitary operators, ii) an efficient optimization procedure for the parameters specifying the unitary operators. While the second element is typically based on variations of gradient descent methods <cit.>,  there is no natural prescription for the first. The circuit ansatz either relies on the Hamiltonian whose eigenstate is being sought alongside a heuristically-chosen mixer Hamiltonian <cit.> or variations of the coupled cluster theory relevant mostly for chemistry problems <cit.>. Neither of the aforementioned are suitable for generic QFT problems whose Hamiltonians, when mapped to registers of qubits in a quantum computer, could involve long-range couplings <cit.> or might not even be Hermitian <cit.>. Furthermore, many interesting properties of QFTs such as mass-ratios and scattering amplitudes remain encoded in highly-excited states <cit.>, which have largely eluded existing hybrid algorithms. Last but not least, restricting the circuit ansatz to a given pool of unitary operators that does not encompass the entire manifold of possible unitary operators necessarily implies that the result so obtained may not be the globally optimum solution which could potentially be found by considering a larger ansatz. 



In light of the above, it is natural to ask: is there a quantum-classical algorithm based on a universal parameterized quantum circuit ansatz that is tractable with established optimization strategies and applicable to a broad set of QFT problems? The goal of this work is to answer this question. The presented solution computes the optimal circuit over all possible two-qubit unitary rotations required to realize an arbitrary L-qubit unitary rotation <cit.>. The unitary operators are chosen to couple neighboring qubits, which is imposed for potential implementation on majority of quantum computing architectures and could be relaxed in a more general setting. The model under investigation affects the parameters of the circuit, but not the ansatz itself. Importantly, the proposed ansatz explores the largest possible manifold of allowed unitary operators and is closer to the globally optimal solution than what could be obtained by considering a more restrictive ansatz. The potential of the algorithm is demonstrated by computing not only ground states of strongly-interacting QFTs, but also highly excited states that correspond to bound-states of topological excitations occurring in these theories. 




  
*Algorithm. The algorithm is implemented on a quantum processor coupled with a classical one (Fig. <ref>). The former is involved in  applying the parametrized circuit and conducting measurements required for the optimization process. The classical processor receives the measurement outcomes, performs the optimization and feeds the outcome back into its quantum counterpart. 

The algorithm begins with the application of N (see Figs. <ref>, <ref> for representative choices of N) layers of unitary operators on L qubits of the quantum processor initialized in the state |ψ_ i⟩. Each of the N layers is built, in principle, out of two sublayers of two-qubit operators [orange rectangles in Fig. <ref>(b)]. Each two-qubit operator is an element of the group SU(4). It can be written, using Cartan's KAK decomposition <cit.>, as a product of four single-qubit SU(2) rotations and an SU(4)-entangler [Fig. <ref>(c)]. The latter when involving qubits i, i+1 equals e^-i(α X_iX_i+1 + β Y_iY_i+1 + γ Z_iZ_i+1) where X_i, Y_i and Z_i are the Pauli operators of the i^ th qubit. The SU(4) entangler can be realized by three controlled-NOT gates and five single-qubit rotations <cit.> [Fig. <ref>(d)]. Finally, each of the four single-qubit rotations depicted in panel Fig. <ref>(c) can be further decomposed into three single qubit rotations by the well-known Euler angles <cit.>. Using the above, it can be shown that the general circuit ansatz is constructed with two-qubit operators [orange ellipses in Fig. <ref>(b, e)] with ten undetermined angles. 


After the application of the N layers, measurements are performed to determine the relevant gradients and the cost-function L(θ⃗), where θ⃗ contains all parameters of the N layer quantum circuit. To obtain the ground state of a given Hamiltonian H, the cost-function is chosen to be L(θ⃗) = ⟨ψ_ f(θ⃗)|H|ψ_ f(θ⃗)⟩. To obtain the n^ th excited state, the cost-function is augmented to L(θ⃗)→ L(θ⃗) + ∑_m < nλ_m|⟨ψ_m|ψ_ f⟩|^2, with λ_m>0, m = 0,…, n-1. This choice amounts to simultaneously minimizing the energy and the overlap to the n-1 lower-energy states that have been already obtained.  

The measurement results are communicated to the classical processor which checks for the convergence of the cost-function for the given choice of N (iteration convergence in Fig. <ref>). If the condition is not satisfied, the parameters of the circuit are updated using an optimization routine. In this work, the quantum natural gradient optimizer <cit.> is used due to its superior performance in obtaining eigenstates of many-body Hamiltonians <cit.>. In contrast to ordinary gradient descent, this  ensures that optimization follows the steepest descent taking into account the geometry of the space of wavefunctions <cit.> and can viewed as the quantum analog of the classical information geometric approach <cit.>. The parameters at the (t+1)^ th iteration are related to the t^ th one by

    θ⃗_t + 1 = θ⃗_t - η g^-1∂ L/∂θ⃗_t,

where η is the learning rate (typically a real number < 1) and g is the Fubini-Study metric <cit.>. The latter is given by the real part of the quantum geometric tensor G:

    G = ⟨∂ψ_ f/∂θ_i|∂ψ_ f/∂θ_j⟩ - ⟨∂ψ_ f/∂θ_i|ψ_ f⟩⟨ψ_ f|∂ψ_ f/∂θ_j⟩.

Note that both the gradients need to be evaluated using measurements for both the cost-function and the Fubini-Study metric <cit.>.
Subsequently, the updated set of parameters are fed back to the quantum processor and the process is repeated until the iteration convergence criterion is satisfied. Once the latter is satisfied, a second layer convergence criterion is verified. This ensures that the number of layers chosen is sufficient to attain the desired degree of convergence. If this condition is not satisfied, the number of layers is grown by 1 and the earlier steps are repeated. Else, the algorithm terminates. When the number of layers is increased from N to N+ 1, the optimized parameters obtained for the N-layer circuit, augmented by an initial guess for the added layer, were used as the choice for the optimization process for N+ 1 layers. This improved the stability of the algorithm.

For N layers and a ring of L qubits, the total number of parameters determined by the above algorithm is 10LN, where the factor 10 originates from the number of free parameters in the two-qubit unitary operator of Fig. <ref>(e). Despite being tractable [see Fig. <ref>(c) for demonstration], conservation of symmetries present in the target Hamiltonian can drastically reduce the number of parameters to be determined. Symmetries can be conserved by modifying the general circuit ansatz of Fig. <ref> and allowing only operators that respect this symmetry. Alternatively, the cost-function can be augmented to ensure that the obtained state lies in the correct symmetry sector. For example, states which are invariant under translation by r lattice site(s) can be obtained by starting with a state and circuit ansatz respecting this symmetry. For r = 1(2), the number of undetermined parameters for N layers reduces to 10N (20N). Alternatively, in order to obtain a ±1 eigenstate of the translation operator T, the cost-function can be augmented as: L(θ⃗)→ L(θ⃗) ∓μ⟨ψ_ f(θ⃗)|T|ψ_ f(θ⃗)⟩, where μ is a positive real number. Similar considerations apply to symmetry operators related to parity/particle number conservation. 





  
*Results. The proposed algorithm is benchmarked using three paradigmatic one-dimensional QFTs. The results were obtained by modifying standard time-evolution routines for matrix product states. In all three cases, the quantum processor considered comprises a periodic ring with a fixed number of qubits with nearest-neighbor connectivity. This is independent of the model under consideration. The results obtained using the optimized quantum circuit are compared with those from density matrix renormalization group (DMRG) and exact computations. 

First, the critical Ising model in a longitudinal magnetic field is analyzed. In the continuum limit, the QFT contains eight mesonic excitations which scatter with a diagonal scattering matrix <cit.>. These mesons can be viewed as an extremal (g→ 1) limit of bound states formed by the Ising domain walls in the ferromagnetic phase g < 1. The magnetic field leads to a linearly growing energy of separation of two domain walls <cit.>, akin to what happens in quarks in quantum chromodynamics <cit.>, see Fig. <ref>(c). The QFT describing this model is integrable and the number of mesons and their mass ratios are given by the rank and the exponents of the Lie algebra E_8 respectively <cit.>. The relevant lattice Hamiltonian, imposing periodic boundary condition, is:

    H_ I = -∑_j = 1^L X_jX_j+1 - g∑_j=1^LZ_j - h∑_j=1^LX_j,

where g and h were chosen to be 1 and ≃0.156 respectively. Figs. <ref>(a) and <ref>(a) together present the results for the lowest eight translation-invariant states obtained using the proposed algorithm. The circuit ansatz was chosen to be invariant under translation by 2 sites. Furthermore, the cost-function was augmented to ensure that only +1 eigenstates of the operator for single-site translation were obtained. Fig. <ref>(a) shows the results for the ground state energy for system-sizes between L = 8 and 20 in steps of 4. The number of layers required to obtain a reasonable approximation of the ground state do not scale with the system-size, which is compatible with the observations of Ref. <cit.>. In fact, N=2 is sufficient to obtain  the ground state with error in the energy < 1% [Fig. <ref>(d)]. Fig. <ref>(a) shows the results for the first seven excited states for L = 8. For the chosen convergence criterion of 10^-3, the maximum error in the obtained energies was 0.58%. In contrast to the ground state, the number of layers required to achieve this precision ranged between L and 3L. The different translation-invariant excited states can be classified as lattice counterparts of the single and multiple meson states existing in the QFT. The mass ratio for the lowest two mesonic excitations [square and diamond markers in Fig. <ref>(a)] obtained for L = 8 obtained using the proposed algorithm is 1.39 which is close to the expected value of 1.41 obtained using exact diagonalization of  the corresponding spin-chain. This should be compared with the QFT prediction <cit.> given by 2cos(π/5)≃ 1.618. 




Next, the three-state Potts model in the presence of a magnetic field <cit.>  is analyzed. In contrast to its two-state counterpart described above, this model exhibits a richer state of excitations. The three states of each Potts spin are encoded in the triplet of states of two qubits: 

    |0⟩ = |↑↑⟩, |1⟩ = 1/√(2)(|↑↓⟩ + |↓↑⟩), |2⟩ = |↓↓⟩.

The relevant Hamiltonian is

    H_ P = -∑_j  = 1^L'(σ_jσ_j+1^† + gτ + h σ +  h.c.),

where σ and τ are the shift and clock matrices respectively <cit.>. Periodic boundary condition was imposed and L' = L/2 is the number of Potts spins encoded in L qubits. For g< 1, h = 0, the model exhibits a ferromagnetic phase with three degenerate vacua and six domain walls [the three shown in Fig. <ref>(d) and those obtained by reflection about the domain wall]. The presence of the magnetic field breaks the degeneracy of the ground space, giving rise to one true vacuum and two false vacua. The domain-walls experience a confining potential as in the Ising case. However, now there are not only mesonic excitations formed by two domain walls, but also baryonic excitations formed by three domain walls [Fig. <ref>(d)]. Notice that H_ P commutes with the ℤ_2 charge operator C which acts trivially on the state |0⟩ and interchanges |1⟩ and |2⟩. Figs. <ref>,<ref> together show the results for g = h = 0.1 for the lowest eight translation-invariant states with eigenvalue +1 for the operator C. The 2-site translation-invariant circuit ansatz was chosen with the C-eigenvalue imposed using the cost-function. As in the Ising case, the ground state is reached with depth independent of the system-size, with errors in energy less than 0.1% for two layers [Fig. <ref>(b, d)]. The excited states were obtained using between L and 3L layers with maximum error in the obtained energies being 2.26% [Fig. <ref>(b)]. The first excited state is one of the false vacua, with the rest comprising mesonic and baryonic excitations. This is verified by counting the average number of domain-walls in the corresponding state. 

Finally, results are presented for the canonical model for quantum electrodynamics in two space-time dimensions: the one-flavor massive Schwinger model <cit.>. After mapping to an array of qubits, the resulting Hamiltonian with open boundary conditions is given by <cit.>:

    H_ MS   = 1/2∑_j = 1^L  - 1(X_jX_j+1 + Y_jY_j+1) + m∑_j = 1^LP_j 
         + g^2/2∑_j = 1^L - 1[∑_k =1^j (-1)^kP_k]^2,

where P_j = [1 + (-1)^jZ_j]/2. In contrast to the Ising and Potts models, this model involves an all-to-all coupling Hamiltonian. To obtain the ground state of this model, the general circuit ansatz of Fig. <ref> is used. Note the existence of the conserved U(1) charge Q, [H, Q] = 0, where Q = ∑_j=1^LZ_j. To test the algorithm under the most challenging of conditions, the ground states were obtained for system sizes between L = 8 and 16 without conserving this charge. For these simulations, m = 0.5, g = 0.3 were chosen. Despite the long-range nature of the model Hamiltonian, reasonable approximations of the ground-state were obtained with circuit-depths that are independent of the system-size with energies within 1% accuracy obtained within application of 2 layers. Excited states were also obtained similar to the Ising and Potts models and are not shown here for brevity.



  
*Conclusion. To summarize, a hybrid quantum-classical algorithm is presented for the investigation of strongly-interacting QFTs. In contrast to existing approaches that rely on heuristic arguments for choosing the parametrized quantum circuit ansatz, a universal ansatz is described relying on Euler's and Cartan's decomposition of single and two-qubit operators. The algorithm is benchmarked by computing eigenstates of lattice models for three paradigmatic one-dimensional QFTs: the Ising, Potts and one-flavor massive Schwinger models. Quantum circuits were obtained for the realization of ground states, false vacua, mesonic and baryonic excitations occurring in these models. 

The presented results demonstrate the power of geometric control theory <cit.> for the analysis of QFTs. While geometric quantum control is known to be the central concept in the search for an optimal circuit for quantum computation <cit.>, previous proposal are centered around minimization of the distance in the Finsler space of unitary operators. In contrast, the current work presents an alternative framework based on the Fubini-Study metric and a universal circuit ansatz. The proposed framework is not only numerically-tractable, but also readily-implementable on current quantum machines, opening the door to hybrid quantum-classical analysis of a large class of QFTs. 




  
*Acknowledgements.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. 





 Paper Title: Quantum Computing Universal Thermalization Dynamics in a (2+1)D Lattice Gauge Theory.



=5

IQuS@UW-21-085,UMD-PP-024-08




niklasmu@uw.edu
InQubator for Quantum Simulation (IQuS), Department of Physics,
University of Washington, Seattle, WA 98195, USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853

Department of Physics and Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742 USA
Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, MD 20742 USA
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA
National Quantum Laboratory (QLab), University of Maryland, College Park, MD 20742 USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

§ ABSTRACT 
Simulating non-equilibrium phenomena in strongly-interacting quantum many-body systems, including thermalization, is a promising application of near-term and future quantum computation.
By performing experiments on a digital quantum computer consisting of fully-connected optically-controlled trapped ions, we study the role of entanglement in the thermalization dynamics of a Z_2 lattice gauge theory in 2+1 spacetime dimensions. 
Using randomized-measurement protocols, we efficiently learn a classical approximation of non-equilibrium states that yields the gap-ratio distribution and the spectral form factor of the entanglement Hamiltonian. 
These observables exhibit universal early-time signals for quantum chaos, a prerequisite for thermalization.
Our work, therefore, establishes quantum computers as robust tools for studying universal features of thermalization in complex many-body systems, including in gauge theories.


Quantum Computing Universal Thermalization Dynamics 


in a (2+1)D Lattice Gauge Theory 

    Marko Cetina
    February 2024
==========================================================================================








§ INTRODUCTION


Thermalization of isolated quantum many-body systems
in e.g., ultra-cold atomic gases <cit.>, trapped ions <cit.>, condensed matter physics <cit.>, cosmology <cit.>, and nuclear and high-energy physics <cit.>, remains a vibrant frontier.  
Most quantum systems thermalize according to the Eigenstate Thermalization Hypothesis (ETH) <cit.>, which
posits that observables eventually attain thermal values after long unitary evolution <cit.>. Yet, probing large quantum many-body systems undergoing thermalization is inherently challenging due to the non-equilibrium nature of the process, which precludes using first-principle classical computational techniques. Recent advancements in quantum information theory and experiment <cit.> have brought this topic within immediate experimental reach, potentially enabling, via quantum simulation <cit.>, verification of important thermalization paradigms, and discovery of novel principles grounded in quantum information science.


While it is commonly posited that quantum chaos and ergodicity are prerequisites for thermalization <cit.>, their demonstration in the context of quantum many-body systems remains somewhat elusive. Indicators of chaos and ergodicity involve measures associated with the eigenvalues of a given Hamiltonian <cit.>, or the properties of its eigenstates <cit.>. A connection has been drawn recently between 

the entanglement of quantum states and quantum chaos, via the so-called entanglement Hamiltonians (EHs) <cit.>. This connection can, in principle, be leveraged to experimentally probe thermalization via quantum simulation. Most studies to date, nonetheless, have stayed in the realm of theoretical exploration <cit.>.



Analog quantum simulation allows one to monitor a quantum system continuously in time. However, current analog quantum simulators have limited programmability and are restricted to probing specific physical models and a limited set of observables <cit.>. In contrast, universal digital quantum computers allow, in principle, the probing of the dynamics of any physical model. The universal control in digital quantum computers enables the use of tomographic techniques to extract a wide range of observables, including entanglement.  
Digitized time evolution, via 

Trotterization <cit.> or other schemes <cit.>, limits the near-term simulations to times that are short compared to those of thermalization. Nonetheless, there exist universal phenomena that are indicative of quantum chaos and ergodicity at earlier times. Probing these phenomena, therefore, is an attractive near-term opportunity for digital quantum computers.

Gauge theories and their lattice formulations are among prime physical models whose simulations will benefit from quantum-computing technology <cit.>. Gauge theories are key in high-energy and nuclear physics <cit.>, condensed and synthetic quantum matter <cit.>, local fermion-to-qubit mappings <cit.>, and quantum-error correction  <cit.>. Studying thermalization dynamics of gauge theories, e.g., in early universe and in high-energy particle collisions, remains challenging using first-principles simulation methods <cit.>. As a first step in experimentally probing thermalization dynamics of gauge theories, we study a Z_2 lattice gauge theory (LGT) in 2+1 spacetime dimensions <cit.> using a digital trapped-ion quantum computer <cit.>. 
We use a chain of fifteen ^171Yb^+ ions to realize a general-purpose fully-connected digital quantum computer with twelve qubits, and use this computer to natively and accurately encode system's initial state, evolve it in time, and measure observables.

Our analysis relies on EH tomography <cit.> in combination with randomized-measurement protocols <cit.> to learn representations approximating non-equilibrium states. The EH is defined as H_ E≡ -log(ρ_A), where ρ_A denotes the reduced density matrix of subsystem A formed by bipartitioning a (pure) quantum state. The utility of EH as a theoretical and experimental tool stems from the observation that, in many cases, it consists of approximately local operators <cit.>. This resemblance to conventional Hamiltonians, i.e., energy operators governing system dynamics, facilitates simple theoretical analysis: If an EH is (k-)local, it can be described by parameters whose number scales polynomially with the system size, unlike the matrix ρ_A which requires an exponential number of parameters. 

Randomized-measurement protocols <cit.> can be used to learn the EH: after repeatedly preparing and then measuring the quantum state in a randomly chosen basis, one can fit a  parameterized EH to the obtained measurement outcomes.

Because the spectrum of H_ E, known as the entanglement spectrum, scales logarithmically with the Schmidt eigenvalues of ρ_A, it is difficult to learn an EH and reproduce the entire entanglement spectrum with arbitrary accuracy. Randomized-measurement protocols may require resolving exponentially small probabilities even for relatively small systems, which is out of reach of current quantum devices and perhaps fault-tolerant devices, too. 
The present experiments can, therefore, only aim at approximating H_ E and accurately reproducing its low-energy spectrum. Applications of this approach include the verification of topological phases <cit.>.

Nonetheless, what if the precise quantitative structure of H_ E were not crucial, but only its statistical properties were? This viewpoint is that of random matrix theory <cit.>, where metrics like level distribution and spectral form factor differentiate between integrable and chaotic dynamics, while being indifferent to quantitative details, with the notable exception of symmetries. This statistical perspective underpins our study. Guided by physical insights regarding the expected operator content of a non-equilibrium EH, we ask whether one can learn, from experimental data, a classical representation of a state to answer a simpler question: does a quantum state exhibit universal signatures of quantum chaos evident in the statistical properties of its EH? Crucially, can we discern this scenario from one where the state lacks chaotic behavior?

To answer these questions, we focus on two observables indicative of quantum ergodic and chaotic behavior: the entanglement(-Hamiltonian) gap-ratio distribution (EGRD) <cit.> and the entanglement spectral form factor (ESFF) <cit.>. The EGRD is predicted to exhibit level repulsion for chaotic states, in contrast to uncorrelated levels in non-chaotic scenarios. Similarly, the ESFF is predicted to display a plateau-ramp structure in chaotic states
 [Strictly speaking, both indicate quantum ergodicity which usually implies quantum chaos.]. To experimentally test these predictions, we initialize our system in a product state and perform a quantum quench by digitally evolving the system for a variable length of time. At short evolution times, we observe an EGRD that is characteristic of uncorrelated states. With increasing evolution time, we observe the onset of level repulsion and a ramp-plateau structure in the ESFF, indicative of quantum chaos. Detailed analysis, including comparison with emulated data, suggests that the observed behavior primarily arises from the thermalization dynamics of the isolated quantum system under simulation, with experimental inaccuracies and shortcomings of our ansatz playing a small, though non-negligible, role. Our work, therefore, establishes present-time quantum computers as robust tools for studying universal feature of thermalization dynamics in complex many-body systems.

This manuscript is organized as follows. In Secs. <ref> and <ref>, we introduce, respectively, the model and the experimental setup, including state preparation, real-time evolution, and measurement. We discuss our tomography scheme and classical post-analysis in Sec. <ref>. Finally in Sec. <ref>, we present our main experimental results, reporting the observation of a behavior consistent with quantum chaos in the structure of the EH of time-evolved states. Several Appendices offer additional insights: Appendix
<ref> provides further details of 
the quantum circuits employed;
Appendix <ref> investigates finite-size, Trotterization, and late-time effects; and Appendix <ref> offers a comprehensive explanation of our EH-tomography protocol.







§ MODEL


We focus on a Z_2 LGT in 2+1 D with the Hamiltonian


    H = ∑_□ W_□ +g ∑_ℓσ_ℓ^z .


Here, W_□≡∏_ℓ∈□σ_ℓ^x is a magnetic-field operator, where □ denotes the elementary (square) plaquettes of the lattice [see fig:model(a)]. σ^x_ℓ and σ^z_ℓ are Pauli operators representing gauge link and electric fields on edge ℓ, respectively. These act on spin-1/2 hardcore bosons residing on the edges of a two-dimensional spatial square lattice. The first term in eq:Z2Hamiltonian corresponds to the magnetic energy, and the second to the electric energy, with coupling g controlling the relative strength of the two non-commuting contributions. In this formulation, the z basis, therefore, represents the electric basis, while the x basis corresponds to the magnetic basis. The Hamiltonian is expressed in dimensionless units and ħ is set to one throughout.

We consider the joint +1 eigenspace of the Gauss-law operators G_j≡∏_ℓ∈ +_jσ^z_ℓ as the physical Hilbert space, where ℓ∈ +_j denote the four links adjacent to a lattice site j. In the subsequent discussion, we consider a quasi-1D chain composed of L_x plaquettes along the x direction with periodic boundary conditions. Fixed boundary conditions are applied along the short side of the chain (with L_y=1 plaquettes) as illustrated in fig:model(b). In this configuration, all Gauss laws contain three links at each site. Consider a short-ribbon operator V_y≡σ^z_ασ^z_β, where α and β are the two opposing links along the inner and the outer circumference at an arbitrary position along the closed chain shown in fig:model(b) [Note that V_y is independent of α and β because, for Gauss law eigenstates, the operator can be acted on by Gauss law operators in such a way that it can be moved around the lattice.]. This operator commutes with the Hamiltonian and defines superselection sectors of the model; we work in the V_y=1 sector (i.e., short-ribbon operators possess eigenvalue one).

As initial conditions, we randomly select Gauss-law–respecting electric eigenstates (z-basis product states) of the form  | Ψ(t=0)⟩≡ | Ψ_0⟩ = ∏_ℓ | ℓ⟩ with | ℓ⟩∈{| ⟩ ,| ⟩}. We then define a subsystem A consisting of L_A ≤ L_x/2 plaquettes. The reduced density matrix ρ_A(t) of A after evolution time t is related to the density matrix ρ(t) of the whole system as


    ρ_A (t) ≡Tr_A̅[ ρ(t)],   ρ(t)≡ U(t) ρ_0 U(t)^†,


where A̅ is the complement of subsystem A shown in fig:model (c) and (d), ρ_0 ≡ |Ψ_0 ⟩⟨Ψ_0 |,  and U(t)≡ e^-iHt is the time-evolution operator.

To directly encode the degrees of freedom of the model described onto the qubits, one requires one qubit per link degree of freedom, or N_q=3L_x qubits in total. To adapt the simulation to the available hardware, we consider a dual formulation of the model that requires only N_q=L_x + 2 qubits. In the dual formulation, all operators, except those containing operators having support  at the boundaries between the subsystems, labeled ℓ_A̅A and ℓ_AA̅, are entirely represented using gauge-invariant (Ising dual) spin-1/2 variables which act solely within the physical Gauss-law subspace; they are denoted by μ_i^x,z Pauli matrices [see purple circles in fig:model(c)]. Concretely, μ_i^x → W_i= ∏_ℓ∈□_iσ_ℓ^x, while electric variables are represented by μ^z_i μ_i-1^z ≡σ_ℓ^z where ℓ is the link between plaquettes i and i-1. The corresponding “bulk” Hamiltonian terms for the subsystem A then reads


    H_A^ dual =  ∑_i∈A_ bulkμ^x_i  + g (∑_⟨ i,j ⟩∈ Aμ^z_iμ^z_j
    +κ∑_i∈ Aμ^z_i),


where κ≡ 1+V_y. Here, i ∈ A_bulk in the sum over μ_i^x indicates that the original plaquette □_i is entirely in A (i.e., does not touch the boundaries). ⟨ i,j ⟩∈ A denotes nearest-neighbor bulk-spin pairs i,j in A. Finally, the sum over μ_i^z runs over all bulk-spin indices i. The terms in the Hamiltonian acting on the complement are defined identically. Importantly, at the boundaries between the subsystems, ℓ_A̅A and ℓ_AA̅ [see the orange circles in Fig. <ref>(c)], the gauge-variant variables of the LGT, as described by eq:Z2Hamiltonian, are retained. Denoting the Ising-spin index at the A side of one of the boundaries to be k, the Hamiltonian terms coupling the subsystems at this boundary are


    H^ dual_A̅A≡μ^x_
        kσ^x_ℓ_A̅A + μ^x_
        k-1σ^x_ℓ_A̅A + gσ^z_ℓ_A̅A ,


with a similar definition for the other boundary. Note that on the plaquette k containing one boundary link, μ_k^xσ^x_ℓ_A̅A≡ W_k, where μ_k^x≡∏_ℓ∈□̃_kσ_ℓ^x, with □̃_i referring to all but non-boundary links of plaquette k, and similarly for the plaquette k-1. The Gauss laws at the boundaries are not eliminated by the duality. The two Gauss laws in the dual model that are independent (one at each boundary) are


    G^ dual_A̅A≡μ^z_
    
    kσ^z_ℓ_A̅Aμ^z_
    k-1 ,


and similarly for the boundary at ℓ _AA̅.
The dual Hamiltonian of the model is the sum of the terms above:


    H^ dual =H_A^ dual +H_A̅^ dual + H^ dual_A̅A+ H^ dual_AA̅ .



This formulation ensures that all gauge-invariant variables of the LGT and its dual have the same expectation values. Importantly, by maintaining the Gauss laws at the boundaries, the entanglement properties of the dual formulation are identical to those of the LGT <cit.>. This is a subtle but crucial distinction from the standard Ising duality <cit.>, which does not preserve the entanglement structure.

The entanglement structure depends on  the symmetries of the reduced density matrix, which stem from the remaining Gauss laws.
Specifically, [S_j,ρ_A(t)] =0,
where in LGT variables, S_j≡∏_ℓ_j ∈ A σ^x_ℓ_j,
with ℓ_j being the two links originating from a boundary site with lattice coordinates j=(j_x,j_y) within A. This is a direct consequence of the state being a Gauss-law eigenstate <cit.>. In addition, [V_x,A,ρ_A(t)] =0 where the long-ribbon operator within A is defined as V_x,A≡∏_ℓ∈𝒫σ^x_ℓ where σ^x_ℓ act on the original degrees of freedom along the path 𝒫 connecting both boundaries within A, see Fig. <ref>(d). Not all symmetry operators are independent.  For the quasi-1D chain of plaquettes shown in fig:model, only one of the two S_j is independent at each boundary. We place one of the boundaries at sites (0,0) and (0,1) and the other at sites (L_A,0) and (L_A,1). We then choose the two independent S_j operators to be S_(0,0) and S_L_A,0, henceforth referred to as  S_AA̅ and S_A̅A, respectively. Note that, when acting on a Gauss-law–respecting state, V_x,A =S_AA̅S_A̅A. Consequently, there are four independent symmetry blocks ρ_A(t)≡⊕_s=1,⋯,4ρ_A,s(t) that will play a role in our analysis.  In the dual formulation, these symmetry operators are simply S_AA̅^ dual≡μ_k^z σ^z_ℓ_A̅A and S_AA̅^ dual≡μ_k'^z σ^z_ℓ_A̅A where k and k' are the first and last plaquettes within A [This discussion is general, independent of the choice of a V_y eigensector.]. We denote the eigenvalue {1,1}, {1,-1}, {-1,1}, and {-1,-1} sectors of {S_A̅A^ dual,S_AA̅^ dual} operators with labels s=1,2,3,4, respectively. 






§ EXPERIMENTAL SETUP


We use a universal trapped-ion digital quantum computer, schematically shown in fig:introoverview(a), to simulate the time evolution of the dual formulation of the Z_2 LGT. We consider L_x=10 plaquettes, focusing on the entanglement structure of a subsystem consisting of L_A=4 plaquettes, as shown in fig:model(c). 

Our experimental system consists of a linear chain of fifteen ^171Yb^+ ions trapped along the 𝐱 axis of a microfabricated ion trap  <cit.> and spaced by approximately 3.7 μm. The twelve LGT spins are mapped to the ions as shown on fig:model(c). The | ⟩ and | ⟩ spin states are respectively mapped to the first-order magnetic-field–insensitive hyperfine | F=1, m_F=0⟩ and | F=0, m_F=0⟩ states of the ^2S_1/2 ground electronic manifold of the ions [Here, we use the quantum-information convention for the spin states, i.e., |↑⟩≡|0⟩ and |↓⟩≡|1⟩.]. The spin states are initialized in the |⟩ state by optical pumping and are measured by coupling to the excited ^2P_1/2, |F=0,m_F=0⟩ state using a 369-nm laser, wherein the presence (absence) of emitted photons differentiates between the bright state | ⟩ and the dark state | ⟩ with an average 0.3% infidelity, limited by off-resonant photon scattering.

The ions are individually addressed by an equispaced array of 1 μm-diameter, 355-nm, laser beams oriented perpendicular to the trap surface (𝐳 axis), together with a global 300 μm × 30 μm beam oriented along the 𝐲 axis, which is parallel to the trap surface (see fig:introoverview(a) and Ref. <cit.>). These beams drive stimulated Raman transition via the ^2P_1/2 and ^2P_3/2 excited electronic states, where a photon is absorbed from the global beam and emitted into an individual beam to flip the qubit from | ⟩ to | ⟩ and vice versa. All spin-manipulation light is produced by a single pulsed laser, modified to control its repetition rate to null 4-photon Stark shifts. The phase and amplitude of the 355-nm beams is controlled by single-channel and 32-channel acousto-optic modulators [Provided by L3 Harris Corporation.].




To minimize addressing errors, crosstalk, and stray coupling to axial motion of the ions <cit.>, single-qubit operations are realized using compound SK1 pulses <cit.> with Gaussian-shaped sub-segments, with typical infidelities of 0.2%. Entangling operations between any two qubits are realized via variable-angle pairwise Mølmer-Sørensen (MS) gates <cit.> employing laser-induced state-dependent forces on the ≈ 3-MHz motional modes of the ion chain oriented along the 𝐲+𝐳 axis. Robust decoupling from the ion motion at the end of each entangling gate is accomplished using amplitude-modulated, detuning-robust pulse waveforms <cit.>. Typical non-unitary errors of fully-entangling MS gates are 1% and consist predominantly of gate-angle errors and Z-flips of the individual qubits <cit.>.

Our ultimate goal is to study thermalization dynamics in real-time non-equilibrium conditions by learning system's entanglement structure. To this end, we first demonstrate the capability of our digital computer to compute the dynamics of the Z_2 LGT model, including initial-state preparation, time evolution, and subsequent measurements. We initialize the system in a randomly chosen electric eigenstates that respects Gauss law (|Ψ_0⟩=|↓↓↓↑↓↓↑↑↑↓↑↑⟩, see fig:model(c) for the qubit mapping, indicated in blue numbers, read from left to right) and simulate its time evolution for g=0.85 using four Trotter steps that separately implement the non-commuting terms of H^dual, i.e., those diagonal in electric (z) or magnetic (x) bases. Our choice of coupling is such that magnetic and electric terms are of similar magnitude, making the model sufficiently non-integrable. For each  evolution time t, we apply single-qubit gates to measure all qubits in either the Bloch x or the z bases and repeat each experiment N_shots=500 times. The measurement results are used to compute single- and two-qubit observables in the Bloch x and z bases and are plotted in Figure <ref> for select observables [For simplicity, the operators whose expectation values are measured are generally noted as Pauli σ operators. When acting on the bulk dual spins, they should be thought of as the Pauli μ operators, while when acting on the boundary spins, they are the Pauli σ operators in the original LGT.]. The statistical errors are determined by a standard bootstrap resampling analysis with N_boots=1000 bootstrap samples. Quantum circuits implementing Trotterized time evolution and measurements are provided in Appendix <ref>. Each Trotter step of evolution uses 12 variable-angle MS gates. 

For most evolution times, the expectation values of the physical operators (dark blue and red) show good agreement with the zero-parameter theory prediction (solid lines). We ascribe the ∼10% deviation in the evolution rate of the system to miscalibrations of the MS-gate angles. Importantly, the expectation values of gauge-invariance–violating operators (i.e., those that do not commute with the Gauss laws at the two boundaries), shown in light blue, are consistent with zero within measurement errors. Figure <ref> of Appendix <ref> provides a close-up of these observables, showing that gauge-invariance violation is about 5% throughout the time evolution. 

To establish the connection between these simulations and the signatures of chaos and thermalization, we now introduce a measurement and analysis protocol that allows the determination of the system's entanglement structure, going beyond low-order correlation functions.






§ ENTANGLEMENT-HAMILTONIAN SPECTRUM AND TOMOGRAPHY


Our main interest lies in the statistical properties of the Schmidt decomposition of the reduced density matrix of the subsystem A, given by ρ_A(t) ≡∑_λ p_λ(t) | λ(t) ⟩⟨λ(t)|, and its associated EH <cit.>


    H_ E(t) ≡ -log(ρ_A(t)) .


The EH's eigenspectrum, called the entanglement spectrum, is defined by {ξ_λ(t) ≡ -log(p_λ(t))}.





The statistical properties of the entanglement spectrum directly inform the thermalization dynamics of the system. One statistical probe is the distribution, P(r), of the gap ratios, r_λ, of the entanglement spectrum <cit.>,


    r_λ≡min(δ_λ , δ_λ-1)/max(δ_λ , δ_λ-1)  ,


where δ_λ≡ξ_λ - ξ_λ-1≥ 0 are the gaps between the eigenvalues ξ_λ of H_ E and r ≡{r_λ}. According to random matrix theory of Hermitian matrices, this quantity distinguishes chaotic from nonchaotic behavior depending on whether the distribution is centered away from zero (level repulsion) or centered around zero (uncorrelated levels) <cit.>. The gap ratios, nonetheless, only capture local correlations of the entanglement spectrum and, owing to their logarithmic nature, treat entanglement eigenvalues ξ_λ≡ -log(p_λ) associated with very small probabilities p_λ≪ 1 no differently than those where p_λ≈ O(1). 

A complementary view of the statistics of the entanglement spectrum that captures global correlations in the level distribution is afforded by the entanglement spectral form factor (ESFF) <cit.>,


    ℱ(θ ;t) ≡⟨1/ℛ^2_A(t)∑_λ,λ' e^iθ[ξ_λ(t) -ξ_λ'(t) ]⟩ ,


where ⟨·⟩ denotes the average over initial states |Ψ_0⟩. Here, ℛ_A ≡lim_α→ 0exp{1/1-αlog(∑_λ p_λ^α)} <cit.>  is the effective rank of H_ E, whose value depends on the state and lies in the range [1, 2^L_A] [In practice, both statistical measures are typically used to describe the properties of physical Hamiltonians, whose rank equals the Hilbert-space dimension. For EHs, we employ regularization suitable for non-full–rank matrices. To address levels with probabilities close to zero within machine precision, we define an effective rank. Only the ℛ_A lowest levels of the EH are included in the analysis. However, experimental constraints, such as finite statistics and device errors, typically lead to EHs that are reconstructed as nearly or exactly full rank, rendering any regularization obsolete. Nonetheless, non-full–rank EHs occur for states computed in exact diagonalization, particularly at early times, that we contrast our data with.].  Starting from an initial flat behavior, a ramp-plateau structure in the ESFF as function of the `Fourier time' θ is known to indicate ergodic behavior and imply quantum chaos <cit.>. 

While in classical computation of a tractably small system, the state and its entanglement are readily available, in a quantum-simulation experiment, these must be inferred from measurements. To tackle this challenge, we constrain the operator content of the EH to obtain an approximate H_E, and utilize an EH-tomography scheme based on randomized measurements <cit.>.   
Concretely, we perform a single-layer, single-qubit randomized-measurement <cit.> of the time-evolved state. 
At the end of the time evolution, this protocol applies one of N_𝒰 different gates 𝒰_j = ⊗_i=0,⋯,N_q-1 u_i,j for j=1,⋯,N_𝒰 consisting of independent single-qubit gates u_i,j sampled from a unitary 2-design <cit.>. For each 𝒰_j, N_shots experiments are performed, measuring all qubits in the Bloch z basis. 

We compare the relative frequencies of the different bitstrings b to the prediction based on an approximate EH inspired by the Bisognano-Wichmann (BW) theorem <cit.>. 
Explicitly, we assume that H_ E(t) is a linear combination of k-local terms 𝒪_i, i.e.


    H_ E (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i .


To find a suitable set of operators { O_i }, we proceed as follows. Starting from the operator content of the physical Hamiltonian in eq:Z2Hamiltonian, containing at most 4-local terms (2-local in the dual formulation), new operators are generated iteratively by forming non-trivial commutators from the existing set. This process can be halted after two iterations (`commutators of commutators') resulting in a maximum of 7-local operators within the LGT framework (or 3-local in the dual formulation). We exclude operators that violate the Gauss laws and those that can be related to others upon application of the Gauss-laws. To ensure Hermiticity, we adjust prefactors and create linear combinations with conjugates [If this iteration were continued, eventually one would end up with an exponentially large set of all possible operators. Our approach is based on a purely heuristic observation that, in practice, only polynomial many (quasi)local operators are required to accurately reproduce the state ρ_A(t) with exponential accuracy <cit.>.]. To find the parameters β_i, a global optimization routine, based on standard Matlab libraries <cit.>, is used to determine the parameterization that most closely reproduces all measurements (refer to Appendix <ref> for details).






§ EXPERIMENTAL STUDY OF THERMALIZATION DYNAMICS OF Z_2 LGT


To study the evolution of the entanglement spectrum in the Z_2 LGT, we repeat the time-evolution experiments shown in Fig. <ref> starting from 6 randomly chosen initial electric eigenstates that satisfy the Gauss laws [We numerically checked that the results remain valid for other randomly chosen states in this basis]. Since these are not the eigenstates of the full Hamiltonian H, the system undergoes non-trivial time evolution when evolved under H. For each evolution time t, symmetry sector, and initial state, we subsequently perform randomized-measurement tomography with g=0.85, N_𝒰=24 bases, and N_ shots=750 bitstring measurements in each basis, and use these measurements to reconstruct the EH as described in the previous section. This procedure yields a set of gap ratios for each initial state, symmetry sector, and evolution time.

In fig:gap-ratio-exp-analysis(a), we plot in orange points the average gap ratio ⟨ r⟩≡∑_r rP̅(r) of the reconstructed EH  as a function of the scaled evolution time g t. The plotted gap ratios are averaged over both the symmetry sectors and the 6 randomly chosen initial states. The black lines correspond to the predicted exact distributions following the Trotterized time evolution. The cyan lines correspond to the EH obtained from an optimal BW-inspired ansatz, where we numerically minimize the
the relative entropy (Kullback–Leibler divergence) between the exact state and the ansatz, corresponding to the limit of infinitely many measurements. A buildup of the level repulsion is discernible as the observed average gap ratio ⟨ r ⟩ evolves from ≈ 0.4 predicted for a non-repulsive Poisson distribution (blue dashed line) towards ≈ 0.6 characteristic of repulsive level statistics of a Gaussian Unitary Ensemble (GUE) <cit.>. Three time regimes (I), (II), and (III) are identified that correspond, respectively, to the evolution of the predicted average gap ratio toward the Poisson-distribution value, toward the GUE value, and to saturation at the GUE value.

In fig:gap-ratio-exp-analysis(b) we plot, for the three ranges of evolution times, the corresponding normalized distribution of the gap ratios, P̅(r), combined over 6 random initial states and 4 symmetry sectors. We observe a clear transition from early-time non-repulsion (Poisson distribution, blue dotted line) in regime (I) to level repulsion (GUE, red) in regime (III). At intermediate times, a distribution is observed between the initial absence of level repulsion and the subsequent emergence of level repulsion.  Error bars and bands denote the variance resulting from averaging over symmetry sectors and initial states. The predicted exact distribution (black line) exhibits a sharper peak around zero at the earliest times due to the reduced density matrix not being full rank. Conversely, the BW-inspired ansatz (cyan line) generally parametrizes a full-rank matrix unless couplings are very finely tuned, resulting in an overestimation of level repulsion in the initial stages. Calculations indicate that the onset of the three regimes remains consistent for any randomly chosen initial states and depends only on the (subsystem-)size when measured in coupling-independent units gt, indicating the universality of thermalization dynamics.






In fig:ESFF-simulator-analysis
we plot the ESSF reconstructed from the data used in fig:gap-ratio-exp-analysis. The three panels of the plot correspond to the three time regimes discussed in relation to the EGRD evolution. The displayed theory curves are averaged over initial states, symmetry sectors, and each of the three time ranges. Starting from an initial flat behavior of the ESSR as a function of θ in panel (I), our data in panels (II) and (III) clearly show the buildup of a ramp-plateau structure, indicating ergodic behavior. In panel (III), we show a fit (purple dotted line) to the observed ramp, indicating a power-law behavior θ^κ where κ = 0.6 ± 0.2, with the fit error determined by changing the fit regime. This exponent is consistent with the results obtained from the numerical analysis of a significantly larger  system in Appendix <ref>. 


Our results demonstrate that, using observables that reveal universal statistical or global features of entanglement spectrum, the onset of chaotic behavior can be robustly traced in the entanglement dynamics of arbitrary initial states. Importantly, time digitization and experimental infidelities in current systems do not significantly impact qualitative features of thermalization dynamics. 






§ CONCLUSIONS


Using a digital trapped-ion quantum computer, we observe early-stage thermalization dynamics in a Z_2 lattice gauge theory in 2+1 dimensions by generating nonequilibrium quantum states and measuring their entanglement structure. As performing quantitative state tomography is extremely challenging and resource intensive, we focused on a simpler question: can a randomized-measurement–based entanglement-Hamiltonian
tomography recover universal properties of the EH that indicate quantum chaos? We experimentally determine the entanglement spectrum of time-dependent quantum states and use two properties of this spectrum, namely the entanglement gap-ratio distribution [Eq. (<ref>)] and the  entanglement spectral form factor [Eq. (<ref>)], as indicators of the emergence of quantum chaos. Our data indicate that both quantities behave as expected: the initially non-repulsive level distribution transitions to a repulsive one. Likewise, a ramp-plateau feature of the ESFF is initially absent but builds up with time. Our results indicate that the timescales for these entanglement-based phenomena mainly depend on subsystem size.

The key technique enabling our analysis is a Bisognano-Wichmann–inspired ansatz for the EH, which allows the parameterization of a nonequilibrium state using a polynomial number of parameters. While this classical parameterization optimally reproduces the observed (randomized) measurement results, in Appendix <ref>, we show that the higher-lying part of the EH spectrum is not quantitatively recovered. Indeed, the measurement cost for precise state tomography still scales exponentially with the subsystem size. Fortunately, to detect the presence of quantum chaos, one needs to only distinguish repulsion from non-repulsion in the EGRD  or identify a ramp-plateau structure in the ESFF. Our work indicates that selecting a sufficiently small subsystem of a much larger, potentially classically nonsimulable system, can sufficiently constrain these observables. 

Our randomized-measurement procedure remains classically simulable because the lattice sizes we considered are manageable with exact diagonalization. Our procedure is also heavily tailored at minimizing the computational load on the quantum computer, avoiding deep quantum circuits and the need for error mitigation, at the cost of classical post-processing. To extend our approach to larger systems, several steps can be implemented in future work:



  –
The employed single-qubit one-layer randomization strategy is symmetry ignorant, i.e., it randomizes regardless of the known symmetry structure of the subsystem density matrix ρ_A(t). Although our scheme is tomographically complete, and enables state reconstruction along with its symmetries, it is inefficient.  Symmetry-conscious protocols, for the LGT in this work and for other models, have been developed <cit.> and could be used in future work.


  –
A major methodological uncertainty is the BW-inspired parameterization of the EH, which is entirely heuristic. While there is research exploring the operator content of the EH of ground, excited, and thermal states <cit.>, further investigations are needed into the applicability and limitations of EH-based schemes for far-from-equilibrium states. Generally, the BW(-inspired) ansätze describe the low-energy regime of the EH well, and incorporating progressively more nonlocal terms  improves convergence into the bulk <cit.> which the EGRD, and to a lesser extent the ESFF, predominantly rely on.  Notably, our analysis reveals that despite the quantitative discrepancy in the bulk, the statistical distribution of the EH appears to be accurately reproduced. This suggests that it may be unnecessary to perform precise state tomography when one's interest is solely in statistical properties. 


  –
Trotterization corresponds to time evolution with an effective, rather than the desired, Hamiltonian. We observe that employing too few Trotter steps leads to poor convergence of our optimization procedure for the BW-inspired ansatz. In this case, more nonlocal operators must be included in the ansatz; constraining these becomes challenging, especially when relying on a limited number of measurements. Since using a large number of Trotter steps increases 
experimental errors, the optimal Trotterization and its interplay with the BW ansatz should be further investigated.


  –
Detailed comparison between the emulator and the experimental data (see Appendix <ref>) highlights the influence of device errors, primarily for the time evolution and tomography steps of our algorithm. The dominant errors in our study are Z-flip errors and over- and under-rotations due to mechanical motion of the ions <cit.>.
Our tomography protocol is especially susceptible to small single-qubit rotation errors during randomization <cit.>.  
Recently demonstrated sympathetic cooling during circuit execution <cit.> would allow the needed measurement circuits to be executed with higher fidelity. The same technique could extend our study to larger systems, later times, and more Trotter steps, e.g., allowing us to directly test the eigenstate thermalization hypothesis in a digital quantum-computing set-up. 



In summary, our results demonstrate that entanglement structure is a measurable quantity in present-day LGT quantum-simulation experiments, and illustrates the potential value of our approach to probe thermalization dynamics and its robust universal features in strongly coupled isolated quantum many-body systems. A compelling future direction is to extend the investigation to later times, to probe aspects such as pre-thermalization <cit.> or fluctuation-dissipation relations <cit.>, once experimental capabilities permit. 
This would also allow for probing the applicability of the Eigenstate Thermalization Hypothesis, e.g., in systems with non-Abelian symmetries <cit.> and other gauge theories <cit.>. Furthermore, the experimental and theoretical tools of this study can be applied in a number of other applications, including obtaining thermodynamic quantities such as work and heat exchanged during nonequilibrium processes <cit.>, and detecting phases of matter, including topological phases <cit.>, in quantum-simulation experiments.






§ ACKNOWLEDGMENT

N.M. thanks H. Froland, A. Polkovnikov, M. Savage, M. Srednicki, X. Yao, T. Zache, P. Zoller, and the participants of the InQubator for Quantum Simulation (IQuS) workshop “Thermalization, from Cold Atoms to Hot Quantum Chromodynamics ”(<https://iqus.uw.edu/events/iqus-workshop-thermalization/>) at the University of Washington in September 2023 for many valuable discussions leading to this work. M.C. thanks L. Feng for valuable discussions and help with the experimental setup leading to this work.
N.M. acknowledges funding by the Department of Energy (DOE), Office of Science, Office of Nuclear Physics, IQuS (<https://iqus.uw.edu>), via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science under Award DE-SC0020970. Z.D., M.C., and T.W. were supported by the National Science Foundation's Quantum Leap Challenge Institute for Robust Quantum Simulation under Award OMA-2120757. Z.D. further acknowledges support by the DOE, Office of Science, Early Career Award DE-SC0020271. This work is further supported by a collaboration between the US DOE and other Agencies. This material is based upon work supported by the DOE, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. 






§ COMPETING INTERESTS

M.C. is a co-inventor on  patents that are licensed from the University of Maryland to IonQ, Inc.













§ CIRCUIT REPRESENTATION FOR STATE PREPARATION, EVOLUTION, AND MEASUREMENT



 


In this Appendix, we provide circuits employed to experimentally realize state preparation, time evolution, and randomized measurement in the Z_2 LGT in (2+1)D within the dual Isnig formulation.

We label qubits with indices q=0,…,L-1, with L=12. Qubit q=0 corresponds to the spin at ℓ_A̅A positioned at the interface between subsystem A and its complement, identifiable by an orange circle in fig:model(c). Qubits q=1,…,4 represent the succeeding dual variables arranged counterclockwise at the center of plaquettes denoted by purple circles in fig:model(c). Following this sequence, qubit q=5 resides at the opposite boundary marked as ℓ_AA̅, and so forth. 










Working in the electric eigenbasis and starting from an all-up spin state, we first perform single-qubit σ^x bit-flip operations to select a randomly chosen product state in the σ^z basis that is consistent with Gauss laws, i.e., G_j |ψ(0)⟩=|ψ(0)⟩ for all j. Recall that in the dual formulation, there are only two Gauss-law operators each located at one of the subsystem's boundaries: G^ dual_A̅A = μ^z_11σ^z_0 μ^z_1 and G^ dual_AA̅ = μ^z_4σ^z_5 μ^z_6. Importantly, any spin configuration in either bulk, A and A̅, is physical, as in the dual formulation, only gauge-invariant degrees of freedom are kept in the bulk.


For time evolution, we utilize a first-order Trotter scheme with time-evolution operator


    U(t)≡ e^-itH^ dual≡∏_δ t U(δ t).


We take δ t to be a variable time step while keeping the Trotter depth fixed. In the simulations conducted in this work, t/δ t= 4, resulting in states and observables closely resembling the exact time-evolved states at early times. However, for late times, Trotter effects become more pronounced.

U(δ t) in Eq. (<ref>) can be written as U(δ t)≡∏_a∈ X,Z,XX,ZZ U_a with U_a ≡  e^-iδ tH^ dual_a; H^ dual_a are the respective 1- and 2-local operators of the dual Hamiltonian in Eq. (<ref>), sorted in X and Z operations, as well as single- and two-qubit entangling gates. While XX, X, and Z are native operations, ZZ entangling operations are realized via basis transformation and usage of the native MS gate,


    R^zz_ij(α) ≡ R^y_i(-π/2) R^y_j(-π/2)    R^xx_ij(α)   R^y_i(π/2) R^y_j(π/2).


Here, R^x/y/z_i(α) ≡ e^-iα/2σ_i^x/y/z and R^xx_ij(α) ≡ e^-iασ_i^xσ_j^x. The experimental errors of our MS gates increase with the absolute value of the gate angle. To minimize the gate error, all MS gates are optimized as follows. First, all angles are mapped to the regime α∈ [-π,π]. Within this range, we make the following substitution


    R^xx_ij(α) →
             R^xx_ij(α)    if |α| ≤π/4
    
             R^xx_ij(α+π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α<0 
    
             R^xx_ij(α-π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α>0
    
              R^xx_ij(α+π)     if α < -3π/4
    
              R^xx_ij(α-π)     if α > 3π/4


so that all MS operations are restricted to the range |α| ≤π/4. 

An example study of the initial-state preparation, time evolution, and measurement in a fixed (Bloch x or z) basis is shown in fig:experimentbasis (b) and (c) of the main text, where we presented measurement of several gauge-invariant and non-gauge–invariant
one- and two-qubit observables. In addition, we present the measurement of several other non-gauge–invariant observables in fig:gauge_invariance. These observables are expected to be zero, but remain non-vanishing because of finite measurements and device errors. A main source of errors are likely coherent errors, related to the over- and under-rotation of single- and two-qubit gates. While the time-evolution circuit only contains gauge-invariant operations for any gate angle employed, the final single-qubit rotations R^y_i(-π/2) and R^x_i(π/2) (that transform all qubits i from the Bloch z basis into the Bloch x and y bases, respectively) can introduce gauge-invariance violation if their rotation angle is set inaccurately. Additionally, errors stemming from initial-state preparation and readout processes can contribute to this violation.






§ FINITE-SIZE DEPENDENCE, TROTTERIZATION, AND LATE-TIME BEHAVIOR


In this Appendix, we perform various tests aiming to extend our study toward larger systems, and to provide an outlook on the expected late-time behavior of non-equilibrium states, features that are inaccessible in current quantum computers. We present a detailed classical  computation of a fairly large system (L=22, L_A=8 with coupling g=1), evolving the state continuously, and computing its entanglement structure using exact diagonalization, hence avoiding a Trotterization error. Such an analysis is possible for the model under consideration, owing to its relative simplicity. This simplicity enables us to achieve results that are fairly insensitive to finite-size effects, even with moderately large systems.  A similar classical computation will be impossible for more involved models or in higher dimensions.

We compute the same entanglement-related observables indicative of quantum chaos as in the main text, starting from several randomly drawn initial product states. The EGRD, i.e., the (normalized) distribution of the gap ratios P̅(r) with r defined after Eq. (<ref>), is computed similarly as in the main text: we consider ten randomly chosen initial states, and combine the gap ratios that are separately computed for each symmetry sector of the respective reduced states and for all initial states. Additionally, we combine the distribution of gap ratios in three time regimes (I), (II), and (III). The result is shown fig:gap-ratio-theory-analysis(a). Further, the mean of the gap ratio, combining initial states and symmetry sectors but not times, ⟨ r ⟩, is shown in fig:gap-ratio-theory-analysis(b) as a function of scaled time gt. 
The distribution in fig:gap-ratio-theory-analysis(b) is peaked near zero first, closely resembling a Poisson distribution. In this regime, the rank ℛ_A≪d_A, where d_A is the Hilbert-space dimension of the subsystem. The effective rank ℛ_A is defined after Eq. (<ref>) and plotted for the present example in fig:gap-ratio-theory-analysis(c) starting from a product state with ℛ_A=1 at t=0. The largest ξ_λ values in this regime correspond to extremely small probabilities p_λ = exp(-ξ_λ), close to or at the level of machine precision. A regularization of the smallest p_λ is required: we manually cut off probabilities below 10^-15 before computing ℛ_A, varying this limit by two orders of magnitude in each direction to provide the blue bands in panel fig:gap-ratio-theory-analysis(c), then using only the ℛ_A lowest levels in the analysis of the EGRD and the ESFF. Gray bands represent the variance with respect to the randomly chosen initial states. In regime (II), ⟨ r ⟩ quickly jumps to a value associated with a Gaussian Orthogonal Ensemble (GOE), until continuing, albeit at much slower pace, towards a  Gaussian Unitary Ensemble (GUE) indicating quantum chaotic behavior. At the end of this stage, ℛ_A= d_A is maximal, and all probabilities p_λ are well above machine precision, eliminating the need for any regularization. Finally,  regime (III) sees the saturation of the gap ratio to GUE level statistics (up to minimal finite lattice-size effects). The beginning and end of the time regimes (I-III) depend on the subsystem size and are chosen as follows, (I): 0 ≤ gt <1.8, (II): 1.8 ≤ gt <5.0, and (III): 5.0 ≤ gt<10.0.





An analogous picture is evident in the time evolution of the ESFF defined in eq:defESFF, and shown in Figs. <ref> (d) and (e). Panel (e) shows the ESFF for various times 0 ≤ gt ≤ 10, starting from a flat distribution at earliest time and showing a plateau-ramp structure at late times. Similarly as in the main text, the ESFF is computed separately for every symmetry sector of the EH to avoid contamination from uncorrelated levels in different sectors, and then averaged over these sectors and over the randomly chosen initial states. Panel (d) shows the time average over regime (III). In this regime, the ramp-plateau is evident. A gray band marks the statistical deviation from the initial-state and symmetry-sector averages. We fit the ramp to the form ∼θ^0.6± 0.1 which is consistent with the experimental data shown in the main text for a much smaller system. The fit error is determined by varying the fit range.

Because the late-time behavior, i.e., g t≫ 1, is inaccessible to current Trotter-based digital quantum simulation, we extend our analysis towards this regime, aiming to elucidate the potential outcomes that future quantum simulators might uncover. A similar analysis has been performed previously in Ref. <cit.>. At late times, it is expected that the entanglement entropy of a subsystem becomes equal to the thermal entropy contained in that subsystem, corresponding to a global Gibbs state with a temperature corresponding to the average energy of the initial state. In fig:gap-ratio-theory-analysis (f), we plot the von Neumann entanglement entropy, for a variety of randomly chosen initial states. The individual curves are color coded relative to their initial average energy density ∼ E_0 = ⟨ψ(0)| H | ψ(0) ⟩, normalized relative to the energy bandwith Δ E,  i.e., the difference between the highest and lowest eigenvalues of the EH. Displayed are only initial states whose energies lie within the interquartile range, representing the central 50% energy spectrum of states. States highlighted in yellow denote the highest energies, while those in blue represent the lowest. We were not able to extend this study to even later times to observe the expected saturation of the von Neumann entanglement entropy, owing to the fact that eventually finite-size effects become large. These results demonstrate that entanglement-entropy saturation is not a practical measure of thermalization in present experiments, demanding long evolution times and exhibiting lack of initial-state insensitivity (i.e., universality).

Shown in fig:gap-ratio-theory-analysis(g) is a separation of the von Neumann entanglement entropy, S_vN, into components, S_vN,s, related to the symmetry structure of  ρ_A = ⊕_s ρ_A,s,


    S_vN = -∑_s p_s log(p_s) + ∑_s p_s S_vN,s,


where the first term is the symmetry component, and the second is the distillable entanglement <cit.>. Here, p_s≡Tr[ρ_A,s]≤ 1 with ∑_s p_s=1, and S_vN,s≡ - Tr[ρ̅_A,slog(ρ̅_A,s)] is the sector-wise entanglement entropy with ρ̅_A,s≡ρ_A,s / p_s. Dashed lines in the plot denote the distillable component, while dotted lines denote the symmetry part. It is evident that both components saturate on different time scales: the symmetry component saturates fairly quickly (albeit still later than the build-up of level repulsion) to its maximal value log(1/p_s) where p_s =1/4 corresponds to equal mixing of the four symmetry sectors of ρ_A. In contrast, the distillable components dominates the late-time behavior of the von Neumann entropy.




While experimental constraints prevent us from accessing late times, fig:vonNeumanExpt illustrates the von Neumann entanglement entropy derived from our experimental data, utilizing the same dataset as in the main text. In this figure, exact classically computed results are denoted by black lines, while cyan crosses represent the optimal BW-inspired parameterization assuming infinite measurements, and orange circles are the experimental data points. The right-hand side of the figure displays two panels: the top panel depicts the symmetry component, while the bottom panel depicts the distillable component. Notably, the observed behavior closely mirrors the (classically computed) findings of the much larger system in fig:gap-ratio-theory-analysis(g). Specifically, the symmetry component saturates at its maximum value of log(4), while the distillable entanglement demonstrates continued growth.  Generally,  the entanglement measured in our experiment via BW-inspired tomography overshoots the exact result. We attribute this discrepancy primarily to  over- or under-rotations within the single-qubit random-bases changes. 






§ ENTANGLEMENT-HAMILTONIAN TOMOGRAPHY


In this section, we outline the randomized-measurement and analysis strategy behind extracting approximate EHs from experiment. We also systematically investigate the performance of the approach with regard to the number of bases sampled, the number of shots performed, and the influence of device errors.




Our tomography protocol consists of two parts: randomizing the state via single-qubit random unitaries drawn from a circular unitary ensemble, followed by classical post-analysis of the extracted measurements to determine an EH that best describes all observables. We use single-qubit random circuits, 𝒰≡⊗_i u_i where u_i is the following single-qubit unitary 



width=0.36

u_i ^     

=

    R^z(γ^1_i)     R^y(γ^2_i)      R^z(γ^3_i)       ,




and for each qubit, the angles γ^1_i, γ^2_i, γ^3_i are drawn according to a circular unitary ensemble (an overall phase is ignored) following Refs. <cit.>. A drawback of this approach is that it does not maintain the symmetry structure of ρ_A, as it randomizes over the whole Hilbert space instead of each symmetry block of ρ_A. Such a symmetry-ignorant randomization mixes different symmetry sectors, resulting in an outcome that is inaccessible to any physical time-evolved quantum state. We choose it, nonetheless, to avoid the larger circuit depth associated with a symmetry-conscious randomization such as that proposed in Ref. <cit.>. We note that a symmetry-conscious scheme would significantly reduce the measurement cost, as the sampling cost would only scale as the size of the symmetry block instead of the Hilbert-space size. It would thus simplify the classical optimization significantly. However, because the single-qubit scheme is tomographically complete, one can still reconstruct, approximately, the symmetry structure from the data.

The ansatz for the EH is a sum of local operators 𝒪_i,


    H_A (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i ,


which are either the physical-Hamiltonian terms or are chosen iteratively from commutators of those terms. β_i are real parameters to be constrained from data. While the BW theorem hints at an optimal choice of operators 𝒪_i for ground states, our ansatz is heuristic, as a systematic ansatz for non-equilibrium states is not known. Our criteria for the operators are that i) they are local, involving operators with support on at most two neighboring plaquettes, ii) they are compatible with the symmetries of the reduced state ρ_A (which will be discussed below), and iii) they are independent, meaning they cannot be transformed into each other through Gauss laws when acting on a physical state (although they are certainly not independent in the algebraic sense).




The operators we use are pictorially represented in Figs. <ref> plus all operators that are generated from those depicted upon translation. Here for the sake of generality, we represent them in terms of Z_2 variables, albeit, in practice, they are represented in the dual formulation in our algorithm. Figure <ref> shows a special operator; this operator (and related examples) relates operators within the subsystem to operators in the complement via Gauss laws. Further, it commutes with ρ_A, and is hence connected to the symmetry blocks of ρ_A. All terms in Figs. <ref> are obtained by considering the physical-Hamiltonian operators and by recursively commuting the physical-Hamiltonian operators, stopping at two recursions (i.e., commutators of commutators). Other selection strategies are also feasible.







Given the ansatz and following Ref. <cit.>, we then minimize the functional


    ⟨∑_b[P_𝒰(b) - Tr[ 𝒰^† |b ⟩⟨ b|  𝒰ρ_A(t ; {β_i}) ] ]^2 ⟩_𝒰


where P_𝒰(b) is the probability to measure a bitstring b in the basis determined by 𝒰 and


    ρ_A(t ; {β_i}) ≡ e^-H_A (t ; {β_i} )/Tr[e^-H_A (t ; {β_i} )] .


is the normalized reduced density matrix parameterized by H_A.
Here, ⟨·⟩_𝒰 is the average over random circuits.
Note that optimizing eq:optimziaton effectively weighs more favorably the largest Schmidt eigenvalues of ρ_A because they, on average, contribute the most to any random observable. Because of this, the optimization more accurately reproduces the low-energy part of H_A. 
The optimization is performed using Matlab's <cit.> non-linear fmincon optimization package with the `sqp' algorithm <cit.>. Convergence and uniqueness of the obtained minimum have been cross checked for several data sets using Matlab's GlobalSearch routine with default parameters <cit.>. All EH couplings are confined to the range β_i ∈ [-50,50]. We check explicitly that the routine does not come close to the boundary of the parameter regime. The EH that is obtained in this way is projected into symmetry sectors, which we then separately analyze. For this projection, the symmetry sector can be read off directly from the row and column numbers of the EH when they are interpreted as binary. The corresponding subblock H_ E,s is then selected for further analysis, where s labels one symmetry sector. For the EGRD analysis, we analyze the gap-ratio distribution separately for every sector, then combine the distributions and re-normalize the total distribution. For the ESFF analysis, we average over the sectors. Finally, EGRD and ESFF shown in the main text and appendices involve a combination/average over initial states. The error band and the error bars in the plots are the standard deviation for the symmetry-sector and initial-state averages. 

An example of the eigenvalue spectrum of the EH, comparing the exact values with those obtained from our procedure, is shown in fig:overviewES. Panel (a) shows the reconstructed entanglement spectrum for several times, comparing exact data (black lines), infinite-measurement results (cyan crosses), emulator data (blue squares), and quantum-computed data (orange circles), for one given initial state. Panel (b) shows the symmetry-resolved spectrum for gt=1.02. Here, s  are the symmetry sectors of ρ_A, see the discussion at the end of Sec. <ref> in the main text. While our analysis effectively describes the low-lying part of the entanglement spectrum, it fails to quantitatively reproduce the higher part associated with very small probabilities. Additionally, the reconstructed spectrum appears more mixed (more entangled) compared to e.g., the emulator data. This discrepancy primarily is due to finite-measurement statistics and device errors. Comparison with the infinite-(ideal-)measurement results also reveals that the BW-inspired parameterization falls short in capturing even quantitatively higher-lying components of the spectrum. It is noteworthy that the limitations of the parametrization become more pronounced at later times where the Trotter-step size is large and even the low-lying part deviates. This is because the effective Trotter Hamiltonian at late times deviates from the known target Hamiltonian from which the BW-inspired ansatz is derived.

In fig:BWgapratiobases, we study the accuracy of our tomography scheme concerning the number of random bases. Each basis is probed with N_ shots=750 samples, employing emulated data to eliminate the effect of device errors. Applying N_𝒰=4 to N_𝒰=128 random bases shows convergence in the lower-lying range of the entanglement spectrum relatively fast. However, the higher-lying part is not reconstructed even with a large sample size. The top panel displays the average EGRD over time, showing consistent behavior across the sample range. Error bars indicate the spread from combining initial states and symmetry sectors, which decrease with increasing N_𝒰. Concretely the error bars are obtained by computing the mean of the gap ratios for every sector and initial states separately, and then computing the standard deviation. 

Finally in fig:BWgapratioshots, we investigate the dependence on the number of measurement of probabilities P_𝒰(s) in each basis (i.e., the number of shots), maintaining a constant number of measurement bases, N_U=24. Once more, convergence is evident in the lower segment of the ES in the lower panel, while the higher portion remains beyond reach even with infinite shots (red octagons). The top panel illustrates the EGRD, indicating consistency, within error bars, with the optimal BW-inspired ansatz result (cyan band).

In principle, given that the scheme is tomographically complete, we anticipate the ability to precisely reconstruct the entire ES with exponential resources. However in practice, we were not able to do so because of the significant bias of the cost function eq:optimziaton towards the low-energy portion of the entanglement spectrum. This bias places considerable strain on the numerical minimization routine, pushing it beyond its numerical-accuracy threshold. Despite this, its accuracy is satisfactory for analyzing experimental data, and we abstained from further optimization attempts. Exploring advanced optimization routines, including machine-learning techniques <cit.>, holds promise for EH tomography in future studies.



 Paper Title: SU(2) gauge theory with one and two adjoint fermions towards the continuum limit.























a.athenodorou@cyi.ac.cy
Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus


e.j.bennett@swansea.ac.uk
Swansea Academy of Advanced Computing, Swansea University, Fabian Way, Swansea SA1 8EN, UK


georg.bergner@uni-jena.de
University of Jena, Institute for Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena, Germany


pbutti@unizar.es
Departamento de Física Teórica, Facultad de Ciencias and Centro de Astropartículas y Física de Altas Energías (CAPA),
Universidad de Zaragoza, Calle Pedro Cerbuna 12, E-50009, Zaragoza, Spain
Instituto de Física Teórica UAM-CSIC, Nicolás Cabrera 13–15, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain


j.lenz@hzdr.de
Swansea Academy of Advanced Computing, Swansea University, Fabian Way, Swansea SA1 8EN, UK
Helmholtz Zentrum Dresden Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany


b.lucini@swansea.ac.uk
Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, UK










§ ABSTRACT 
We provide an extended lattice study of the 2 gauge theory coupled to one Dirac fermion flavour (=1) transforming in the adjoint representation as the continuum limit is approached. This investigation is supplemented by numerical results obtained for the 2 gauge theory with two Dirac fermion flavours (=2) transforming in the adjoint representation, for which we perform numerical investigations at a single lattice spacing value, which is analysed together with earlier calculations. The purpose of our study is to advance the characterisation of the infrared properties of both theories, which previous investigations have concluded to be in the conformal window. For both, we determine the mass spectrum and the anomalous dimension of the fermion condensate using finite-size hyperscaling of the spectrum, mode number analysis of the Dirac operator (for which we improve on our previous proposal) and the ratio of masses of the lightest spin-2 particle over the lightest scalar. All methods provide a consistent picture, with the anomalous dimension of the condensate γ_* decreasing significantly as one approaches the continuum limit for the  = 1 theory towards a value consistent with γ_* =, while for = 2 the anomalous dimension decreases more slowly with β. A chiral perturbation theory analysis show that the infrared behaviour of both theories is incompatible with the breaking of chiral symmetry. 



11.15.Ha 12.60.Nz




2 gauge theory with one and two adjoint fermions towards the continuum limit
    Biagio Lucini
    August 5, 2024
============================================================================










§ INTRODUCTION


Large anomalous dimensions have been advocated in beyond the Standard Model strong dynamics to reconcile the proposed fundamental mechanism of electroweak symmetry breaking with experimental observations. A primary example is provided by walking technicolor, whereby a large anomalous dimension of the chiral condensate would explain the suppression of flavour-changing neutral currents <cit.>. An additional case in point is partial top compositeness in a theory in which the Standard Model Higgs boson is originated by the global symmetry breaking in a novel strong interaction <cit.>. In the case of such a Composite Higgs model, a large anomalous dimension of the chimera baryon (a baryon-like composite state containing fermions in the fundamental and in a two-index representation) is needed to enhance the mass of the top quark to the electroweak scale (see., e.g., <cit.>). Large anomalous dimensions are expected to appear in non-Abelian gauge theories near the onset of the conformal window, a phase in which the theory is approximately scale-invariant in the infrared. The opening of the conformal window is determined by the structure and the size of the gauge group, and by the fermion content of the model (specifically, the gauge group representations under which the fermion matter fields transform and the number of fermion flavours in each representation). The onset of the conformal window being a strong coupling problem, analytically only semi-quantitative estimates are possible (see, e.g., <cit.>).

As for other non-perturbative questions in non-Abelian Quantum Field Theories, numerical simulations of the relevant model(s) discretised on a spacetime lattice provide in principle a robust framework to investigate the infrared conformal regime and its onset, enabling us to access crucial quantities such as the value of the anomalous dimension itself. Triggered by the experimental activity that led to the discovery and the characterisation of the Higgs boson <cit.>, the resurgent interest in understanding whether strongly interacting systems beyond the Standard Model can provide a viable fundamental mechanism of electroweak symmetry breaking has spawned a significant number of lattice studies aimed at unveiling the phase diagram of gauge theories for given group families and fermion representations in terms of the size of the gauge group and the number of flavours in each representation (see, e.g.,  Refs. <cit.> and  Refs. <cit.> for a necessarily incomplete set of investigations targeting the problem of near-conformality respectively in the context of walking technicolour and top partial compositeness). To this purpose, a number of methods aimed at determining with sufficient accuracy the anomalous dimension of relevant operators or the (scheme-dependent) coupling at which the theory is infrared conformal have been proposed and studied. Among them, popular approaches include hyperscaling, dilaton perturbation theory, scaling of the eigenvalues of the Dirac operator, Schrödinger functional, and the gradient flow. Some relevant references to this body of work include <cit.>, with relevant reviews provided for instance in <cit.>. 

While strongly interacting theories beyond the Standard Model provide a clear-cut phenomenological motivation to study near-conformality, the underlying question of the possible phase structure of gauge theories is more fundamental and wider-reaching. The existence of an asymptotically free confined phase and of a Coulomb phase can be inferred directly from the sign of the β-function as a function of the gauge group size and structure, of the number of flavours and of the representation of the gauge group under which the latter transforms, including cases in which fermions transforming in different representations are present. The higher end of the conformal window, before the loss of asymptotic freedom, can be exposed in perturbation theory, giving rise to the well-known Banks-Zaks infrared fixed point <cit.>. Outside perturbation theory, the conformal window is well understood in the presence of a high degree of supersymmetry <cit.>, which imposes strong constraints to the β-function. In non-supersymmetric quantum field theory, lattice calculations remain the most robust method to investigate the lower end of the conformal window, in which strong coupling plays a crucial role.

In this paper, we investigate two well-known gauge models, both based on gauge group 2, coupled respectively with one and two fermions transforming in the adjoint representation of the gauge group, with the aim of providing further insides on their infrared behaviour and investigate quantitatively the value of the anomalous dimension in the continuum limit. The system with number of fermionic flavours = 2, whose physical relevance relies on the observation that it could provide a minimal template for a realisation of walking technicolour,  has been studied systematically for well over a decade on the lattice, using a wide array of methods. The consensus reached is that the system is infrared conformal with a small anomalous dimension, which exclude it from the class of phenomenologically relevant models. However, a robust determination of the anomalous dimension is still lacking. Tensions in the obtained data are shown for instance in <cit.>.

The system with = 1  has been considered in the context of semiclassical analysis <cit.> and volume independence, leading to the conjecture of emergent fermion symmetry also in the non-supersymmetric case <cit.>. A numerical study of this system was  first provided by some of us in Ref. <cit.>,  where it was shown that, despite common lore, the system is not in the expected confining and chiral symmetry breaking phase; moreover, albeit at fixed lattice spacing, this study was the first to determine an order one anomalous dimension for the chiral condensate, which would fulfill one of the requirements of walking technicolour. Subsequent analytic work based on 't Hooft anomaly matching <cit.> showed that the model has an interesting phase structure in which topology plays a key role. An interesting observation made in <cit.> is the emergence of a light composite fermion in the spectrum. Subsequent numerical studies <cit.> have confirmed that the system does not have the conventional properties of a confining and chiral symmetry breaking theory. However, while more likely, the near-conformal scenario has not proved to provide a striking fit to the data. 
Additionally, the expected light fermion in the spectrum has not been observed, and the anomalous dimension has been found to have a strong dependency on the lattice spacing. 
In contrast to earlier results, a recent study based on an overlap lattice action <cit.> favours rather a chiral symmetry breaking scenario. This lattice formulation provides an improved representation of the continuum symmetries, but so far the considered lattices have been rather coarse. A similar result might be deduced from the study of a mixed adjoint-fundamental theory <cit.> with Wilson clover fermions, but again the parameter range has been very limited. Finally, the system with N_f=1 includes only two Goldstone bosons resulting from the breaking of chiral symmetry. This makes it phenomenologically unviable to account for the standard model electroweak symmetry breaking. However, the system can be extended to a physically viable setup through the ultra minimal walking technicolor (UMWT) model <cit.>, which also includes fermions in the fundamental representation.

The current status of understanding of these two models calls for additional simulations on larger lattices, closer to the continuum and to the chiral limit. In alignment with this demand, the purpose of our work is to extend the numerical results of previous investigations in order to provide a more solid understanding of the infrared dynamics of the target models. Therefore, we provide a range of new numerical results for 2 gauge theories with = 1 and = 2 adjoint fermion flavours at larger couplings, larger lattices and smaller masses. For continuity with our previous studies, we employ the Wilson action for the gauge part of the models, and we use the standard Wilson discretisation of fermions. This simple discretization allows us achieve very large lattice sizes at small fermion masses and combine a large set of data in our final analysis that include also those produced in previous investigations. We aim to get to the maximum point that can be achieved with our current approach. At the same time we want to investigate to what extend the same uncertainties that we have observed for =1 also appear in case of the seemingly well investigated =2 theory. The main tools we employ for the characterisation of the model are the behaviour of the chiral condensate, the scaling of the mass spectrum and the scaling of the eigenvalues of the Dirac operator. 

The rest of the paper is organised as follows. In Sec. <ref> we provide a quick overview of the continuum theories of interest and their lattice counterpart employed in this work. Sec. <ref> is dedicated to a discussion of the methodology and the extraction of the main results, with different physical observables of interest discussed in different subsections. Conclusions can be found in Sec. <ref>. Preliminary results of our study have already appeared in <cit.>. 




§ SUMMARY OF THE CONTINUUM AND LATTICE THEORY



The Lagrangian of the 2 gauge theory with  adjoint Dirac flavours in Minkowski space, is defined as:

    ℒ =    - 1/2Tr( G_μν(x) G^μν (x) ) 
       + ∑_α=1^ψ_α (x) ( i D - m ) ψ_α(x) ,

with D = (∂_μ + i g A_μ(x) )
	γ^μ, γ_μ being the Dirac matrices, A_μ(x) =
	∑_a T^a A_μ^a(x) with a = 1,2,3, and the T_a the Lie algebra
	generators of 2 in the adjoint representation. The field strength tensor is defined as G_μν = ∂ _μ A_ν(x) - ∂
	_ν A_μ(x) + i g [A_μ(x), A_ν(x)], with g being the
	gauge coupling of the theory, and the trace is taken
	over the gauge group. Notation and conventions are explained in Ref. <cit.>.

We may re-express each Dirac flavour ψ_α(x) in terms of two Majorana flavours ξ_α±(x), by making use of the projections

    = ψ_α + C ψ_α^/√(2) ,
         = ψ_α - C ψ_α^/√(2)i ,

such that

    ψ_α =1/√(2)(  + i )  .


Eq. (<ref>) may then be re-expressed as

    ℒ =    - 1/2Tr( G_μν(x) G^μν (x) ) 
       + 1/2∑_α =1^∑_k∈{+,-}_α k(x) ( i D - m ) ξ_α k(x) ,

exposing the 2 chiral symmetry in the action. The adjoint representation being real, the expected chiral symmetry breaking pattern is 

    2↦2  .


In this work, we consider the cases =1,2.
We use the same computational setup as our previous work <cit.>: we generate dynamical gauge configurations using the standard Wilson plaquette action and the standard Wilson fermion action

    S=S_G + S_F ,

where 

    S_G = β∑_p Tr[ 1 - 1/2U(p)  ]  ,


    S_F =  ∑_x,yψ(x) D(x,y)ψ(y) ,

 is the number of flavours, U(p) is the traced parallel transport of the links U_μ(x) around a plaquette p, and D(x,y) is the Wilson Dirac operator

    D(x,y) = δ_x,y 
      - κ   [ (1-γ_μ) U_μ (x) δ_y,x+μ.
       +  . (1+γ_μ) U^†_μ (x-μ) δ_y,x-μ] .

For full details we refer the reader to Ref. <cit.>.
This generation is performed using importance sampling methods: the Hybrid Monte Carlo (HMC) <cit.> algorithm for =2, and RHMC <cit.> for =1. A change from our previous work is the software used to perform these algorithms: while for heavier ensembles we continue to make use of HiRep <cit.>, for lighter ensembles we use Grid <cit.>, which enables significantly more powerful GPU-based resources to be used.[In each case minor modifications were made to the base code referenced above. In the case of HiRep, these are available in the  branch of the fork at <https://github.com/edbennett/HiRep>, commit https://github.com/edbennett/HiRep/tree/92bbe4d25feb80ce645bb9225d8bd04b1d67347a. In the case of Grid, they are in the  branch of the fork at <https://github.com/edbennett/Grid_epcc>, commit https://github.com/edbennett/Grid_epcc/tree/48ca3a244d3f6ce0caa6510096778f07d80588b3.] Computation of specific observables using these configurations is discussed in the relevant subsections of the next section.




§ RESULTS










In the following sections we provide an overview of the main results obtained at a wide range of lattice volumes, masses, and coupling constants, as summarised in Tables <ref>, <ref>, and <ref>. An investigation explaining why we have chosen these parameters to avoid the bulk phase and retrieve the desired continuum physics can be found in our previous investigations <cit.>. In this work the parameter range is significantly larger than in any previous work. In fact, in this work we have pushed the boundary of computational  feasibility with the chosen approach.

Our main goal is to include in particular the dependence on the gauge coupling β. The large parameter space requires first some general considerations to avoid a breaking of center symmetry indicated by the Polyakov loop or topological freezing. Afterwards we present the results for the scaling of the particle spectrum and the mass anomalous dimension.




 §.§ Expectation value of Polyakov loops and center symmetry


On a finite lattice with compact directions, when the direction μ is large enough, the model is characterised by a linear realisation of the four center symmetries 

    ℤ_μ(2): U_μ(x) →(1 - 2 δ_x_μ, x̂_μ) U_μ(x)  ,

which multiplies by z = -1 all the links U_μ(x) in the lattice slice identified by the fixed μ-th coordinate x_μ = x̂_μ. Here, x̂_μ is a conventionally fixed value. 

The ℤ_μ(2) symmetries can be spontaneously broken at small volumes, signaling severe finite-size effects.  
To verify that the finite lattice extent has not caused the centre symmetry of the group to be broken, we compute the expectation value of the Polyakov loop

    P_μ = 1/N∑_x_⊥Tr(∏_i=0^L_μ-1 U_μ(x_⊥,x_i)) .

A broken centre symmetry would be indicated by two peaks in the histogram of this observable, while the preserved symmetry shows a single peak centered at ⟨ P_μ⟩=0. We show a sample of such histograms in Fig. <ref>. In all cases we observe this symmetry to be unbroken.


	







 §.§ Topological aspects


The formal definition of the topological charge <cit.> for a gauge field is given by the four-dimensional Euclidean integral of the topological density across space-time:

    Q = 1/32π^2∫ d^4 x  ϵ_μνρσ[G_μν(x)G_ρσ(x)]  .

In practical terms, any appropriate lattice discretization of the topological charge density that results in the correct continuum expression can be employed to assess the lattice equivalence. Using the standard clover representation of the topological charge density, Eq. (<ref>) takes the summation form:

    Q_L = 1/32π^2∑_xϵ_μνρσ[C_μν(x)C_ρσ(x)] ,
 
where C_μν(x) is the standard cloverleaf discretization of the gauge field tensor G_μν(x):

    C_μν(x)   =  Im/4[
    U_μ(x)U_ν(x+ a μ̂)U_μ^†(x+a ν̂)U_ν^†(x) 
        +   
    U_ν(x)U^†_μ(x-aμ̂+aν̂)U_ν^†(x-a μ̂)U_μ(x-a μ̂) 
        +    U^†_μ(x-a μ̂)U^†_ν(x-a μ̂-a ν̂)
        U_μ(x-a μ̂-a ν̂)U_ν(x-a ν̂) 
        +   
    U^†_ν(x-a ν̂)U_μ(x-a ν̂)U_ν(x+a μ̂-a ν̂)U^†_μ(x)
    ] .


An issue lattice simulations of gauge theories face is the development of significant correlations among topological sectors at reaching the continuum limit. This phenomenon, known as topological freezing, occurs because the probability distribution for the creation of instantons and anti-instantons from the vacuum is suppressed as the continuum limit is approached <cit.>. Consequently, the total topological charge cannot fluctuate adequately, causing the Markov chain to become trapped in specific topological sectors. Consequently, the simulation might fail to thoroughly explore every topological sector efficiently and thus be ergodic. In practice, as the continuum limit is approached, pronounced correlations emerge and contribute to an increment of the necessary number of configurations needed to attain statistically significant vacuum expectation values for physical observables.



This issue is a common obstacle in all Markov-chain algorithms where updates are performed locally.  To investigate whether this phenomenon occurs for the values of β chosen in our work, we have closely observed the autocorrelation time of the topological charge in our simulation runs.


Monte Carlo assessments of the topological charge Q encounter challenges due to ultraviolet fluctuations that obscure the inherent topological structure. To address this issue, smoothing techniques like gradient flow <cit.> can be applied to mitigate these fluctuations.


Regarding our simulations for =1, in Fig. <ref>, we illustrate the Monte Carlo evolution of the topological charge Q for the ensembles DB4M13, DB5M8, DB6M9, and DB7M10 along with their respective histograms. For DB4M13 DB6M9, and DB7M10 we provide histories for three different threads of configuration production.

These above four sets of configurations correspond to the smallest fermion masses at the four largest values of β. It is noticeable that none of the ensembles exhibit an exceptionally long autocorrelation time for the topological charge Q, indicating that topological freezing is not a concern within the specified parameter range. The simulations explore a substantial number of topological sectors, and the resulting histograms align well with a Gaussian distribution centered at zero. This strongly suggests an effective sampling of topological observables.

Turning now to the case of =2, in Fig. <ref> we present the Monte Carlo histories of the topological charge Q for the ensembles Nf2DB2M6 and Nf2DB2M7 along with their corresponding histograms. Once more, these two sets of configurations refer to the smallest masses as well as to the largest possible volumes. Clearly, and in contrast to the case of =1, the topological charge appears to fluctuate infrequently indicating large auto-correlation times and suppressed ergodicity. As a result, Q probes only a few topological sectors. Bearing in mind that the sampling is not ideal as we move towards smaller masses for =2 and β=2.35 we should be cautious of any low-ergodicity effects contaminating our measurements.
Detailed findings concerning the autocorrelation time and the fitted form of the Q histogram are provided in the data release associated with this work <cit.>.

In addition to verifying the reliability of the simulations, this study also offers additional insights into the physical properties of the theory. The topological susceptibility χ defined as
    
    χ = ⟨ Q^2 ⟩  -  ⟨ Q ⟩^2 /V ,

provides information regarding the potential infrared conformal scenario for the theory. When an infrared (IR) conformal theory is deformed by a fermion mass, it transitions to a confining phase. In this case, the theory should behave similarly to the equivalent quenched theory. A suitable quantity to verify this scenario is the topological susceptibility, as it should align with the results of pure SU(2) Yang–Mills theory <cit.>.


In Fig. <ref>, we display the topological susceptibility in units of the string tension, χ^1/4/√(σ), as a function of the string tension in lattice units, a^2 σ for both SU(2) with =1 and =2 adjoint fermions. The results are compared to the corresponding quantity in pure 2 Yang–Mills theory for the quenched = 0 (am →∞) case. The topological susceptibility for =1 and β≠ 2.4, for =2 excluding the very first two points for β=2.35 show good agreement among these three theories, with any discrepancies attributed to higher-order scaling corrections of a^2 σ. Clearly, for =1 and β = 2.4, the topological susceptibility exhibits large negative deviations as we approach a^2 σ→ 0, which are understood as underestimated values attributed to critical slowing down. Similarly, the same phenomenon occurs for =2 at β=2.35 for the two smallest values of a^2 σ. 



In addition, the gradient flow facilitates the establishment of scale parameters, namely, t_0 and w_0, which can be determined with a high degree of precision using the following prescription: first, set ℰ(t) as
        
    ℰ(t) = t^2 ⟨ E(t) ⟩   where E(t) = 1/4 B^2_μν (t) .

In this expression B_μν represents the field strength derived from flowing G_μν. We designate the scale t_0 as the specific value of t at which the following identification occurs:
    
    ℰ(t) |_t=t_0(c) = ℰ_0 .

In a similar manner we define w_0 in the following way:
     
    t d/dtℰ(t) |_t=w^2_0(c) = 𝒲_0 .


In the above definitions, the parameters ℰ_0 and 𝒲_0 are selected to ensure that the pertinent condition, either a ≪√(8t_0)≪ L or a ≪√(8w_0)≪ L, is fulfilled. Choosing a small value of c tends to result in larger lattice artifacts, while larger c typically leads to increased autocorrelations <cit.>. In our specific scenario, we set c to 0.2, aligning with the common choice of c = 0.3 in QCD, assuming a scaling with N as described in Ref. <cit.>. G_μν is computed via the clover plaquette C_μν.

    







	


    


 §.§ Glueball Masses


Glueball masses can be determined by employing the standard decomposition method on a Euclidean correlator that incorporates an operator referred to as ϕ(t) <cit.>. This decomposition procedure hinges on expressing these physical states in the framework of the Hamiltonian, designated as H, and its corresponding energy eigenstates:

    ⟨ϕ^†(t=an_t)ϕ(0) ⟩    =  ⟨ϕ^† e^-Han_tϕ⟩
        = ∑_i |c_i|^2 e^-aE_in_t
       t→∞=
    |c_0|^2 e^-aE_0n_t .

In the above context, E_0 signifies the ground state energy. The summation mentioned is constrained to states with non-zero overlaps, satisfying the condition c_i = ⟨ vac | ϕ^† | i ⟩≠ 0. The quantum characteristics of the operator ϕ must align with those of the specific state under examination; i.e., glueball states. Identifying the ground state relies on two crucial factors: the strength of its correlation with the state created by ϕ and the rate of exponential decay, as indicated in Eq. (<ref>). Improving this correlation involves devising operators that effectively capture the essential traits of the state. To enhance the overlap onto the states under investigation, the GEVP technique <cit.> is employed on a set of operators ϕ_i derived from various lattice loops at different blocking levels <cit.>. This process utilizes correlation matrices, denoted as C_ij = ⟨ϕ_i^† (t) ϕ_j (0) ⟩, where i,j=1,...,N_ op, in conjunction with GEVP. Here, N_ op represents the number of operators used.

To formulate an operator projecting onto a glueball state, we generate an ordered product of 2 link matrices along a loop that allows continuous contraction, followed by the calculation of its trace. The real (imaginary) part of this trace corresponds to positive (negative) charge conjugation C=+(-). To ensure the operator possesses zero momentum, we sum over all spatial translations of the loop. Additionally, we consider all possible rotations of the loop, combining them in manners consistent with the irreducible representations (ℛ) of the rotational symmetry group. For the creation of operators with both parities (P=±), we construct the parity inverse for each loop and then combine them in appropriate linear combinations. In Fig. <ref>, we present a selection of the paths utilized in building our basis.


The distinct representations R of the discrete subgroup of cubic rotations within the complete rotation group are labeled as A_1, A_2, E, T_1, T_2. The A_1 representation is a singlet and exhibits complete cubic rotational symmetry, encompassing the J=0 state in the continuum limit. Similarly, the A_2 representation is also a singlet. The E representation constitutes a doublet, while both T_1 and T_2 representations are triplets. In the lattice context, the three states corresponding to the triplet of T_1 and T_2 are degenerate. To address this, their values are averaged, treating them as a single state when estimating glueball masses. The same approach is applied to the E doublets, where their mass estimates are averaged.

The rotational symmetry representations outlined above are rooted in our cubic lattice formulation. As we approach the continuum limit, these states will converge to continuum glueball states falling into representations of continuous rotational symmetry. Consequently, they will form degenerate multiplets comprising 2J + 1 states, where J denotes the spin of the states. When determining the continuum limit of the low-lying glueball spectrum, it is more informative to assign states to a specific spin J instead of representations of the cubic subgroup. The latter provides a less precise “resolution," as it maps all spins J = 1, 2, 3, …, ∞ to just 5 cubic representations. For low values of J (J=0,1,2), the distribution of the 2J + 1 states can be characterized as A_1 → J=0, T_1 → J=1, and E, T_2 → J=2.



 §.§ Torelon masses and the string tension


Similar to the glueball mass calculation, we can determine the mass of the closed spatial Polyakov loop, also known as a torelon, which wraps around a compactified spatial direction. Torelon states are characterized by quantum numbers such as angular momentum-spin J, transverse parity P_⊥, and longitudinal parity P_||. The ground state, with J^P_⊥, P_||=0^++, allows for the easiest extraction of the string tension <cit.>. 

The mass of the torelon is determined by computing the ground state energy m_T(L) of a flux-tube with length L=N_s, which forms a closed loop by winding once around a spatial compactified torus. We utilize Eq. (<ref>), where the operator ϕ represents the product of 2 link matrices along a non-contractible closed path, which wraps around the spatial torus once. We use the most basic form of such an operator, the elementary Polyakov loop. This is the path-ordered product of link matrices in a spatial direction, winding once around the torus along the afforementioned spatial direction. Once again, we utilize smearing and blocking techniques to improve the projection onto physical states. We also sum over translations along the spatial-torus as well as over the transverse directions within the time slice to project onto zero longitudinal and transverse momentum, respectively. As all spatial lattice extents are identical, we average the correlation function over the three spatial directions.

Confining torelons, exhibit behavior resembling that of bosonic strings <cit.>. It is expected that the closed flux-tube behaves akin to a confining bosonic string, whose spectrum can be approximated with reasonable accuracy by the Nambu-Goto string model <cit.>. The ground state mass of the closed bosonic string is described by the expression:

    m_T(l) = σ l √(1 - 2 π/3 σ l^2).

Here, σ represents the string tension and l=a L indicates the length of the torelon. Therefore, we calculate the string tension across the torelon mass spectrum. Results for the string tension are shown in Fig. <ref>.





 §.§ States involving fermions




Mesons and baryons are sourced by operators of the form

    O(x) = ψ(x) Γψ(x) ,

where Γ is a product of Dirac γ matrices giving the symmetry of interest.
As discussed above,
this theory has a chiral symmetry breaking pattern
2→2,
and we choose to label the states observed in these terms.
The remaining subgroup 2 in case of =1 is equivalent to U(1) (refer to as U(1) baryon symmetry).
For convenience,
in Table <ref> we summarise this convention.

Similarly to the case for glueballs above,
correlation functions of these operators
converge at large time to an exponential decay
with the ground state mass in the exponent

    ⟨ O_X^† (t) O_X(0) ⟩t→∞= f_X· e^-m_X t ,

where m_X and f_X
are the ground state mass and decay constant of
the state sourced by O_X(x).
In a finite volume with antiperiodic boundary conditions,
this then becomes

    ⟨ O_X^† (t) O_X(0) ⟩t→∞= f_X [e^-m_X t+e^-m_X (T-t)] ,

Inverting Eq. (<ref>) defines an effective mass m_eff,
and by finding where m_eff has a plateau at large time,
we may identify the region that is free from the effect of excited states.
We then fit the correlator to Eq. (<ref>)
to obtain the mass and decay constant of the state.

Additionally,
as the bare Wilson fermion mass acquires an additive renormalisation,
we define a renormalised fermion mass—the PCAC mass—via the axial Ward identity <cit.>.

In this work we continue to use HiRep to implement 
the computation of correlation functions.
The fits are performed using the  <cit.> library,
which computes uncertainties using the Γ method <cit.>,
automatically accounting for autocorrelations in the data.

One particular difference to theories in the fundamental representation is 
the appearance of bound states formed out of gluons combined with
an adjoint fermion field.
One particular of these states is a field 
(also called hybrid fermion,
and denoted as ğ in plots),
which corresponds to the so-called gluino-glue field in 𝒩=1 supersymmetric Yang-Mills theory.
The corresponding operator is 

    O(x) = ∑_μνσ_μν[ F^μν(x)λ(x) ],

where σ_μν=1/2[ γ_μ, γ_ν], the field strength is represented by a clover plaquette and smearing is applied to improve the signal.
Further details of the techniques can be found in corresponding publications of supersymmetric Yang-Mills theory <cit.>.
This correlation function is
computed separately from the spectrum of mesons and baryons
using the FlexLatSim code <cit.>,
but analysed with the same workflow using .



We present the spectrum of masses and decay constants respectively
of the =1 theory,
including mesons, baryons, glueballs, the string tension, and the hybrid fermion,
in Figs. <ref> and <ref>,
in terms of the gradient flow scale w_0.
For the case =2,
we present the spectrum in terms of the string tension √(σ)
in Fig. <ref>;
in this case we do not use w_0,
as in the majority of ensembles
the lattice extent is insufficient to extract it.

In particular,
we draw attention to two features of the spectrum.
The first,
highlighted in Fig. <ref>,
is that,
as observed in our previous work
and in other theories in or near the conformal window,
the scalar glueball is lighter than the 2^+ scalar baryon,
the would-be pion in a QCD-like theory.
Indeed,
this effect becomes more pronounced at higher values of β.
Secondly,
as can be seen in Fig. <ref>,
in the =2 theory
the state is also lighter than the 2^+ scalar baryon.

















 §.§ Determination of the mass anomalous dimension





  §.§.§ Finite-size hyperscaling of particle spectrum


In a mass-deformed conformal theory in a finite volume, we expect the masses M of states to hyperscale with the deforming mass (which we approximate by ) and the lattice extent L as

    L a M = f (L(a)^1/1+γ_*) .


While we do not a priori know the functional form of f(L(a)), we may look for a value of γ_* that causes the data to collapse onto a single universal curve. As in Ref. <cit.>, we may make this determination more systematic by defining the curve collapse as the minimisation of the function

    P(γ_*) = 1/N_o∑_ℓ∈{L}∑_i∈ S_ℓ(L_i aM_i - f_ℓ(L_i^1+γ_* am_PCAC i))^2 ,

where {L} is the set of lattice extents, S_ℓ is the set of data having lattice extent L_i ℓ but lying such that L_i^1+γ_* is in the range spanned by data with lattice extent ℓ, N_o is the number of points summed over, and f_ℓ is an interpolating function across the data having lattice extent ℓ, which we choose to be piecewise linear. We apply this procedure for the data for each value of β in both =1 and 2.










The results of this analysis are plotted for a selection of β values considered for =1 in Fig. <ref> and for =2 in Fig. <ref>, and the resulting values of γ_* tabulated in Tabs. <ref> and <ref> respectively. As observed in our previous work <cit.>, we see the anomalous dimension in the =1 theory continue to decrease as β increases; for =2 we obtain γ_*= from the analysis of the mode number, which is only moderately smaller than the result γ_*=0.371(20) obtained previously using similar techniques <cit.> for β=2.25.




  §.§.§ γ^* from the mode number of the Dirac operator

The other method we employ to calculate the mass anomalous dimension is studying the low-lying mode number of the Dirac operator. A similar computation to extract γ^* was performed in previous publications <cit.>, to which we will refer to for additional details. However, the final analysis of the data has been improved and follows a slightly different strategy. For this reason, it is useful to review some general concepts, that will be useful in the following as they will guide the statistical analysis of the data.

The quantity of interest is the dependence of the spectral density ρ of the Dirac operator on its eigenvalues ω. In fact, in an infrared conformal gauge theory, at the IR fixed point, the leading behaviour of ρ(ω) is dictated by the renormalization group (RG) equations and proves to be power-like in ω, where the anomalous dimension γ^* appears as an exponent <cit.>. Equivalently, a more accessible quantity is the integral of the spectral density, called mode number, i.e. the number of eigenvalues of the Dirac operator lower than a certain mass threshold Ω^2 (divided by the volume). As argued in Refs. <cit.>, on the lattice, the mode number in lattice units a^-4ν̅, can be expressed as a function of the threshold aΩ as

    a^-4(Ω) ≈ a^-4_0(m) + A[(aΩ)^2-(am)^2]^2/1+γ_* .

where a^-4ν̅_0, A, am and γ^* are parameters to be fitted to the data. In Eq. (<ref>), the ≈ sign signals the fact that the equation is valid only where the dynamics is dominated by the IR fixed point, namely a region ω_IR<ω<ω_UV of low eigenvalues insensitive to the fermion mass. The parameters appearing in Eq. (<ref>) are of physical relevance and contain information that can be incorporated into the analysis of the data. Specifically:

    
  * The overall additive constant ν̅_0 represents the integral of the spectral density in the region [0,ω_IR],
    where the fermion mass drives the theory away from the IR fixed points, spoiling scale invariance. In this region, no analytical models are available for ρ, and therefore its integral is represented by a constant ν̅_0. As the fermion mass is responsible for this behaviour, this region shrinks in the chiral limit, and, as argued in Ref. <cit.>, it can be parameterized as
    
    ν̅_0(m)∝ M^4

    where M is some mass in the spectrum vanishing in the chiral limit.
    
  * The mass parameter m can be related to the bare PCAC mass as
    
    m = Z_A  .
 


The procedure for the extraction of the anomalous dimension consists of the following. First, the mode number ν̅ has to be computed through spectral projection and the Chebyshev expansion method as described in <cit.>. We have confirmed consistency between both methods and the main part of the results has been obtained with the Chebyshev expansion. This provides the mode number for several values of the threshold aΩ on a subsample of well-decorrelated configurations. Then a fit of Eq. (<ref>) to the data has to be performed to extract the value of the anomalous dimension γ^*. In doing so two main issues have to be addressed. 

The first issue is the choice of the window [aΩ_min,aΩ_max]. This range is expected to reflect the region of validity of Eq. (<ref>) and its location is not known a priori and has to be determined empirically from the data. In principle, when choosing an inappropriate window, i.e. a region where Eq. (<ref>) is not valid, the low quality of the choice of aΩ_min and aΩ_max is expected to be signalled by a bad-χ^2 fit. In practice, the usage of the simple χ^2 as an indicator of the quality of the fit is not always a good discriminator. In fact, if the window is small enough and centred on high eigenvalues, the fit converges with a good χ^2 to unphysical values of the parameters.[As pointed out in Ref. <cit.>, the spectral density has an asymptotic behaviour dominated by asymptotic freedom, given by ρ∝ω^3. If the fitting region is small enough, the fit of Eq. (<ref>) to the data can give an acceptable χ^2, leading to a vanishing value of am and a γ^* with no physical meaning.] For this reason, fits in a low-eigenvalues window are to be preferred, although a second following issue is expected to be triggered; the high degree of non-linearity of the function in Eq. (<ref>) makes the fit highly unstable, the minimization algorithm often drives towards a region in the parameter space where ((aΩ)^2 - (am)^2)<0 and leads to a breakdown of the fitting procedure. In these cases, the fit parameters can be bounded or reparameterized to avoid this problem, although, we observed that in some cases the algorithm finds other local minima, and the parameters converge to extreme (unphysical) values. 

In previous publications <cit.>, these two issues were dealt with by repeating a series of bootstrapped fit to an extensive range of possible window [aΩ_min,aΩ_max] and thus seeking by eye for a region of stability. 



Coefficients were then extracted by averaging over the windows inside the stability region, where its fit is weighted with the relative portion of converged fit in a bootstrap sample, as an indicator of stability. In this work, we adopt a different approach, which reduces the number of arbitrary choices in the analysis and frames the procedure within a more solid statistical framework. To address the problem of the stability of the fit, we constrain fit parameters to physically reasonable ranges by means of Bayesian constraints with Gaussian priors <cit.>. The procedure consists in the minimization of the augmented chi-square χ_aug^2 = χ^2 + χ^2_pr, defined as

    χ^2(𝐩) = ∑_i,j[ν_i-f_i(𝐩)]C^-1_ij[ν_j-f_j(𝐩)] ,

where the index i runs over all the data points within the fitting window, ν_i represents the mode number data computed on the lattice for a specific mass threshold aΩ_i, f_i(𝐩) corresponds to the model function in Eq. (<ref>) applied to a specific value of aΩ_i within the window for given values of the parameter vector 𝐩=[a^-4ν̅_0,A,am,γ^*], and C is the full covariance matrix estimated from the Monte-Carlo data. In addition to that, we consider uncorrelated Gaussian priors for the parameters p̃_n, σ̃_p_n that contribute to the global loss function as

    χ^2_pr(𝐩,{p̃_n,σ̃_p_n}) = ∑_n=1^4 (p_n-p̃_n)^2/σ̃_p_n^2 .

The selection of their central value is performed in a physics-informed way. Following Eq. (<ref>), the prior of am is centred to the value of a defined in Sec. <ref> times a coefficient which we set to 1.[This coefficient should correspond to the renormalization factor Z_A. We do not have a determination, but we assume it to be of order 1.] 
On the base of Eq. (<ref>), the prior of a^-4ν̅_0 is centred to the lattice estimate of the (fourth power of the) scalar baryon mass times a coefficient of order 1. The priors for A and γ^* are set to 1(10) and 0.5(5.0), respectively.
It is to be stressed that priors are only meant to improve the convergence of the fit as they supposedly prevent the minimization algorithm from being stuck in some local minimum. The choice of their numerical values should not influence the final result. To ensure the validity of this assumption, as it is common in this Bayesian framework, the width of the prior should be made large enough to accommodate significant fluctuations from the central value, i.e. choosing a so-called flat prior. Following this criterion, the widths of the priors are set to account for a 1000% relative error in the choice of the central values of γ^* and A's priors, 100% for am and a^-4ν_0. In order to ensure the positivity of a^-4ν̅_0 and am, we use a log-normal distribution for their prior.

We repeat the constrained fit for a large number of possible windows by varying aΩ_min∈[0.03,0.1] and aΩ_max∈[0.08,0.18]. 

As an example of the results of the fit, in Fig. <ref> we display the anomalous dimension γ^* extracted from a fit, as a function of the parameters aΩ_min and aΩ_max for the ensemble  in a three-dimensional space with coordinates (aΩ_max,aΩ_min,γ^*).

In order to seek a stable region in an automatized way, we calculate the gradient of the surface generated by our γ^* value and we filter those points in which the norm of the gradient is below a certain threshold, in order to automatize the identification of the stable region. In the upper panel of Fig. <ref> we assign a colder colour to those points whose gradient is smaller, i.e. corresponding to a “flatter” and therefore stabler region. The final result has to be given by averaging the coefficients over the different fitting windows in the stable region. This last scenario is analogous to the issue of data truncation arising in other contexts such as hadron spectroscopy in lattice QCD. In the last few years, Bayesian model averaging techniques to address this problem have started to raise a lot of interest <cit.>, and have been used in several published results. Similar to our case, the goal is to address the problem of the selection of a subset of data for a fit and avoid being overly conservative in the estimation of the systematic error, by re-formulating the problem as a model selection problem. 
Our proposal is to apply the methodology suggested in <cit.> by assigning to each window a so-called (modified) Akaike information criteria (AIC), defined as

    AIC = χ_aug^2(𝐩^⋆) + 2N_cut

where χ_aug^2(𝐩^⋆) is the augmented chi-square for the “best-fit” parameters 𝐩^⋆
and N_cut is the number of available points minus the number of points in the fit.[In Ref. <cit.>, the weight factor differs for an extra factor k in the exponent, k being the number of parameters in the fitting function. Since in our case the number of fitted parameters is the same for every window, we drop it since it corresponds to an overall factor that disappears in the normalization.] A weight factor w=e^-AIC/2 is then assigned in the final average to each window in the stable region.

In order to illustrate how the different windows are weighted in the model average, in the lower panel of Fig. <ref> we colour our points proportionally to their weight, calculated as in Eq. (<ref>). Using the same colour scale, in the left (right) hand side panel Fig. <ref> we depict the value of γ^* at fixed higher (lower) end of the window aΩ_max as a function of the lower (higher) one. As it is visible from the example, the model average tends to prefer bigger windows with good χ^2_aug, as expected.

In Table <ref> we report the values obtained for all the ensembles analyzed.



  §.§.§ γ_* in the continuum limit




We have previously observed,
and confirmed in the previous subsection,
that for the =1 case,
the value of γ_* is not stable as the coupling β is changed.
We have sufficiently many values of β
to attempt a continuum limit extrapolation.
Since the mode number method described above
allows computation of γ_* for individual ensembles,
and there is a strong dependence of the gradient flow scale w_0 / a on the fermion mass am,
one might expect to be able to study dependence of the per-ensemble γ_*
as a function of a / w_0,
and extrapolate to the a / w_0 → 0 limit.
However,
as we show in Fig. <ref>,
γ_* does not vary sufficiently with am to align with
the trend observed as β is changed.
Instead,
we must use a consistently-defined scale for each β value.



We adopt,
ad hoc,
two different prescriptions for this scale and the subsequent fitting form,
so that we can have some estimate of possible systematic effects.
Firstly,
we consider the behaviour of γ_* obtained from the mode number
as a function of 1/β directly,
shown in Fig. <ref>.
We observe that the data begin to flatten off as β decreases.
We fit these data using the ad hoc form

    γ_*(β) = d_0 + d_1 e^ - d_2β ,

which is inspired by assuming a linear correction in a,
with the latter parameterized with an exponential in β,
as suggested by the leading perturbative behaviour. 
This gives a continuum limit value of
γ_* =,
with
χ^2/dof =.
The high value of the χ^2/dof, which is mostly due to the lowest β point, suggests that we are still too far from the continuum limit for Eq. (<ref>) to provide an accurate description of the data. It is also possible that the errors on the extracted values of γ_* are somewhat underestimated, or that an anaccounted systematic error on γ_* affects our determination. Investigating the source of this discrepancy is beyond the scope of this work. However, we remark that adding a higher order term in the exponent reduces χ^2/dof to around 7, which is still far from an acceptable value. 





The second approach we consider is to take the chiral limit value for a / w_0.
This necessarily assumes that the theory is not in the conformal window,
where we would expect this value to be zero;
rather than infrared conformal, here we assume that the theory is near-conformal. We extrapolate a / w_0 as a function of a using a quadratic Ansatz,

    a/w_0 = a/w_0^χ + B (a) + C(a)^2 ,

where a/w_0^χ is the extrapolated chiral limit value of a/w_0.
This extrapolation is shown in Fig. <ref>.
As β increases,
a/w_0^χ approaches but does not reach zero, in agreement with our assumption of some residual global symmetry breaking. We then consider γ_* as a function  of a/w_0^χ, which we plot in Fig. <ref>.
We impose a quadratic Ansatz,

    γ_*(w_0^χ) = γ_*^cont. + c_0 (a/w_0^χ) + c_1 (a/w_0^χ)^2 ,

which gives a resulting continuum limit of γ_*^cont. =.
As these data lie closer to the continuum limit,
the uncertainty on the extrapolated value is smaller
than the extrapolation in 1/β above.
(There may however be systematic uncertainties not accounted for
due to the choice of fit form.)
The result is consistent with the limit imposed by the fit to Eq. (<ref>). Note however that in the case discussed here the value of γ_* would be scheme dependent.

Taking the mean of these two estimates,
weighted by their statistical errors,
gives our final estimate for the continuum limit value
of the mass anomalous dimension of the =1 theory,
γ_*=.

For the =2 theory,
we do not have data for sufficiently many values of β
to perform an extrapolation;
however,
we observe that the values obtained in Table <ref>
are consistent within uncertainties
between the two values of β considered,
suggesting they already may be close to the continuum value.





 §.§ Comparison with chiral perturbation theory


In addition to,
and to prove a contrast with,
the hyperscaling case discussed above,
we also probe how well a
chiral perturbation theory Ansatz describes
the data in the infrared limit.
We fit the mass of the 2^+ scalar baryon,
the pseudo-Nambu-Goldstone boson of the (potentially) broken chiral symmetry,
with the functional form:

    M̂_2_s^+ =   
        2Bm̂_PCAC
        (1
        + Lm̂_PCAC
        + D_1m̂_PCAClog(D_2 m̂_PCAC)) 
       + W_1 a
       + W_2 a^2/w_0^2 ,

where hatted quantities represent
the equivalent unhatted quantities
normalised by the gradient flow scale w_0.
We perform a simultaneous fit
across all values of β for which data is available,
and then for each ensemble compute the fit value for M̂
given the ensemble's observed values for
 and w_0.

The results for =1
are plotted in Fig. <ref>:
similarly to our previous work <cit.>,
we see deviations from this form increasing
as the value of a decreases
or the value of β increases.
The results for =2,
shown in Fig. <ref>,
meanwhile,
show less significant deviations.
The amount that can be drawn from this is however limited,
as only a single value of β is considered,
and the number of degrees of freedom is reduced
due to the small number of ensembles for which w_0 can be computed.










 §.§ R ratio


In this section, we examine our recently acquired data along with older results for 2 with =1 as well as with =2 flavors in the adjoint representation. Our aim is to explore universal characteristics that could delineate the broad parameter space of those strongly interacting gauge theories, in particular exposing their anomalous dimension as we approach the continuum. 

As shown in <cit.>,  in an appropriate regime, the ratio

    R = M_2^++/M_0^++  ,

where M_2^++ represents the mass of the lightest spin-2 composite state and M_0^++ denotes the mass of the lightest spin-0 state, has a universal behaviour that is determined by the value of the mass anomalous dimension γ_*, Therefore, the ratio R serves as a useful tool for characterizing general properties within an infrared (IR) conformal gauge theory. Several noteworthy characteristics render the observable R particularly significant. Crucially, R exhibits scheme-independence, and remains unaffected by the internal global symmetries of the theory. Hence, the comparison of R values computed in theories featuring vastly different internal symmetries and symmetry-breaking patterns provides an additional approach to the determination of γ_* that proves especially valuable for gauge theories incorporating fermionic fields, where the intricacies of chiral symmetry and its breaking introduce non-trivial, model-dependent features.

Prior to presenting numerical results for R, let's briefly outline the anticipated behavior of a conformal theory under deformation by a fermion mass m. For a small, finite deforming mass region, spectral masses are expected to scale as M ∝ m^1/Δ where Δ=1 + γ_⋆ represents the scaling dimension. Since a finite lattice volume imposes an absolute infrared cutoff, an appropriate scaling variable to consider on a finite box with typical dimension L is x = L m^1/Δ.  Consequently, the mass of a generic i^ th  state acquires a dependence on x, which can be expressed as LM_i = f_i (x), with f_i an a priori unknown function. At leading order in x, for a state with mass M_0, this yields LM_0 ∝ x, and through back substitution, L M_i = f_i (LM_0).  Consequently, when considering mass ratios, the explicit dependence on the lattice extent cancels out, enabling comparison of data across different volumes:

    M_i/M_j = f_i(LM_0)/f_j(LM_0) .

We anticipate that the behavior of R will exhibit four distinct regimes. For large values of m, R tends to align closely with the value observed in pure 2 Yang-Mills theory, determined in lattice studies as R = 1.44(4)<cit.>. Conversely, at small m, the regime that is realised will hinge on the lattice size L. In sufficiently large volumes, the observed physics displays confining properties due to the mass deformation. Conversely, at very small volumes, the theory transitions to the so-called “femto-universe" regime <cit.>, where previous studies suggest R approaches unity <cit.>. However, for L large enough and m small enough, an intermediate regime of R exists where conformal behavior can be observed. This intermediate region is of particular interest as in it R  can be extrapolated to obtain a chiral limit value that provides a universal signature of conformal behaviour. Note that whether the intermediate region is observed in a given simulation is a consequence of the choice of the lattice parameters, since a same value of x can be obtained by tuning m and L to very different regimes.  

Given the universal behaviour of R, to gain insights on the anomalous dimension, we opt to contrast our findings with a toy model constructed using principles of gauge-gravity duality <cit.>. This calculable model, developed within the bottom-up approach to holography, incorporates only an operator with a scaling dimension Δ deforming the theory. Leveraging the obtained results for the anomalous dimension, we can compare the measured ratio with expectations derived from the toy model utilizing the corresponding value of Δ. Further elucidation on this concept is available in Ref. <cit.>.

In Fig. <ref>, we display our findings regarding R across seven distinct values of the coupling β for 2 at =1. The data for the initial four plots have been previously showcased in Ref. <cit.>. Our results are juxtaposed with the R ratio for the pure 2 Yang-Mills theory (depicted by the blue dashed line) and with predictions derived from the gauge-gravity inspired toy model (shown as the upper band). It is worthy to note that the conformal prediction varies for each value of β due to differing measured values of γ_*. Additionally, for β=2.05, the width of the band differs slightly from that presented in Ref. <cit.> due to a different method of extracting γ_* resulting in a more conservative estimation of the error: in Ref. <cit.>, the results were derived using the Dirac mode number; in contrast, our current investigation employs the finite-size hyperscaling Ansatz.

For large values of LM_0^++, the data for β=2.05 up to β=2.3 tends toward the pure gauge prediction, in line with expectations. As LM_0^++ decreases, approaching the smallest values of β, the calculations become consistent with the conformal plateau, particularly for larger R. At β=2.1 and, to a lesser extent, β=2.15, there is some indication of behavior diverging from confinement, although the ratio does not rise sufficiently to match the value predicted by the gauge/gravity model. Notably, for the highest values of β, the results for R do not significantly exceed the Yang-Mills value, even for large LM_0^++, possibly indicating an increasing difficulty in realising the regime that is relevant for near-conformality in this observable as β increases. Interestingly, at the highest two values of β and lower LM_0^++, there is clear evidence of R decreasing towards R ≈ 1, indicating a femto-universe behavior; In this region, the mass of the ground scalar glueball as well as the tensor glueball become approximately degenerate due to the lattice box geometry.

Overall we observe that for =1, the value of R decreases from a scenario with a large anomalous dimension  (R ≫ 1) as the value of β increases and approaches the continuum limit. In fact, any signals of conformality become weaker and are eventually lost, which aligns with the idea that as the continuum is approached, the anomalous dimension decreases. This effect increses the difficulty in tuning the simulation to the intermediate region that is relevant for determining the anomalous dimension using the universality of R. Followingly, for large values of LM_0^++, the ratio R converges to the pure Yang-Mills prediction of R=1.44(4), enhancing the scenario of existing universality. Finally, for small values of L M_0^++ the R-ratio appears to be aligned with the femto-universe scenario.

Similarly, in Fig. <ref>, we present our results for the ratio M_2^++/M_0^++ at inverse couplings of β=2.25 and β=2.35. The findings for β=2.25 were previously discussed in detail in Ref. <cit.>. Our current study extends these earlier results by including data for β=2.35, attempting for a more comprehensive analysis of the behavior of this mass ratio under varying coupling conditions.

In general, the results for the case of =2 exhibit a significantly higher degree of scatter, making it difficult to draw clear and robust conclusions. This behavior is illustrated by a large number of data points aligning with the 2 pure gauge prediction, while many other points lie significantly lower, approaching unity. We consider these latter data points to be affected by significant finite volume effects, which aligns with the femto-universe hypothesis.

Bearing the above in mind, let us attempt to understand our numerical findings. First, for large values of LM_0^++, the data for β of 2.25 tends to align with the pure gauge prediction, as expected. As LM_0^++ decreases, the calculations become consistent with the conformal plateau prediction; it appears that the latter is realised for values of R ranging for LM_0^++=8 - 10. This feature appears to weaken as we move closer to the continuum. Namely, for β=2.35 we cannot see an agreement with the conformal model anymore at intermediate values of LM_0^++ apart from one data-point for the very low values of L M_0^++ which appears to be an outlier. Instead, we observe that most of the data points are surrounding the pure gauge theory prediction. This suggests once more that the anomalous dimension weakens as we approach the continuum limit.











§ CONCLUSIONS


Armed with the purpose of advancing the characterisation of the infrared behaviour of 2 gauge theories with = 1 and = 2 Dirac fermion flavours transforming in the adjoint representation, we have performed an extended study of the former model in bare parameter space, with the intent of extrapolating the results to the continuum limit. For the considered values of the parameters, we have provided a detailed study of the mass spectrum, which has enabled us to extract the fermion condensate anomalous dimension using the expected hyperscaling behaviour for (near-)conformal systems. In addition, the anomalous dimension has been investigated making use of a mode number analysis of the Dirac spectrum and of the universal behaviour of the ratio of the mass of the lightest spin-two particle over the mass of the scalar. With different efficacy, all methods give consistent results and point towards heavy corrections to the anomalous dimension as the continuum limit is approached. Using ad hoc Ansätze, we have extrapolated the anomalous dimension to zero lattice spacing. Neglecting the systematics, including the one coming from the extrapolation, which can be somewhat large, our final result for the = 1 theory is γ_* =. In contrast with earlier findings at larger lattice spacings, this value is significantly smaller than one.  Barring very large lattice artefacts, a scenario that would need to be investigated further, the strong dependence of the anomalous dimension on the lattice spacing seems to exclude a clear signature of infrared conformality. We have also performed a chiral perturbation theory analysis, which shows that a regime with detectable chiral symmetry breaking is also excluded. This however does not exclude the possibility that the infrared dynamics of the model will result in a small breaking of chiral symmetry. Putting together both the near-conformal and the chiral perturbation theory analysis, the most likely conclusion is that the theory is in the chiral symmetry broken phase, but rather close to the sill of conformality.

Our calculations for the = 2 theory are more limited in scope, as we have only two lattice spacing values available, one from an earlier study and one investigated explicitly in this work. Combining the corresponding data with an analysis that parallels the = 1 case, within the limitations of the information that can be extracted, we note a milder dependence of γ_* on the lattice spacing and again albeit milder an intrinsic difficulty of chiral perturbation theory to describe the results. A likely conclusion is that the = 2 model lies inside the conformal window, and has a small anomalous dimension.

After over 15 years of chasing the infrared behaviour of those models, this very important quest appears to be far from over. However, our work has pushed the boundaries of what can be done with state of the art resources using a few millions hours of A100 GPUs with the formulation of the action adopted so far. Future investigations call for the use of actions that have smaller lattice spacing corrections, and can hence produce more physical results at a relatively coarse lattice spacing, and are more amenable to be pushed towards the chiral limit. To carry on this line of investigation, the possibilities that we are currently exploring include Möbius domain wall fermions (see, e.g., <cit.>) and Pauli-Villars bosons <cit.>.  



A.A. was supported by the Horizon 2020 European research infrastructures
programme “NI4OS-Europe” with grant agreement no. 85764, by “SimEA" project funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 810660, as well as by the “EuroCC" project funded by the “Deputy Ministry of Research, Innovation and Digital Policy and the Cyprus Research and Innovation Foundation" as well as by the European High-Performance Computing Joint Undertaking (JU) under grant agreement No. 101101903. 

P.B. acknowledges support by the project H2020-MSCAITN-2018- 813942 (EuroPLEx) and the EU Horizon 2020 research and innovation programme and by the Grant DGA-FSE grant 2020-E21-17R Aragon Government and the European Union - NextGenerationEU Recovery and Resilience Program on “Astrofísica y Física de Altas Energías" CEFCA-CAPA-ITAINNOVA. 

The work of E.B. and J.L. has been supported by the UKRI Science and Technology Facilities Council (STFC) Research Software Engineering Fellowship EP/V052489/1. The work of E.B., J.L., and B.L. has been supported in part by the EPSRC ExCALIBUR programme ExaTEPP (project EP/X017168/1). E.B. and B.L. have been partly supported by the STFC  Consolidated Grant No. ST/T000813/1. B.L. received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 813942. 

G.B. is funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. 432299911 and 431842497.

This work used the DiRAC Extreme Scaling service Tursa at the University of Edinburgh, managed by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC service at Edinburgh was funded by BEIS, UKRI and STFC capital funding and STFC operations grants. DiRAC is part of the UKRI Digital Research Infrastructure. We acknowledge the support of the Supercomputing Wales project, which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government.






  
*Open access statement
For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.



  
*Research Data Access Statement
The data generated for this manuscript can be downloaded from Ref.<cit.>,
in addition to data from previous work already available from Ref. <cit.>
and the analysis code from Ref. <cit.>.



apsrev4-2.bst





 Paper Title: Fully strange tetraquark resonant states as the cousins of $X(6900)$.













yaoma@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

wlwu@pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

lu.meng@rub.de
Institut für Theoretische Physik II, Ruhr-Universität Bochum,  D-44780 Bochum, Germany 

chenyanke@stu.pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

zhusl@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China


§ ABSTRACT 

We conduct systematic calculations of the S-wave fully strange systems with “normal" (J^P C=0^++, 1^+-, 2^++) and “exotic" (J^P C=0^+-, 1^++, 2^+-) C-parities, which are the 
strange analogue of the fully charmed tetraquark state X(6900).
Within a constituent quark potential model, we employ the Gaussian expansion method to solve the four-body Schrödinger equation and the complex scaling method to identify resonant states. We obtain a series of resonant states and zero-width states in the mass range of 2.7 to 3.3 GeV, with their widths ranging from less than 1 MeV to about 50 MeV. Their rms radii strongly indicate that they are compact tetraquark states. Among these states, the T_4s,2^++(2714) may be the most likely one to be observed experimentally. We urge the experimental exploration of the 2^++ s s s̅s̅ state around 2.7 GeV in the ϕϕ channel. Since the lowest S-wave s s s̅s̅ state is around 2.7 GeV, the compact P-wave s s s̅s̅ states are expected to be heavier. Hence, ϕ(2170) and X(2370) are unlikely to be compact tetraquark states.





Fully strange tetraquark resonant states as the cousins of X(6900)
    Shi-Lin Zhu 0000-0002-4055-6906
    August 5, 2024
==================================================================






§ INTRODUCTION
 

Since 2003, the interest in multiquark states has been reignited by the experimental discovery of a series of candidates (for recent reviews, see Refs. <cit.>). These discoveries have significantly enriched the hadron spectrum. The study of their properties and structures enhances our understanding of the non-perturbative nature of the strong interaction.

Inspired by their many strangeness decays, there are several promising candidates for the fully strange tetraquark states. 


 
  * The state ϕ(2170) (was Y(2175)) was first reported in 2006 by the BaBar Collaboration in the process of e^+ e^-→ϕ(1020) f_0(980) <cit.>. Later it was confirmed by the BES and BESIII Collaborations in the J / ψ→ηϕ f_0(980) process <cit.>, and the Belle Collaboration in the e^+ e^-→ϕπ^+π^-and e^+ e^-→ϕ f_0(980) processes <cit.>. 
 
 
  * The state X(2370) was first observed in 2010 by the BESIII Collaboration in the π^+π^-η^' invariant-mass spectrum <cit.>. Later it was confirmed in 2019 by the BESIII Collaboration in the K K̅η^' invariant-mass distribution <cit.>, which suggests that the X(2370) likely contains four valence s or s̅ quarks.

 
  * The pseudoscalar state X(2500) was first observed in 2016 by the BES Collaboration in the process J / ψ→γϕϕ <cit.>, together with the scalar state f_0(2100), and three tensor states, the f_2(2010), f_2(2300) and f_2(2340).

 
  * The X(2239) was seen in 2018 by the BESIII Collaboration in the cross section of e^+ e^-→ K^+ K^- <cit.>.
 
 
  * An evidence for a structure X(2100) in the ϕη^' mass spectrum of the J / ψ→ϕηη^' decay was reported in 2019 by the BESIII Collaboration <cit.>.

 
  * Another structure in the ϕ f_0 cross section around 2.4 GeV, labelled as X(2436) <cit.>, was reported in 2007 by the BaBar Collaboration <cit.>. It is also seen by the BES Collaboration <cit.> and Belle Collaboration <cit.>, and evidenced by the BESIII Collaboration <cit.>. However, the latest BESIII  results indicate that its statistical significance is no more than 2σ <cit.>.


Among the aforementioned possible T_s s s̅s̅ candidates, all the established ones have negative parity. The existence of the positive parity states remains uncertain. Additionally, the LHCb Collaboration recently discovered a fully charmed tetraquark candidate X(6900) <cit.>, which was later confirmed by the CMS and ATLAS <cit.>. 
The QCD Lagrangian leads to the same quark gluon interaction if one omits the quark mass difference. The discovery of X(6900) implies the possible existence of the strange analogue T_s s s̅s̅. 

Inspired by these experimental observations, various theoretical investigations have been conducted on the fully strange tetraquark systems. Regarding the ϕ(2170) state with J^P C=1^–, some authors used the QCD sum rule <cit.>, quark model <cit.> and diquark-antidiquark scheme <cit.> to interpret it as an s s s̅s̅ tetraquark state, although other interpretations were not  excluded <cit.>. Similarly, the X(2370) state was also explained as an s s s̅s̅ tetraquark state with 0^-+ <cit.>. The pseudoscalar state X(2500) was interpreted as a P-wave s s s̅s̅ tetraquark state of J^P C=0^-+ <cit.>. In Ref. <cit.>, the X(2500), X(2239), f_2(2300) were assigned as 0^-+, 1^–, 2^++ s s s̅s̅ tetraquark states, respectively. Moreover, the structure X(2100) was interpreted as a 1^+- s s s̅s̅ state <cit.>. In Ref. <cit.>, the resonant states X(2100) and X(2239) were treated as the axial-vector and vector tetraquarks with the quark content s s s̅s̅, respectively. The structure X(2436) was explained as a P-wave s s s̅s̅ tetraquark state of J^P C=1^– <cit.>. The s s s̅s̅ tetraquark states with other quantum numbers were also expolred  <cit.>.

So far, there is no experimental conclusion regarding the existence of positive parity s s s̅s̅ states, and theoretical investigations on these states are also limited. Although some theoretical calculations on the fully strange tetraquark system in quark models have been conducted <cit.>, they are relatively sparse and less refined compared to those on tetraquark systems containing heavy quarks. The studies in Refs. <cit.> focus exclusively on the diquark-antidiquark picture, obtaining only the states above the dimeson thresholds. These states would become scattering states or resonant states once the dimeson configuration is incorporated. Only the resonant states have physical significance and should be identified using nontrivial approaches. In Refs. <cit.>, the s s s̅s̅ tetraquark resonant states were investigated via the real scaling method. This method detects resonant states by avoiding the cross pattern of energy levels across a parameter controlling the spatial extension of the wave function, which is usually quite rough. 

To address these issues, we will conduct systematic calculations of the positive parity S-wave fully strange system. We will employ the Gaussian expansion method (GEM) <cit.> to solve the four-body Schrödinger equation within a nonrelativistic constituent quark potential model. Additionally, we will adopt the complex scaling method (CSM) <cit.> to identify genuine resonant states from meson-meson scattering states. In our recent works, we have incorporated both the dimeson and diquark-antidiquark configurations to carry out benchmark calculations for tetraquark bound states <cit.> and resonant states <cit.>. We will continue to include both configurations for the fully strange tetraquark system. In this study, we will investigate both the “normal" C-parity states (J^PC=0^++, 1^+-, 2^++) and the “exotic" C-parity states (J^PC=0^+-, 1^++, 2^+-), with the “exotic” C-parity systems referring to those that cannot be composed of two S-wave ground quarkonia.

This paper is arranged as follows. 
In Sec. <ref>, we introduce the theoretical framework, including the Hamiltonian in the constituent quark potential, the construction of the wave function, the complex scaling method, and the approach to analyzing the spatial structures.
In Sec. <ref>, we present the numerical results for the properties of the fully strange resonant states.
Finally, we give a brief summary and discussion in Sec. <ref>.




§ THEORETICAL FRAMEWORK
 



 §.§ Hamiltonian
 

The nonrelativistic Hamiltonian of a four-quark system reads

    H=∑_i^4(m_i+p_i^2/2m_i)+∑_i<j=1^4 V_i j ,

where m_i and p_i are the mass and momentum of quark i.

In this study, we will employ the AL1 constituent quark potential model <cit.> containing the one-gluon-exchange (OGE) interaction and a linear confinement term.
The AL1 potential reads 

    V_i j=-3/16   λ_i^c ·λ_j^c(-κ/r_ij+λ r_i j-Λ
       +8πκ'/3m_im_jexp(-r_ij^2/r_0^2)/π^3/2r_0^3s_i·s_j),






where r_0=A(2m_im_j/m_i+m_j)^-B. λ^c represents the SU(3) color Gell-Mann matrix and s_i is the spin operator of quark i. The parameters of the potential have been determined by fitting the meson spectra in Ref. <cit.> and are listed in Table <ref>. The calculated meson masses in AL1 model are shown in Table <ref>. 





















 §.§ Wave function construction
 

The wave function bases of the sss̅s̅ tetraquark states can be expressed as the direct product of spatial wave function ϕ, spin wave function χ_s, and color wave function χ_c.

    ψ=𝒜(ϕ⊗χ_s⊗χ_c),

where 𝒜 is the antisymmetrization operator, representing the exchange of identical quarks. For the sss̅s̅ system, 𝒜=(1-P_12)(1-P_34), where P_i j permutes the ith and jth (anti)quarks. 


For the spatial wave function, the Gaussian expansion method (GEM) <cit.> is employed. Namely, the spatial wave function is expanded using the following basis:

    ϕ_n l m(r)=√(2^l+5 / 2/Γ(l+3/2) r_n^3)(r/r_n)^l e^-r^2/r_n^2 Y_l m(r̂),

where the r_n is taken in geometric progression, r_n=r_0 a^n-1. Y_l m is the spherical harmonics.

For a four-body system, one can work on the Jacobi coordinates to exclude the center-of-mass motion. In principle, one could construct a complete basis functions using any type of Jacobi coordinates. However, to get rid of the complexity from the angular momentum, we use only S-wave bases constructed in different types of Jacobi coordinates to partially compensate for higher partial waves (see Ref. <cit.> for details). In this work, we include two types of Jacobi coordinates, the diquark-antidiquark structure and the dimeson structures, as shown in Fig. <ref>.


In our calculation, we choose n_max=12 bases for each coodinate, and the settings of the basis parameters are 

    {[ r_0=0.4 fm, r_n_max=2.0 fm              s-s  or s̅-s̅; r_0=0.4 fm, r_n_max=2.0 fm               (s s)-(s̅s̅); r_0=0.4 fm, r_n_max=1.3 fm                       s-s̅; r_0=0.5 fm, r_n_max=4.5 fm              (s s̅)-(s s̅) ],.

where we use relatively large r_n_max for the coordinate between two s s̅ clusters to better describe the meson-meson scattering states and possible molecular states.

For the spin wave function, the bases for total spin S=0,1,2 are

    S=0:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_0]_0; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_0 ],. 
        S=1:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_1]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_0]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_1 ],. 
        S=2:[ [(s_1 s_2)_1(s̅_3 s̅_4)_1]_2 ].


Two optional color wave function bases sets are

    color-I: {[ [(s_1 s_2)_3̅(s̅_3 s̅_4)_3]_1;  [(s_1 s_2)_6(s̅_3 s̅_4)_6]_1 ],.


    color-II: {[ [(s_1 s̅_3)_1(s_2 s̅_4)_1]_1; [(s_1 s̅_3)_8(s_2 s̅_4)_8]_1 ]..

Since both sets are orthogonal and complete, using either set will yield exactly the same results <cit.>.

Besides the antisymmetrization operation in Eq. (<ref>), since the s s s̅s̅ system carries definite C-parity, it is necessary to add (or subtract) the C-transformation of each term in Eq. (<ref>) to ensure that the overall wave function has the desired positive (or negative) C-parity. The specific C-transformation is

    [(q_1 q_2^')^s_1_c̅(q̅_3 q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       .(-1)^S-s_1-s_2[(q_1 q_2^'))_c̅^s_2(q̅_3 q̅_4^')_c^s_1]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-I set, and

    [(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       (-1)^s_1+s_2[(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-II set. The particle q and q' can be different quarks, although they both represent the same s quark in the s s s̅s̅ system.




 §.§ Complex scaling method
 

The complex scaling method (CSM) is a direct approach to obtain the energies and the decay widths of resonant states in a many-body system by performing an analytical continuation of the Schrödinger equation <cit.>. This is achieved by carrying out a complex rotation on the coordinate r and momentum p, given by

    U(θ) r=r e^i θ,    U(θ) p=p e^-i θ.

Under the rotation, the Hamiltonian in Eq. (<ref>) becomes

    H(θ)=∑_i=1^4(m_i+p_i^2 e^-2 i θ/2 m_i)+∑_i<j=1^4 V_i j(r_i j e^i θ).

Meanwhile, for the resonant states with pole positions within the range of the rotated angle, their wave functions become normalizable by integration, thereby solvable through localized Gaussian bases in the same way as bound states. As a result, solving the complex-scaled Schrödinger equation will simultaneously yield the eigenenergies of bound states and resonant states within the rotated angle.

A typical pattern of the solved eigenenergies in the complex energy plane is: The bound states lie on the negative real axis of the energy plane. The continuum states align along beams originating from thresholds with Arg(E)=-2 θ. The resonant states with mass M_R and width Γ_R are located at E_R=M_R-i Γ_R / 2, and only those within |Arg(E_R)|<2 θ can be solved. The positions of the bound and resonant states remain unchanged with the variation of the rotation angle.  One can find more details in Refs. <cit.>.



 §.§ Spatial structure


The root-mean-square (rms) radius is a good physical quantity for reflecting the spatial structure of the tetraquark states, and a widely used criterion for distinguishing between compact and molecular tetraquark states.

The conventional definition of the rms radius under CSM is 

    r_i j^rms,C≡Re[√((Ψ(θ)|r_i j^2 e^2 i θ| Ψ(θ))/(Ψ(θ) |Ψ(θ)))],

where the Ψ(θ) is the obtained complex wave function of the tetraquark state. The round bra-ket represents the so-called c-product <cit.> defined as

    (ϕ_n |ϕ_m) ≡∫ϕ_n(r) ϕ_m(r) d^3r,

without taking complex conjugate of the bra-state. This procedure ensures the function inside the integral is analytic, thereby the expectation value of the physical quantity remains stable as the rotation angle changes. The rms radius calculated from the c-product is generally not real; however, its real part can still reflect the internal quark clustering behavior if the resonant state is not too broad, as discussed in Ref. <cit.>.

Nevertheless, for tetraquark states containing identical quarks, this definition may fail to identify the molecular nature of their spatial structure due to the antisymmetrization of identical quarks, as discussed in Ref. <cit.>. To eliminate this ambiguity, we have proposed a new approach to calculate the rms radii of the fully heavy Q Q Q̅Q̅ system in Ref. <cit.>, which can also be applied to the s s s̅s̅ system. In this new approach, the total wave function is non-orthogonally decomposed into

    Ψ(θ) =     ∑_s_1 ≤ s_2[[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ).
        - [(q_2 q̅_3^')_1_c^s_1(q_1 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_3, r_4;θ) 
        - [(q_1 q̅_4^')_1_c^s_1(q_2 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_4, r_3;θ)
        + .[(q_2 q̅_4^')_1_c^s_1(q_1 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_4, r_3;θ)]
    
    =    𝒜[ ∑_s_1 ≤ s_2[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ)]
    ≡   𝒜Ψ_13,24(θ),

where s_1, s_2 sum over all spin configurations with total spin S. The particle q and q' can be different quarks, although they both represent s quark in the s s s̅s̅ system. We use the decomposed wave function Ψ_13,24(θ) to define the new rms radius

    r_ij^rms,M≡Re[√((Ψ_13,24(θ)|r_i j^2 e^2 i θ| Ψ_13,24(θ))/(Ψ_13,24(θ) |Ψ_13,24(θ)))] .

This newly defined rms radius can better describe the spatial structure of the molecular-type tetraquark states containing identical quarks. We will provide the results for both definitions of the rms radius in our calculation.




§ NUMERICAL RESULTS
 



 §.§ States with “normal" C-parity


The “normal” C-parity s s s̅s̅ systems refer to those that can be composed of two S-wave ground quarkonia, namely, the  systems with J^PC=0^++,1^+-,2^++. The complex eigenenergies yielded are shown in the upper panel of Fig. <ref>. We choose different complex scaling angles θ=9^∘,12^∘,15^∘ to distinguish bound states and resonant states, which remain stationary as the angle changes, from the continuum states. The markers that align along beams originating from meson-meson thresholds with Arg(E) = -2 θ are the continuum state eigenenergies. The results show that no bound state is found below any of the thresholds, while a series of resonant states are obtained and marked out by black circles. Their complex energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. For convenience, we label the tetraquark resonant states obtained in our calculations as T_4s, J^PC(M), where M is the mass of the state.










The masses of the resonant states are in the region (2.7,3.3) GeV, and their widths range from a few MeV to several tens of MeV. For the resonant states above the 1S2S di-meson threshold, our calculation actually underestimates their widths because the decay width of the 2S meson has not been accounted for. The rms radii of all the obtained states approximately match the sizes of the 2S mesons. Furthermore, the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states.



In tetraquark systems, there are abundant dimeson thresholds. Each of these thresholds represents a branch cut connecting two different Riemann sheets. In our calculations, we must choose appropriate rotation angle to make the pole of interest emerge. Meanwhile, due to numerical reasons, the pole close to the continuum solutions will be affected by them, as illustrated in Refs. <cit.>. Therefore, to reduce uncertainties, the rotation angle should be chosen such that the continuum states are not too close to the resonant state. In practices, we adjust the angle to make each pole and its rms radii stable. 






Comparing the fully strange systems with ccc̅c̅ using the AL1 model in our previous work <cit.>, we observe a similar overall pattern. However, the fully strange systems exhibit an additional quasi-bound state with 2^++ below the ϕϕ(2S) threshold. This difference may stem from the mass-dependent color-magnetic term in the potential.

For the 0^++ states, the five resonant states can all decay into η'η', ϕϕ, η'η'(2S), ϕϕ(2S) final states, although there may be some effects if the mixing between the I=0 η(nn̅) and η'(ss̅) is taken into account. The T_4s,0^++(2917) and T_4s,0^++(3133) have an additional decay channel into η'η'(3S). The higher T_4s,0^++(3175) and T_4s,0^++(3248) have two additional channel into η'η'(3S) and η'(2S)η'(2S). For the 1^+- states, the five resonant states can all decay into η'ϕ, η'ϕ(2S) and η'(2S)ϕ. The  T_4s,1^+-(3142) has an additional decay channel into η'ϕ(3S). And the higher T_4s,1^+-(3228) and T_4s,1^+-(3237) have two extra η'ϕ(3S) and η'(3S)ϕ decay channels. For the 2^++ states, the lowest T_4s,2^++(2714) below the ϕϕ(2S) threshold have only one decay channel ϕϕ. Other two 2^++ resonant states T_4s,2^++(2993) and T_4s,2^++(3164) have an additional decay channel into ϕϕ(2S). The T_4s,2^++(3266) has an extra decay channel to ϕϕ(3S).

Among these states, the state T_4s,2^++(2714) is the most likely one to be observed experimentally. Its mass is the lowest, and its decay channel ϕϕ predominantly decays into K^+K^-K^+K^-, which are all charged particles that can be detected directly by detectors, leading to high detection efficiency and resolution.




 §.§ States with “exotic" C-parity


The “exotic” C-parity systems refer to those that cannot be composed of two S-wave ground quarkonia, namely, the system with J^PC=0^+-,1^++,2^+-. The complex eigenenergies yielded are shown in the lower panel of Fig. <ref>. We obtain a series of resonant states and zero-width states, whose energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. The rms radii approximately match the sizes of the 2S mesons. The newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

For the J^PC=0^+- and 2^+- systems, there is no S-wave di-quarkonium threshold. So the calculated states appear as bound states on the complex energy plane. However, there may actually be higher-wave thresholds below these states, which are not included in the calculation. If the higher-wave mesons are included, the properties of these states may change and their widths may no longer be zero.

For the J^PC=1^++ system, the masses of the resonant states range in (2.7,3.3) GeV, and their widths are small, with the largest being only a few MeV. Generally, one would expect the widths of these resonant states to arise from their decays into lower-energy meson-meson channels. However, their narrow widths suggest that such decays are suppressed. The reason for the suppression has been discussed in detail in Ref. <cit.>. Simply put, when the rms radii between different i,j quarks are roughly equal, the transition operator of the color-electric term becomes approximately proportional to the identity operator. Therefore, the color mixing transition is suppressed. In the s s s̅s̅ system, however, the contribution from the color-magnetic term is not as significantly suppressed by the quark mass as it is in the heavy quark sector. Therefore, the transition can still occur via spin mixing, which, although limited, can lead to slightly larger widths of these resonant states. All three J^PC=1^++ resonant states can decay via ϕϕ(2S), though this process is suppressed.








§ SUMMARY AND DISCUSSION
 

We calculate the mass spectrum of the S-wave fully strange tetraquark systems with “normal" C-parity J^PC=0^++,1^+-,2^++ and “exoitic" C-parity J^PC=0^+-,1^++,2^+- using the AL1 quark potential model. We employ the GEM to solve the four-body Schrödinger equation, and the CSM to identify genuine resonant states from meson-meson scattering states.

We obtain a series of resonant and zero-width states in the mass region (2.7,3,3) GeV, with widths ranging from less than 1 MeV to around 50 MeV. Apart from the state T_4s,2^++(2714), all the others lie above the M(1 S) M^'(2 S) dimeson thresholds.

We calculate two kinds of rms radius for all states, the conventional rms radius r_ij^rms,C and the newly defined rms radius r_ij^rms,M based on the decomposed wave function. For all states, both kinds of rms radii approximately match the sizes of the 2S mesons, and the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

Among these states, the state T_4s,2^++(2714) is the most likely to be observed experimentally due to its lowest mass and its decay into ϕϕ, which predominantly produces directly detectable charged K^+K^- pairs, ensuring high detection efficiency and resolution.

Recalling the experimental candidates for s s s̅s̅ tetraquark states, their masses are all in range of 2.1 to 2.5 GeV, whereas the lowest S-wave resonant state in our calculations is around 2.7 GeV. This suggests the states like ϕ(2170) and X(2370) are unlikely to be compact s s s̅s̅ tetraquark states. Since the lowest S-wave state is already as high as 2.7 GeV, it's unlikely that a P-wave compact state would be lower than the S-wave ones. However, they could still be P-wave molecular-type resonant states. In this work, a state T_4s,2^++(2714) is found near the threshold of ground ϕ combined with a radially excited ϕ(2S), although it does not exhibit a molecular spatial structure. Introducing P-wave thresholds might yield molecular-type resonant states near the threshold of an S-wave meson combined with a P-wave meson, which could be lower in mass. This possibility awaits careful investigation in the future.

We emphasize that the fully charmed and fully strange tetraquark resonant states share many common characteristics. Their internal structures and underlying dynamics are nearly the same. All of them are compact states. The discovery of X(6900) implies the existence of its fully strange cousins, which can be searched for in the ϕϕ or ϕϕ(2S) channels. While X(6900) can only be produced at LHC now, the fully strange tetraquark resonances may be observed at BESIII, Belle II and LHC. The future discovery of these states shall shed light on both the charmed and strange compact tetraquark states.     












We are grateful to Zi-Yang Lin, Jun-Zhang Wang and Liang-Zhen Wen for the helpful discussions. This project was supported by the National Natural Science Foundation of China (11975033 and 12070131001). This project was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project ID 196253076-TRR 110). The computational resources were supported by High-performance Computing Platform of Peking University.










 Paper Title: Highly-efficient quantum Fourier transformations for some nonabelian groups.


FERMILAB-PUB-24-0241-SQMS-T













emm712@gwu.edu



Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA

Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Fermi National Accelerator Laboratory, Batavia,  Illinois, 60510, USA



Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA


egustafson@usra.edu
Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA


§ ABSTRACT  
    Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. 
While the standard abelian QFT is well-studied, important variants corresponding to  nonabelian groups of interest have seen less development.
    In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, 𝔹𝕋, 𝔹𝕆, Δ(27), Δ(54), and Σ(36×3). 
   For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations.  
    Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.

Highly-efficient quantum Fourier transformations for certain nonabelian groups
    Erik Gustafson
    February 2024
==============================================================================





§ INTRODUCTION


Fourier transforms have numerous applications in both classical and quantum computing.   
Classical applications range from signal analysis and compressed sensing, to coding theory, physics, and number theory 
<cit.>.  
In terms of quantum algorithms, it forms an important subroutine for Shor's algorithm <cit.>, the hidden subgroup <cit.> and hidden shift problems <cit.>, phase estimation <cit.>, and – of particular interest here – quantum simulation of high energy physics (HEP) <cit.>. 


Shor's algorithm and its extensions for 
prime factoring and discrete logarithms rely on the ability of quantum computers to efficiently solve the hidden subgroup problem for finite abelian groups using the quantum Fourier transform (QFT).
Beyond special cases, 
it is unknown if efficient QFT circuits imply an efficient solutions to the related hidden subgroup problem, a question connected to numerous deep open problems in quantum computing <cit.>. Indeed, the known nonabelian fast Fourier transforms are often more complex <cit.> than their abelian counterparts and developing efficient algorithms remains an active area of research both classically and quantumly <cit.>. Furthermore, even for particular classes of groups where asymptotically efficient quantum circuit are known, detailed resource analysis of specific cases can yield dramatic improvements over the general techniques. 

The need for fast QFTs for nonabelian groups is 
particularly acute in the quantum simulations of HEP, where previous work found that the naive Fourier transform represented ≳ 95% of the total gate cost <cit.>. Important targets for HEP applications are the QFTs for crystallographic subgroups of SU(2) <cit.> and SU(3) <cit.>, and beyond those cases SU(N) <cit.> more generally. 
As we will see below, the reduction in resource costs for HEP simulations from fast QFTs provide multiple orders of magnitude improvement. While we focus on groups of interest to HEP, we anticipate our work to have broader algorithmic applications. A recent example <cit.> utilized the QFT over the symmetric group <cit.> within generalized quantum phase estimation <cit.> to derive quantum algorithms and complexity results concerning computing structural properties of the group known as Kronecker coefficients. 


Mathematically, Fourier transforms are defined over a specific group – either abelian or nonabelian, continuous or finite – and correspond to a unitary transformation from the group algebra to a complex vector
space whose basis vectors are the matrix elements of the group's irreducible representations (irreps). For a finite group G, the discrete Fourier transform (DFT) can be defined as

    f̂(ρ) = √(d_ρ/|G|)∑_g ∈ G f(g) ρ(g),

where | G | is the size of the group, d_ρ is the dimensionality of the representation ρ, and f is a function over G. Implemented naively, the abelian DFT over ℤ_n has a classical complexity of O(n^2). Through repeated application of a subgroup-decomposition, it is possible to reduce the classical complexity to O(n log(n)) with a fast Fourier transform (FFT) using the Cooley-Tukey method <cit.>, though a similar algorithm was also developed much earlier by Gauss <cit.>.

A generalization of the Cooley-Tukey method to non-abelian groups which constructs the FFT through subgroup-decomposition was first identified by Diaconis and Rockmore <cit.>. A naive approach, that would compute Eq. (<ref>) directly would yield an asymptotic scaling of O(| G |^2). On the other hand, the Diaconis-Rockmore algorithm, like the Cooley-Tukey algorithm, relies on the fact that one could write any irrep of a group element as ρ(g) = ρ(ℓ) ρ(h) whenever g =ℓ h where h ∈ H 
belongs to a subgroup of H ⊂ G, and ℓ is an element of the left transversal. With this decomposition, the problem essentially reduces to computing the Fourier transform of the subgroup H. In this way, Diaconis and Rockmore showed that one can reduce the runtime of computing the Fourier transform of the symmetric group S_N from O(| S_N |^2) to O(| S_N |log| S_N |)[where |S_N|=N!].

Given the unitarity of the Fourier transform, it is possible to implement the DFT and FFT as quantum algorithms. The simplest example is that of simultaneously applying a Hadamard gate to each qubit corresponds to the QFT over the group ℤ^⊗ n_2. The usual quantum Fourier transform (QFT) circuit over the abelian group ℤ_N essentially uses this idea with the chain of subgroups ℤ/2^nℤ⊃ℤ/2^n-1ℤ⊃…⊃ℤ/2ℤ⊃{1} <cit.>. Similar QFT circuits can be constructed over other non-abelian groups, as in the early work by Hoyer <cit.>, Beals <cit.>, and Pueschel et al. <cit.>. The term quantum Fourier transform (QFT) is sometimes used to refer only to the quantum implementation of the FFT for the group ℤ_2^n but in this work we will use QFT to refer to any algorithm for performing Fourier transformations over general finite groups, G. For the quantum circuits, we will use 𝔘^G_FT and 𝔘^G_FFT to represent the naive and fast Fourier transforms respectively. In this case we consider “naive" to mean explicitly building the unitary matrix corresponding to the Fourier transform and then compiling it using available software libraries, such as Qiskit <cit.>.

In this work we will construct 𝔘^G_FFT for specific discrete subgroups of SU(2) and SU(3) relying on the methods of <cit.>. We briefly review the general construction of fast fourier transforms in Sec. <ref>.  Sec. <ref> discusses the basic gates for qubits and qutrits we will use to construct 𝔘^G_FFT.  The next two sections cover specific 𝔘^G_FFTs. Sec. <ref> is devoted to the crystallike subgroups of SU(2), binary tetrahedral[It is worth noting that 𝔹𝕋 is isomorphic to the Clifford group] (𝔹𝕋) and binary octahedral (𝔹𝕆), and Sec. <ref> for the SU(3) subgroups Δ(27), Δ(54), and Σ(36×3). Comparison of using 𝔘^G_FFT vs 𝔘^G_FT in fiducial quantum simulations are presented in Sec. <ref>, following by concluding remarks and discussion of future research directions 
in Sec. <ref>. 



§ FAST FOURIER TRANSFORMATION ALGORITHM



Before beginning the discussion of the fast Fourier transformation it is crucial to define several technical terms that will be used throughout this work: left (right) transversal, inner conjugate, and left (right) regular representations.
Let G be a group, and H a subgroup. For any g ∈ G, the left (right) cosets of H in G are the sets obtained by left (right) multiplication by g. The left (right) transversal for a subgroup H of some group G is then the set of representative elements of each left (right) coset such that exactly one element of each coset appears in the transversal.
An example of this would be numbers { 1, i} form the right transversal of the group ℤ_2 (with elements { 1, -1}) in ℤ_4.

A subgroup H ⊂ G is said to be normal when conjugating any element of H by an element of G returns another element of H. In particular, this is true of any element t in the transversal,

    t^-1 h t = h', with  h,h'∈ H

The property of possessing a normal subgroup is a crucial feature of the method of Pueschel et al. <cit.>, which allows us to build the Fourier transform for the parent group using this method. An important operation in this regard is the inner conjugation by an element of the transversal t for a given representation, ϕ, of a subgroup H,

    ϕ^t(h) = ϕ(t^-1 h t).

Representations for which ϕ^t(h) = ϕ(h) for all t are said to be extendable. When building the Fourier transform for a group G from a normal subgroup H, one must build up the irreps of the group G from those of its subgroup H, and in the process, the extendable representations incur trivial phase factors.

The final operation we need to consider is the left regular representation. This left regular representation is a functional operator L(g'), which takes as input a group element, g, and performs the permutation:

    L(g')|g⟩ = |g'g⟩,

where g,g'∈ G. The right regular representation can be similarly defined. The regular representations are important operations as they must be block diagonalized by the QFT. In fact, Pueschel et al. <cit.> define a QFT as a unitary change of basis of the Hilbert space that block diagonalizes the regular representations.


Deriving 𝔘^G_FFT requires an algorithmic construction through a systematic extension of Fourier transforms along a series of nested subgroups <cit.>.There are numerous paths, given by the lattice of subgroups, one could take to develop FFT for non-abelian groups as shown in Fig. <ref> for the example of 𝔹𝕆. Regardless of the path through group space chosen, the general procedure is to find and construct a quantum circuit for a unitary operator which naturally extends the FFT from one finite group in the series to the next <cit.>.




While multiple methods for constructing these Fourier transforms exist, we will use the method 
proposed in Ref. <cit.>.
The algorithm works by appending a list of elements, denoted a right transversal, to an existing group whose QFT is known, e.g. ℤ_4, and then using the effects of the right transversal to extend the irreps of the subgroup to those of a parent group, e.g. ℚ_8 or ℤ_8.  The structure for these circuits is shown in Fig. <ref>.



In this method, the quantum circuit for a given Fourier transformation can be broken up into six pieces that form a recursive algorithm: the subgroup fast Fourier transform (FFT), two irrep permutations (P and C), a twiddle matrix (T), a cyclic fast Fourier transform over the transversal (DFT), and a phase-kickback operation (Φ).

The FFT operation is performed first on the quantum register corresponding to the subgroup.
This operation takes the group element basis into a mixed basis; the subgroup operations exist in the irrep basis while the transversal elements are still in the group element basis. This is the portion of the algorithm that allows for a recursive generation as each successive subgroup can be broken down in this way.

The P and C operations are operations that permute the subgroup irreps such that the book-keeping and quantum circuits are easily manipulated. P takes irreps and permutes them in such a way that they are identifiable for easy bookkeeping.
The C operation then takes irreps which are not invariant under conjugation by the transversal, i.e. ρ^a(t g t^-1) ≠ρ^a, and adjusts them so they are nested together.
This will be important later for the phase-kickback operation, Φ. 

The twiddle operation, T is an operation controlled on the transversal register that either mixes the conjugate sets to make a larger dimensional irrep or transforms an existing n-dimensional irrep under the explicit representation. This is immediately followed by DFT on the transversal register which mixes the irreps and ensures that the transversal component is moved to the irrep basis. The final step involves a phase kickback that applies certain phases to the given irreps so that they are the actual irreps and not an isomorphism. This is important as even though irreps are really only meaningful within the Fourier transform up to unitary equivalence, in the block diagonalized regular representation, the same irrep must appear as many times as its dimensionality along the block-diagonal. These pieces together form the recursive structure necessary to build a nonabelian quantum Fourier transform. 
    

In this work we construct 𝔘_FFT^G for five cases not currently provided in the literature.  These are 𝔹𝕋,𝔹𝕆 and Δ(27),Δ(54),Σ(36×3) which correspond to subgroups of SU(3) and SU(2) respectively.  The nested subgroup paths we will construct are:

    1→Z_3×Z_3 →Δ(27)→Δ(54)→Σ(36×3)
       1→Z_2 →Z_4 →Q_8 →BT→BO.

The 𝔘_FFT^G for 𝔹𝕋, 𝔹𝕆, and Σ(36×3) provide the resource cost reductions for quantum simulations of lattice gauge theories for these respective groups <cit.>.  





§ PRIMITIVE GATES


Throughout this work we consider two quantum architectures: the standard qubit-based and a heterogeneous qubit-qutrit one.  Our consideration of qubit-qutrit devices is motivated by the non-abelian groups studied here, which contain parts that are naturally trinary-valued. In this section, we discuss the primitive gates we will use to construct the FFTs.  

The qubit gates include the usual arbitrary U(2) rotations (including X,Y,Z), CNOT, and the 3-qubit Toffoli. While CNOT+U(2) forms a universal set <cit.>, we use Toffoli for compression of circuits in presenting them.  Ultimately, all these circuits will be decomposed into a fault-tolerant gate set. For most cases, we will decompose all the operations to the lowest possible form; the primary exception will be arbitrary two-qubit rotations for which efficient decompositions are readily available <cit.>. 

For qubit-qutrit gates, we use the set of gates provided in Ref. <cit.>. The one qutrit gates include the generalization of the Pauli gates to qutrits,

    X_a,b   = |a⟩⟨ b| + |b⟩⟨ a| + ∑_c≠ a,b |c⟩⟨ c|
    
        Z_a   = ∑_c=0^2 (1 - 2 δ_c, a) |c⟩⟨ c|,

and the qutrit equivalent of the Hadamard gate,

    H_3 = 1/√(3)[     1     1     1;     1   ω_3 ω_3^2;     1 ω_3^2   ω_3;       ],

where ω_3=e^2π i / 3. Similar to the qubit Hadamard, this corresponding to a FFT on ℤ_3. Another important single qutrit operation is implemented by the clock-shift gate and its inverse

    χ = [ 0 1 0; 0 0 1; 1 0 0;   ],    χ^-1 = [ 0 0 1; 1 0 0; 0 1 0;   ] .


The final qutrit gates are the T_3 and S_3=T_3^3 which are the generalizations of the T and S gates for qubits that are important for fault-tolerant compilation:

    T_3=Diag(1,ω_9,ω_9^8),   S_3=Diag(1,ω_3,ω_3^2)

where ω_n=e^2π i/n is the n-th root of unity.

For entangling gates, we consider the controlled sum (controlled increment) gate which performs modular addition on the qutrits and is denoted as a controlled χ. For each of these qutrit operators, we also desire a qubit decomposition. 

Because of the mismatch in dimensionality between qutrits and qubits, there is some freedom in how to embed a single qutrit in two qubits. We choose to assign states according to their decimal integer values, such that 

    |0⟩_3 ≡ |00⟩_2,     |1⟩_3 ≡ |01⟩_2 ,     |2⟩_3 ≡ |10⟩_2.

and the |11⟩_2 state is forbidden.  This embedding is nice because it allows for easy construction of the two-qubit unitary from the qutrit one by simply direct summing the qutrit unitary with 1_1.  For example, the qubit H_3 is



    H^ qubit_3 = H_3^ qutrit⊕1_1=1/√(3)[     1     1     1     0;     1   ω_3 ω_3^2     0;     1 ω_3^2   ω_3     0;     0     0     0  √(3) ].

This unitary can be decomposed using available transpilers such as Qiskit to 3 CNOTs and 14 R_z gates. We also denote an additional H_3 gate mapping for qubits which appears in the derivation of the 𝔹𝕋 circuits; this operator is

    (H_3^ qubit)' = 1⊕ H_3^ qutrit = 1/√(3)[  √(3)     0     0     0;     0     1     1     1;     0     1   ω_3 ω_3^2;     0     1 ω_3^2   ω_3 ].


The qubit version of the permutation gate χ defined in Eq. (<ref>) is provided in Fig. <ref>. If this operator needs to be controlled then one applies this operator once provided the product of the controls is 1 modulus 3, the inverse operator is applied if the product of the controls is 2 modulus 3, and left alone if the product of controls is 0 modulus 3. The most efficient way to do this for more than one control is to compute the modulo 1 and modulo 2 values into two ancillae and then apply χ or χ^† controlled on the respective ancilla. 



The two remaining circuit operators that need decomposition are the phase operation Z_2 and the permuation operator X_1,2. The phase operation is given by

    Z_2 = Diag(1,1,-1,1),
 with its circuit found in Fig. <ref>. The X_1,2 gate is

    X_1,2 = [ 1 0 0 0; 0 0 1 0; 0 1 0 0; 0 0 0 1;   ],

and is identifiable as the usual SWAP gate (See Fig. <ref>).

A final crucial component for transforming qutrit gates to qubit ones is how to implement control on the resulting multi-qubit states. In theory, this just involves on controlling on the states |00⟩_2, |01⟩_2, and |10⟩_2 respectively. However, because the |11⟩_2 state is forbidden, some control operations can be simplified. Specifically, for the |01⟩_2 and |10⟩_2 states, one only needs to control on the second or first register respectively because the |11⟩ state is never occupied. When necessary, we also use the qudit |d⟩ notation when considering pure matrix representations and the qubit |ab⟩ notation when mapping to quantum circuits. Now that we have cleared this ambiguity.

These rules are sufficient for decomposing the circuits provided in the main text. 





§ QFT FOR SUBGROUPS OF SU(2)


As discussed above, there are numerous paths to construct an FFT for 𝔹𝕋 and 𝔹𝕆.  We will pursue the path 

    1 →ℤ_2→ℤ_4 →ℚ_8 →𝔹𝕋→𝔹𝕆

as it correspond most closely to the results from <cit.>. 
The consequence of this choice will become apparent following the discussion of the group structures, their presentations in terms of generating elements, and irreps.


 §.§ Structure and irreps of 



Before delving into the Fourier transformations, it is important to list out the structure of the principal finite groups of interest for SU(2): ℚ_8, 𝔹𝕋, and 𝔹𝕆. 
Each of these groups forms a subgroup in a nested series which extend from one subgroup to the next via a right transversal.
The groups ℚ_8, 𝔹𝕋, and 𝔹𝕆 can respectively be represented with the following ordered products

    f =    (-1)^a 𝐣^b 𝐤^c 
    
    g =    (-1)^a 𝐣^b 𝐤^c 𝐮^d
    
     h =    (-1)^a𝐣^b𝐤^c𝐮^d𝐭^e.

The generators which appear in these above equations have the following matrix presentation in the faithful representation:

    𝐣   = [  0  1; -1  0;    ], 𝐤 = [  i  0;  0 -i;    ],
    𝐮   =[   -η   -η;  η^* -η^* ], 𝐭 = 1/√(2)[  1 -i; -i  1;    ]

where η = 1 + i/2, 0≤ a, b, c, e≤ 1, and 0≤ d ≤ 2.     

Additional details about 𝔹𝕋 and 𝔹𝕆 can be found in Refs. <cit.> however we quote here the important information regarding the irreps, denoted ξ^m for ℚ_8, ρ^m for 𝔹𝕋 and ρ̅^m for 𝔹𝕆. The irreps for all three groups are listed in Tabs. <ref>, <ref>, and <ref> with ω_3 = e^2π i / 3. 






    




 §.§ Fourier Transformation for 


 

The construction of 𝔘_FFT^𝔹𝕋 requires the identification of the six constituent gates: ℚ_8 FFT, P, C, T, DFT, and Φ.
The first, the FFT_ℚ_8 was derived in <cit.> and is presented in  Fig. <ref>. Given the choice of implementation for 𝒯_𝔹𝕋 that will be taken below, we will see that the C and P transformations are unnecessary.

We thus need to connect the irrep vectors of ℚ_8, |ξ^a⟩, with their corresponding computational basis states.
These states are explicitly

    |0⟩_𝐤|0⟩_𝐣|0⟩_-1 =  |ξ^1⟩,    |1⟩_𝐤|0⟩_𝐣|0⟩_-1 = |ξ^3⟩
     
    |0⟩_𝐤|1⟩_𝐣|0⟩_-1 =  |ξ^2⟩,  |1⟩_𝐤|1⟩_𝐣|0⟩_-1 = |ξ^4⟩
    { |a⟩_𝐤|b⟩_𝐣|1⟩_-1: 0≤ a,b≤ 1} =  |ξ^5⟩,

where the corresponding irreps are enumerated in Table <ref>. The only remaining step to generate the other gates is to identify the sets of inner conjugates. For 𝔹𝕋, we can identify the inner conjugacy transformations using the results from Refs. <cit.>:

    𝐮 (-1)^a 𝐣^b 𝐤^c 𝐮^2 =     (-1)^a 𝐣^c𝐤^b + c
    𝐮^2 (-1)^a 𝐣^b 𝐤^c 𝐮 =     (-1)^a 𝐣^b + c𝐤^b.

These transformation rules immediately tell us what the pairs of inner conjugates are; ξ^1 and ξ^5 are left invariant under this transformation, i.e. these irreps are extendable. 
Meanwhile, we can identify that (ξ^2)^𝐮(f) = ξ^4(f) and (ξ^2)^𝐮^2(f) = ξ^3(f). This implies that {ξ^2, ξ^4, ξ^3} form a set of inner conjugates and are not extendable representations.

The irreps {ξ^2, ξ^3, ξ^4} when combined with the transversal, { 1, 𝐮, 𝐮^2}, make the 3d irrep of 𝔹𝕋, ρ^7. 
This allows us to write down the twiddle operation T. This operation will end up being a controlled operation on the ℚ_8 register of the form 

    T_𝔹𝕋 = ∑_d=0^2 |d⟩⟨ d| ⊗𝒯_𝔹𝕋^d

where

    𝒯_𝔹𝕋 = 1_1⊕χ⊕ (𝐮_2⊗1_2).

The quantum circuit which implements this transformation is highlighted in green in Fig. <ref>. The first two Toffoli gates perform the cyclic permutations, while the controlled 𝐮 operation performs a controlled rotation on the |⟩_𝐣 register corresponding to the presentation of 𝐮 in the ρ^4 irrep show in Eq. <ref>.
 This leaves the implementation of Φ which can be broken down similar to T_𝔹𝕋 in Eq. (<ref>) as

    Φ_𝔹𝕋 = ∑_d=0^2 |d⟩⟨ d| ⊗ϕ_𝔹𝕋^d,

where

    ϕ_𝔹𝕋 = Diag(1,1, ω, ω^2, 1, 1, 1, 1).

The phase kickback here needs to be applied to the irreps which form a conjugate set, in this case ξ^2, ξ^3, ξ^4 but leaves the other irreps unchanged. 
The complete quantum Fourier transformation for  then strings these gates together and is shown in Fig. <ref> with the phase kickback highlighted in blue. It is important to note the following, the H_3' gates do not need to be controlled on the |⟩_𝐮 register since it forms a basis transformation taking the permutations of the Toffoli gates to phases. 





 §.§ Fourier Transformation for 


The construction of the Fourier transformation for  will follow that of , with the subgroup FFT being that of .  Next, we can identify the irreps after 𝔘_FFT^𝔹𝕋 with their corresponding basis vectors. 
The 1d irreps correspond to the basis vectors

    |0⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^1⟩
    
     |1⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^2⟩
    
     |2⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^3⟩.

The 2d irreps corresponding to sets of quadruplets

    { |0⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^4⟩
    { |1⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^5⟩
    { |2⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^6⟩
 
where 0≤ a,b≤ 1. 
Finally the 3d irrep will correspond to the nonaplet

    { |d⟩_𝐮 |1⟩_𝐤|0⟩_𝐣|0⟩_-1,    
    
           |d⟩_𝐮 |0⟩_𝐤|1⟩_𝐣|0⟩_-1,    
    
           |d⟩_𝐮 |1⟩_𝐤|1⟩_𝐣|0⟩_-1}    = |ρ^7⟩

for 0≤ d ≤ 2. We next identify the inner conjugates under 𝐭 from the expression

    𝐭 (-1)^a 𝐣^b 𝐤^c 𝐮^d 𝐭^-1 = 
    (-1)^a'𝐣^c + δ_d,1 + δ_d,2𝐤^b + δ_d, 1𝐮^2d

where a' is an unimportant polynomial of the c, b, δ_d,1, and δ_d,2. 
We note that the nontrivial effect of inner conjugation by 𝐭 is 𝐮↔𝐮^2.
It then follows that  

    (ρ^1)^𝐭 = ρ^1,   (ρ^4)^𝐭   = ρ^4,   (ρ^7)^𝐭 = ρ^7
    
    (ρ^2)^𝐭   = ρ^3
    
    (ρ^5)^𝐭   = ρ^6

and thus the inner conjugates are {ρ^2, ρ^3} and {ρ^5,ρ^6}. This implies that ρ^2 and ρ^3 will mix to create the 2d irrep of 𝔹𝕆, ρ̅_3. Meanwhile the 2d irreps ρ^5 and ρ^6 will combine to generate the 4d irrep ρ̅_8.

Everything is now in place to construct the twiddle operator, T_𝔹𝕆, and the phase kickback Φ_𝔹𝕆.
Similar to , we can write decompose the twiddle matrix

    T_𝔹𝕆 = ∑_e=0^1 |e⟩⟨ e| ⊗𝒯_𝔹𝕆^e.

The only remaining issue is to identify what the matrix 𝒯_𝔹𝕆 is. We can identify this by evaluating the effects on the basis states for 𝔹𝕋:


    𝒯_𝔹𝕆   = 1_1⊕ X⊕ (1_3⊗ [-(𝐮_2⊗1_1)])⊕
       [  1/√(2)       0       0       0       0       0 -i/√(2)       0       0       0       0       0;       0       0       0       0       0       0       0       0       0       0       1       0;       0       0       0       0       0       0       0       0       0       0       0       1;       0       0       0  1/√(2)       0       0       0       0       0 -i/√(2)       0       0;       0       0       0       0       0       0       0      -1       0       0       0       0;       0       0       0       0       0       0       0       0      -1       0       0       0; -i/√(2)       0       0       0       0       0  1/√(2)       0       0       0       0       0;       0       1       0       0       0       0       0       0       0       0       0       0;       0       0       1       0       0       0       0       0       0       0       0       0;       0       0       0 -i/√(2)       0       0       0       0       0  1/√(2)       0       0;       0       0       0       0       1       0       0       0       0       0       0       0;       0       0       0       0       0       1       0       0       0       0       0       0;         ]
 
This operator can be translated to a quantum circuit with a modicum of work into the one shown in the green highlighted portion of Fig. <ref>. In most cases the ancilla present in the quantum circuit will be unentangled either through running the Fourier transformation in reverse or by otherwise uncomputing the values they hold.

The only remaining component necessary is the phase-kickback

    Φ_𝔹𝕆 = ∑_e=0^1|e⟩⟨ e|⊗ϕ^d_𝔹𝕆

The phase kickback gets applied to the original states corresponding to |ρ^3⟩ and |ρ^6⟩ irreps.
This yields the operator 

    ϕ_𝔹𝕆 =    Diag(1, 1, -1) ⊕1_9⊕
       Diag(1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1).

We provide the full quantum Fourier transformation for 𝔹𝕆 in Fig. <ref>. This circuit includes 2 ancilla which are eventually uncomputed when the transformation is run in reverse. 





§ QFT FOR SUBGROUPS OF SU(3)


For the case of SU(3), there are two motivations for the particular groups we investigate. So-called flavour groups are important for model-building in particle physics to understand the mass splittings between different flavor generations, e.g. see Ref. <cit.>. Δ(3n^2) and Δ(6n^2) are prime examples of flavour groups. Other groups, for example Σ(ϕ×3), have found interest for simulations in quantum simulations for lattice gauge theorys in HEP <cit.>. For the work here, it is nice that Δ(3n^2) form normal subgroups of SU(3)



 §.§ Structure and irreps of 



The group elements, g ∈Δ(27), h ∈Δ(54), and f ∈Σ(3× 36), can be written as

    g    = ω_3^p 𝐂^q 𝐄^r
    
        h    = ω_3^p 𝐂^q 𝐄^r 𝐕^2s
    
        f    = ω_3^p 𝐂^q 𝐄^r 𝐕^2s + t

respectively, where 0≤ p, q, r ≤ 2 and 0≤ s, t≤ 1 <cit.>. 
Each of these subgroups is extended to a larger subgroup of SU(3) according to

    1→Z_3×Z_3 Δ(27)Δ(54)Σ(36×3)


where the elements above the arrows indicate the right transveral used to extend the group.
We also provide the irreps for Δ(27), Δ(54), and Σ(3× 36) are denoted ξ^a, ρ̅^a, and ρ^a respectively and provided in Tabs. <ref>, <ref>, and <ref>.
In addition, we also need to enumerate the irreps of ℤ_3×ℤ_3. We denote these irreps as χ^3a + b + 1 with 0≤ a, b≤ 2. These irreps have the following presentation
    
    χ^3a + b + 1(ω_3^p 𝐂^q) = e^2π i / 3 (a p + q b)

    where p and q correspond to the generators ω_3 and 𝐂 in Eqs. (<ref>) to (<ref>).
    

    
    
    
    


 §.§ Fourier transformation for 


We need to first identify the operator FFT which generates the Fourier transformation for ℤ_3×ℤ_3. 
This quantum circuit is simply the tensor product of a discrete abelian quantum Fourier transformation across each of the individual ℤ_3 groups.
This leaves us with the irreps corresponding to the following basis vectors:

    |00⟩ =    |χ^1⟩    |01⟩ =    |χ^2⟩    |02⟩ =    |χ^3⟩
    
        |10⟩ =    |χ^4⟩    |11⟩ =    |χ^5⟩   |12⟩ =    |χ^6⟩
    
        |20⟩ =    |χ^7⟩    |21⟩ =    |χ^8⟩    |22⟩ =    |χ^9⟩


Now we need to identify which irreps form the conjugate pairs from <cit.> and which are trivially extended.
This involves calculating what χ^3a + b + 1(𝐄^r ω_3^p 𝐂^q 𝐄^-r) is equal to. 
We find that {χ^1, χ^2, χ^3} are trivial, i.e., χ^m(𝐄^r ω_3^p 𝐂^q 𝐄^-r) = χ^m for m=1,2,3, while {χ^4, χ^5, χ^6} and {χ^7, χ^8, χ^9} form inner conjugates.
We can now write down the operator for T_Δ(27),

    T_Δ(27) = ∑_r=0^2 |r⟩⟨ r|⊗𝒯_Δ(27)^r

where,

    𝒯_Δ(27) = 1_3⊕χ^-1⊕χ

Additionally we can write down the phase kickback matrix,

    Φ_Δ(27) = ∑_r=0^r |r⟩⟨ r| ⊗ϕ_Δ(27)^r

with

    ϕ_Δ(27) = 1_3⊕Diag(1, ω_3^2, ω_3) ⊕Diag(1, ω_3, ω_3^2).
 
These pieces in turn give us the necessary unitaries to construct the 𝔘_FFT^Δ(27). The operator in Eq. (<ref>) is given by a inverted controlled sum gate, while the operator in Eq. (<ref>) is a inverted controlled phase gate. We provide the full quantum circuit for qutrits in left panel of Fig. <ref> with each of the components highlighted and labeled. 
After this Fourier transformation we can identify the computational basis states with their corresponding irreps:

    |000⟩ =     |ξ^1⟩,     |001⟩ =     |ξ^2⟩,     |002⟩ =     |ξ^3⟩,
    
        |010⟩ =     |ξ^4⟩,     |011⟩ =     |ξ^5⟩,     |012⟩ =     |ξ^6⟩,
    
        |020⟩ =     |ξ^7⟩,     |021⟩ =     |ξ^8⟩,    |022⟩ =     |ξ^9⟩,


    {|1ab⟩: 0≤ a,b≤ 2 } =     |ξ^10⟩
    {|2ab⟩: 0≤ a,b ≤ 2} =     |ξ^11⟩



While this qutrit decomposition for the fast Fourier transformation is the most straightforward to right down, it may not necessarily be the most efficient way as fault tolerant hardware is developed. In this context it is important to have a purely qubit based version of this fast Fourier transformation.




We provide the quantum circuit for 𝔘_FFT^Δ(27) using a homogenous qubit register in the right panel of Fig. <ref>. This circuit is the functional equivalent of the one provided in Fig. <ref>. Each of the three qutrits is broken down into a two qubit subregister. 

The main non-trivial component of the twiddle operation is how the multi-controlled shift operation is applied. We use the same rule as discussed in the previous section to map onto the first ancilla.
We compute the operators that correspond to a product equal to 2 modulo 3 on the first ancilla and a product equal to 1 modulo 3 on the second register. 
This reclaimable scratch register computation is done to minimizes the T-gate count for the quantum simulation. 
This operator ends up being repeated again for Φ_Δ(27). 
In total this circuit requires 24 Toffoli operations.  
The remaining T-gates come from the decomposition of the H_3 gates discussed in the prior subsection. As mentioned above each of these operations requires 14 R_z rotations and 3 transversal CNOTs. 
This gives us a total T-gate count of:

    168 + 80.5 log_2(1 / ϵ),

where ϵ is the target synthesis error. The circuit has a maximal T-gate width of 4 and requires 2 ancilla; which corresponds to a temporary 33% qubit memory overhead increase for the scratch registers. 







 §.§ Fourier transformation for 


As the derivation follows similarly to the previous ones, we will abbreviate it going further. We first need to identify the irreps of Δ(27) which form conjugate pairs.
We can identify this by observing how the operator 𝐕^2 transfroms an arbitrary element of Δ(27); we find that

    𝐕^2 ω_3^p 𝐂^q 𝐄^r 𝐕^2 = ω_3^p 𝐂^2q𝐄^2r.

This result implies that we have the following conjugate pairs from the Δ(27) irreps:

    {ξ^2, ξ^3}, {ξ^4, ξ^7}, {ξ^5, ξ^9}, {ξ^6, ξ^8}, {ξ^10, ξ^11}

These conjugate pairs allow us to identify the operator Φ_Δ(54) which is given by

    Φ_Δ(54) =    ∑_s=0^1 |s⟩⟨ s| ⊗
    (
       1_2 ⊕ -1 ⊕1_3 ⊕ -1_3⊕1_9⊕-1_9)^s.






The twiddle matrix, T_Δ(54), is 

    T_Δ(54) = ∑_s=0^1 |s⟩⟨ s|⊗𝒯_Δ(54)

where

    𝒯_Δ(54) =    X_1,2⊕(X⊗ X_1,2) ⊕1_3⊕
       (X⊗1_3)⊕1_3⊕(X⊗1_3).

The first two terms of matrix sum in 𝒯_Δ(54), correspond to the trivial irrep and the mixing of the other 1d irreps. The last terms in the sum are the mixing introduced by 𝐕^2 to the 3d irreps ξ^10 and ξ^11.

The total fast Fourier transformation is provided in on the left panel of Fig. <ref>. In this figure both the twiddle and phase kickback operations highlighted as well as the Fourier transformation across the transversal. We now proceed to the details of the qubit decomposition for this fast Fourier transformation. Because the 𝐕^2 register only requires a qubit, there is only the decomposition of the qutrit operations that are important. 

The decomposition to qubit gates is substantively simpler than in the target case for Δ(27). The controlled X_1,2 operations are mapped to controlled swaps while the Z_2 operations are mapped to controlled CZ gates. This decomposition using the results of the appendix provides us with the circuit in the right panel of Fig. <ref>. This circuit uses the same 2 ancilla as in the Δ(27) circuit. The decomposition requires 6 Toffoli gates for T_Δ(54) and 12 for Φ_Δ(54) giving a total of 126 T gates on top of the Δ(27) costs. This yields a total T gate cost of 

    294 + 80.5 log_2(1 / ϵ).

This requires 5 ancilla qubits, the majority of which occur in the phase kickback operation. 



 §.§ Fourier transformation for 

In order to progress from  Δ(54) to Σ(36×3), we need to construct a basis for the irreps of Δ(54).  We take the following elements:

    |ρ̅^1⟩   =  |0000⟩      |ρ̅^2⟩=|1000⟩
    
        |ρ̅^3⟩   ={ |0100⟩,|0200⟩, |0210⟩,|0020⟩}
    
        |ρ̅^4⟩   ={ |1100⟩,|1200⟩, |1210⟩,|1020⟩}
    
        |ρ̅^5⟩   ={ |0010⟩,|0020⟩,|0120⟩,|0220⟩}
    
        |ρ̅^6⟩   ={ |1000⟩,|1110⟩, |1120⟩,|1220⟩}
    
        |ρ̅^7⟩   ={ |0ab1⟩: 0≤ a,b ≤ 2}
    
        |ρ̅^8⟩   ={ |1ab1⟩: 0≤ a,b ≤ 2}
    
        |ρ̅^9⟩   ={ |0ab2⟩:  0≤ a,b ≤ 2}
    
        |ρ̅^10   =⟩{ |1ab2⟩:  0≤ a,b ≤ 2}.

We also need the following result from inner conjugation of Δ(54) by 𝐕:

    𝐕 (ω_3^p 𝐂^q 𝐄^r 𝐕^2s)𝐕^-1   = ω_3^p + 2qr𝐂^2r𝐄^q𝐕^2s.

This tells us that the inner conjugates are {ρ̅^3,ρ̅^4}, {ρ̅^5,ρ̅^6}, {ρ̅^7,ρ̅^8}, {ρ̅^9,ρ̅^10}
This immediately allows us to identify that shuffling and transformation between the states needs to occur according to the representations in Tab. <ref>. In total this gives us the twiddle operations

    T_Σ(36×3) = ∑_t=0^1 |t⟩⟨ t| ⊗𝒯_Σ(36× 3)

with

    𝒯_Σ(3× 36) = [ 1 0; 0 i;   ]⊕(1_4 ⊗χ_4)⊕𝒱⊕ (𝒱^*)

where χ_4 is the 4d generalization of the clock-shift operation

    χ_4=[ 0 1 0 0; 0 0 1 0; 0 0 0 1; 1 0 0 0;   ]

and

    𝒱 = 1_6⊗ (-iH_3).

Addtionally we can immediately identify the phase kickback operation as

    Φ_Σ(36× 3) = ∑_t |t⟩⟨ t| ⊗(⊕_j=0^27[  1  0;  0 -1 ])^t.

The quantum circuit using a mixed qutrit-qubit system is provided in Fig. <ref>. The pure qubit decomposition into a fault tolerant cirucit is provided in the remainder of this subsection.
The Σ(36×3) fast Fourier transformation shown in Fig. <ref> has the decomposition to a 8 qubit register as shown in Fig. <ref>. 
Unlike the other two circuits where the twiddle matrix and phase kick back were approximately equal in T-gate costs. The twiddle matrix is the most expensive portion of the Σ(36× 3) fast Fourier transformation. The first controlled S matrix requires 8 clean ancilla qubits for the scratch register. This scratch register bloat is only necessary at one small point for the controlled-S gate. The explicit C^nNOTs that appear in the circuit require 20 Toffoli operations in total. There is an implicit set of 14 Toffoli gates within the controlled-U operation which is shown explicitly in Fig. <ref>.  

The final place where T-gate synthesis presents is in the controlled-V and controlled-V̅ case which follows from the derivation of Ref. <cit.>. The first step is to diagonalize the V operator, which will require 32 R_z gates when dressing both sides of the diagonalizing operator. When diagonalized, the operator V has in the qubit space, the diagonalized matrix,

    V_diag. = Diag(-i,i,1,1),

while V̅ has the diagonalized form

    V̅_ diag. = Diag(i,-i,1,1).

These controlled operations together form a diagonal matrix which can be implementation with 4 R_z and 6 CNOT gates using the methods of Ref. <cit.>. Therefore, in total the controlled V and V̅ incured a cost of 36 R_z gates and no Toffoli gate. 
This all together brings the total number of additional T-gates on top of the Δ(54) gates to 238 + 36.8 log_2(1 / ϵ) for a total T-gate count of

    532 + 117.3 log_2(1 / ϵ).







        
    


§ OUTLOOK FOR QUANTUM SIMULATIONS FOR QFTS


The results from this work allow us to make both concrete predictions for existing groups as well as extrapolations about the costs of simulations for various other nonabelian groups of import. The cost comparison for various decompositions are provided in Tab. <ref>.

For one method of quantum simulation <cit.>, there are four primitive gates that must be constructed: the inverse, multiplication, trace and Fourier tranform gate;  which the Fourier transform dominating the resource costs. The main cost for quantum simulations involves mapping a total target synthesis error ϵ̅_T^I to a total number of T-gates N_T^fid. This comes from a target cost function C_T^I. Greater details about these methods and the finer nuances of these calculations can be found in Refs. <cit.>. 
The T-gate cost provided in this work can be considered quasi-optimal and an approximate lower bound for the total resources necessary for implementation.  For the groups 𝔹𝕋,𝔹𝕆, and Σ36×3, the total simulation cost are presented in Tab. <ref>.

This reduction in cost for the 𝔹𝕋 Fourier transformation still leaves it as the most costly operation but nevertheless reduces the overall simulations costs for pure gauge viscosity simulations proposed in Ref. <cit.> by an additional order of magnitude and by 11 orders of magnitude compared to Ref. <cit.>. 
Meanwhile 𝔘_FFT^𝔹𝕆 is nearly two orders of magnitude cheaper than 𝔘_FT^𝔹𝕆. This brings the cost down for Fourier transformation to be on par with the trace and multiplication gates in terms of T-gate cost, bringing the total simulation cost down by 2 orders of magnitude. Even greater reductions are found with 𝔘_FFT^Σ(36×3) which reduces the cost of the gate by a factor of nearly three orders of magnitude at the expense of doubling the qubit overhead; resulting in a total reduction in simulation cost by a factor of 580. 









§ DISCUSSION


In this work, we have derived fast quantum Fourier transforms for a number of nonabelian groups of interest in high energy physics and beyond. A potentially interesting next step relevant to a wide variety of systems would be the exploring applications and tradeoffs involving qudits or mixed qubit and qudit systems. For example the symmetric and alternating groups, 𝒮_n and 𝒜_n, accelerate in size quite quickly with the number of elements being n! and n!/2 where n is the number of elements being “permuted". This factorial scaling indicates certain qubit memory may be wasted. For example with the symmetric group of the amount of “wasted" states can be between 20-45% for n ≤ 20. This might demonstrate that there are hardware advantages to the use of qudit based systems for these sorts of problems.

The general structure of these fast Fourier transformations for smaller groups is paramount for the construction of fast Fourier transformations for larger ones. Hence results of this paper are applicable to further (super)groups in the same way. In particular the results of Σ(36×3) will be crucially important for the groups, Σ(72×3), Σ(216×3), and Σ(360×3), which are crucial for simulations of quantum chromodynamics. Additionally, it is important to note that our formulation of the quantum Fourier transform relied on a particular encoding of the discrete groups, and therefore modifications may be required for other encodings <cit.>.  For quantum simulations of lattice gauge theories which use other digitizations, the implementation of fast Fourier transforms requires more investigation.

It worth also noting that the notions of approximations to the standard ℤ_n quantum Fourier transformation exist <cit.>. An fascinating direction is to develop analogous constructions for so-called approximate quantum Fourier transformations for nonabelian groups, as well as identify potential applications where such approximate implementations yield potential advantages over exact implementations. 

One may wish to further explore applications of novel insights related to nonabelian QFTs to the HSP. Various papers have considered quantum algorithms for the nonabelian HSP, see Refs. <cit.>,  and this work has demonstrated explicitly how to constructions the fast Fourier transformation for nontrivial example groups; a crucial step forward toward efficient algorithms for HSP. The research performed in this work demonstrates that nonabelian fast Fourier transformations can be constructed efficiently for fault tolerant qubit based hardware and can offer orders-of-magnitude reduction in T-gate cost for quantum simulation. 

    The authors would like to thank Namit Anand, Aaron Lott, and Norman Tubman for helpful comments and suggestions. We are grateful for support from NASA Ames Research Center. Erik Gustafson, Stuart Hadfield, and M. Sohaib Alam were supported by the NASA Academic Mission Services, Contract No. NNA16BD14C and NASA-DOE interagency agreement SAA2-403602, and through the Superconducting Quantum and Materials System Center (SQMS) under contract number DE-AC02-07CH11359. This material is based on work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center 
    under contract number DE-AC02-07CH11359 (Henry Lamm). Edison Murairi is supported under the grant number DE-FG02-95ER40907. Fermilab is operated by Fermi Research Alliance, LLC under contract number DE-AC02-07CH11359 with the United States Department of Energy.






