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REVIEW 3 major objections 4 minor 73 references

Entanglement entropy dynamics of non-Gaussian states in free boson systems: Random sampling approach

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A random sampling shortcut evaluates the matrix permanent with about $2^{0.2N}$ samples instead of $2^N$, putting 100-site Rényi entropy dynamics in reach.

desk verdict Useful empirical methods paper; the 2^{0.2N} cost claim is real but extrapolated, and the paper slightly oversells the system-size reach. read the letter →

arxiv 2411.10085 v2 pith:WQJKQ22X submitted 2024-11-15 quant-ph cond-mat.quant-gas

Ryui Kaneko , Daichi Kagamihara , Ippei Danshita This is my paper · ORCID
classification quant-phcond-mat.quant-gas PACS 03.67.Mn02.70.Uu05.30.Jp
keywords matrixpermanentRényientanglemententropyfreebosonsrandomsamplingquantumquenchcharge-density-wavestatedynamicscomputationalcost
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a random sampling estimator for the matrix permanent that appears in the second Rényi entanglement entropy of free bosons after a quench from a charge-density-wave state. Because this insulating initial state is non-Gaussian, the entropy formula reduces to a permanent of size $N_{\mathrm{s}}$, whose exact evaluation costs $\mathcal{O}(2^{N_{\mathrm{s}}})$ terms; the authors show numerically that a Glynn-estimator sample mean needs only $\mathcal{O}(2^{\alpha N_{\mathrm{s}}})$ samples with $\alpha \approx 0.2$ in both one and two dimensions. This brings system sizes of $N_{\mathrm{s}}=100$ in 1D and $10\times10$ in 2D within reach, and the 1D curve at $N_{\mathrm{s}}=40$ matches the exact permanent result. The method is still exponential, but the much slower growth makes entanglement-entropy dynamics in free boson systems tractable beyond the previous few-tens-of-sites limit.

What carries the argument

The load-bearing object is the Glynn estimator, $\mathrm{Gly}_{\mathbf{r}}(A) = \prod_{k=1}^{n} r_k^* \prod_{i=1}^{n} \sum_{j=1}^{n} a_{ij} r_j$, evaluated on independent random phases $r_j = e^{i\theta_j}$ with $\theta_j$ uniform in $[0,2\pi)$. Its expectation equals the permanent because only permutation pairings survive the random-phase average, so the exact $2^n$-term permanent sum is replaced by a sample mean over $N_{\mathrm{total}}$ vectors. For the physical matrix $A$ in Eq. (14), built from single-particle correlations after the quench, sample values have positive and negative parts that nearly cancel at long times, and the required sample count is controlled by the fitted coefficient $c(N_{\mathrm{s}})$; the paper's numerical finding is that this coefficient grows like $2^{0.2N_{\mathrm{s}}}$ rather than $2^{N_{\mathrm{s}}}$.

What would settle it

Compute the coefficient $c(N_{\mathrm{s}})$ from the empirical variance of the Glynn estimator at sizes beyond the fitted range ($N_{\mathrm{s}}\ge80$ in 1D and larger 2D systems) at the same long-time points, fixing the target with an independent high-accuracy estimate, and check whether $\log_2 c(N_{\mathrm{s}})$ keeps the slope $\alpha\approx0.2$. Also rerun the $N_{\mathrm{s}}=100$ case with the paper's nominal $N_{\mathrm{total}}=2^{0.2N_{\mathrm{s}}+12}$ samples and compare the observed block-bootstrap error with the target; a visibly larger error means the fitted scaling law underestimates the sampling cost.

Watch

Extended reading notes

Core claim

The central claim is that the sample-count constant in the statistical error $\sigma_{S_2}/N_{\mathrm{s}} = \sqrt{c(N_{\mathrm{s}})/N_{\mathrm{total}}}$ obeys $c(N_{\mathrm{s}}) = 2^{\alpha N_{\mathrm{s}} + \mathrm{const.}}$ with $\alpha_{\mathrm{1D}} = 0.219(6)$ and $\alpha_{\mathrm{2D}} = 0.20(8)$ in the fitted size range. Equivalently, the random sampling estimator evaluates the permanent $\mathrm{perm}\, A$ of Eq. (14) with about $2^{0.2N_{\mathrm{s}}}$ samples instead of the $2^{N_{\mathrm{s}}}$ terms required by the exact permanent formulas. The authors demonstrate this at time $tJ = 2N_{\mathrm{s}}$ in 1D and $tJ = 2L_x$ in 2D, and they use the resulting speedup to compute Rényi entropy dynamics up to $N_{\mathrm{s}}=100$ in 1D and $10\times10$ in 2D, with the $N_{\mathrm{s}}=40$ 1D curve reproducing the exact permanent computation.

Load-bearing premise

The load-bearing premise is that the fitted scaling laws $c_{\mathrm{1D}}(N_{\mathrm{s}}) = 2^{0.219 N_{\mathrm{s}} - 8.8}$ and $c_{\mathrm{2D}}(N_{\mathrm{s}}) = 2^{0.20 N_{\mathrm{s}} - 13}$ keep holding at larger sizes and all times, although they come from a limited fit and no variance bound, so if the true exponent grows, the claimed efficiency and the 100-site curves fail.

Editorial extensions

If this is right

  • At $\alpha \approx 0.2$, the sampling cost $2^{0.2N_{\mathrm{s}}}$ grows far more slowly than the $2^{N_{\mathrm{s}}}$ exact-permanent cost, so sizes of order 100 sites are reachable on a single CPU core, with the $10\times10$ 2D case taking under a day.
  • In 1D, entropy density curves for $N_{\mathrm{s}} \ge 40$ overlap within error bars after the quench, indicating volume-law scaling and showing that the previous exact $N_{\mathrm{s}}=40$ limit already captured the thermodynamic-limit behavior.
  • In 2D, the entropy density grows linearly for $tJ \lesssim 0.3\sqrt{N_{\mathrm{s}}}$, consistent with earlier results for Gaussian initial states, and then develops size dependence while approaching a value near $0.3$ in both dimensions.
  • The procedure applies to any initial state that is a product of local Fock states and to any noninteracting hopping Hamiltonian, because only the single-particle eigenfunctions enter the correlation matrix.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the empirical $\alpha\approx0.2$ scaling holds when tested at larger sizes, the same random-phase Glynn estimator could cheapen other permanent evaluations with structured matrices, such as those in boson sampling, but the paper does not claim this.
  • The paper's conjecture that $\alpha$ is set by the entanglement entropy per particle predicts a testable relation: estimating the sampling cost for other Fock initial states with different per-particle entropy should show $\alpha$ shrinking as the entropy density drops.
  • The chain of inequalities in Sec. IV defines cheaper entropy-like quantities that bound $S_2$ from below; using those quantities as cross-checks or preconditioners at large $N_{\mathrm{s}}$ could reveal whether the sign-problem-like cancellation, rather than raw variance, is what the method is really fighting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a random-sampling Monte Carlo estimator for the matrix permanent, based on the Glynn formula, and applies it to compute the second Rényi entanglement entropy dynamics after a quantum quench from a CDW state in free boson systems. The estimator identity E[Gly_r(A)] = perm A is proved in Eqs. (22)-(28). The authors estimate statistical errors by blocking and bootstrap, and find numerically that the variance constant c(Ns) grows as 2^{alpha Ns - beta} with alpha_1D = 0.219(6) and alpha_2D = 0.20(8), much smaller than the 2^{Ns} cost of exact permanent evaluation. Using Ntotal = 2^{0.2 Ns + 12} samples, they compute entropy dynamics for Ns up to 100 in 1D and 10x10 in 2D, and validate the 1D Ns=40 curve against an exact permanent calculation.

Significance. If the empirical cost scaling holds, the method is a genuinely useful numerical tool: it extends tractable system sizes for non-Gaussian bosonic entanglement dynamics from a few tens of sites to about a hundred sites, and the exact Ns=40 benchmark in Fig. 6(a) gives non-trivial validation of the estimator. The paper is also careful in its use of blocking and bootstrap to estimate statistical errors, and it explicitly connects the sampling difficulty to the negative-sign-problem analogy. However, the central claim of an O(2^{0.2 Ns}) sampling cost rests entirely on an empirical fit over a limited size range and a single time point, with no accompanying variance bound or large-N validation. The 2D exponent carries a large uncertainty (0.20 +/- 0.08), so the generality of the alpha ~ 0.2 conclusion is not firmly established.

major comments (3)
  1. [Sec. III B, Eqs. (35) and (40); Sec. III C] The extrapolation of the fitted scaling c(Ns) = 2^{alpha Ns - beta} to larger system sizes and to all times is load-bearing but unsupported. The fits are made at the single time tJ = 2Ns (1D) and tJ = 2Lx (2D), over Ns <= 60 (1D) and Ns <= 120 (2D), and the 2D exponent is only 0.20 +/- 0.08. The choice Ntotal = 2^{0.2 Ns + 12} in Sec. III C, which underlies the Ns = 100 and 10x10 entropy curves, is valid only if the same alpha applies at those sizes and at all plotted times. Since no variance bound is derived and no check at Ns > 60 (1D) or Ns > 120 (2D) is given, I ask the authors to provide additional evidence: for example, compute c(Ns) at intermediate un-fitted sizes (e.g., Ns = 80 in 1D), or test the stability of alpha under deletion of the smallest fitted points, or give a heuristic variance estimate based on the distribution of Re p(m). Without such a check, the efficiency gain and the error bars for the largest systems are conditional on an untested premise.
  2. [Sec. III A, Figs. 2(b) and 3(b); Sec. III C] At long times the estimator is rare-event dominated: for Ns = 40 and tJ = 2Ns, Fig. 2(b) shows that typical |Re p(m)| ~ e^{-16} while the mean perm A is ~ e^{-12}, so the expectation value is controlled by rare large samples. The blocking/bootstrap procedure with fixed Nblock = 2^10 estimates the standard error from 2^10 block averages. For Ns = 100, where Ntotal = 2^{32}, each block contains 2^22 samples, but the bootstrap then resamples only 2^10 block means; the authors do not demonstrate that this captures the heavy tail. I request a diagnostic: compare the bootstrap error with the direct empirical standard error of the sample mean, and vary Nblock (e.g., 2^6, 2^8, 2^10, 2^12) at the largest sizes to show that the estimated sigma is stable. Without this, the error bars in Figs. 6(b) and 7 may be underestimated.
  3. [Sec. III C, Fig. 6(a)] The exact match at Ns = 40 validates the estimator identity and the error-estimation procedure for that size and over the whole time range, but it does not validate the cost scaling law c(Ns) = 2^{alpha Ns - beta} at larger Ns. The manuscript would be strengthened by a benchmark at an intermediate un-fitted size (e.g., Ns = 60 or 80 in 1D) against a more expensive but reliable computation (for instance, a larger exact permanent, or a random-sampling run with substantially more samples), to confirm that the Ntotal chosen from the fitted alpha indeed yields the claimed statistical error. This is important because the paper's headline result is the extension to 'more than 100 sites'.
minor comments (4)
  1. [Eq. (34) and Fig. 4 caption] The caption states that c1D(Ns) 'represents the number of samples required to achieve a given statistical error', but c is defined as the variance of the entropy density, so Ntotal = c / (sigma_S2/Ns)^2 up to a factor. Please clarify the relation between c and the required sample number in the text.
  2. [Figs. 1 and 2] The notation '□ ln[|Im p(m)|]' and similar appears with placeholder square symbols. Please replace with the intended mathematical symbols (e.g., -ln[|Im p(m)|]).
  3. [Sec. II, sentence around Ref. [52]] The statement that 'the equivalent sampling procedure itself is proposed in Ref. [52]' may overstate the novelty; the Glynn estimator with random phases is a standard identity (see Refs. [47,53-55]). Please rephrase to indicate that Ref. [52] applies the same idea in a different context, rather than proposing the estimator.
  4. [Abstract and Introduction] The abstract says the method allows calculations 'for more than 100 sites', but the largest 1D system shown is Ns = 100 and the 2D system is 10x10 = 100. Please either provide results for Ns > 100 or rephrase to 'up to 100 sites'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the random-sampling estimator is proven unbiased in Eqs. (22)-(28), the alpha ≈ 0.2 exponent is an empirical fit to statistical errors rather than an input, and the Ns = 40 curve is cross-checked against a brute-force permanent calculation.

full rationale

The paper's derivation chain is self-contained for its central claim. The estimator identity perm A = E[Gly_r(A)] is proven in Eqs. (22)-(28) by direct expansion, not assumed. The permanent formula for the Rényi entropy, Eq. (14), is taken from the authors' previous work [33], but that citation supplies a published mathematical expression and is not used to forbid alternatives; the new contribution is the sampling estimator, whose cost law is measured. The scaling law c(Ns) = 2^(alpha Ns - beta) (Eqs. (35) and (40)) is obtained by fitting statistical errors of S2/Ns versus Ntotal from independent simulations; it is not defined in terms of the final entropy values that are later reported. The later entropy curves use Ntotal = 2^(0.2 Ns + 12) as a sample budget, but the entropy values themselves are unbiased sample means, not numbers read off the fit. The Ns = 40 comparison against the exact permanent result provides an external-in-algorithm check via the brute-force permanent from [33]. Thus no load-bearing step reduces to its own inputs. Self-citations to [33] are present but non-load-bearing and independently checkable. The extrapolation of the fitted alpha to Ns = 100 and to all times is a statistical or correctness risk, not a circularity, because the paper does not claim the scaling is derived from first principles.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central efficiency claim rests on four fitted scaling parameters, two hand-chosen sampling parameters, and an extrapolated empirical scaling law. The physics input is standard free-boson evolution plus the authors' prior permanent formula. No new physical entities are introduced.

free parameters (6)
  • alpha_1D = 0.219(6)
    Exponent in c1D(Ns) = 2^(alpha_1D Ns - beta_1D), Eq. (36); central to the claimed O(2^(alpha Ns)) cost.
  • beta_1D = 8.8(3)
    Intercept in the c1D fit, Eq. (37); affects absolute sample count but not the exponential rate.
  • alpha_2D = 0.20(8)
    Exponent in c2D(Ns) = 2^(alpha_2D Ns - beta_2D), Eq. (41); large relative uncertainty.
  • beta_2D = 13(4)
    Intercept in the c2D fit, Eq. (42).
  • Nblock = 2^10
    Manual choice for blocking analysis in Sec. III A; influences bootstrap error estimation.
  • Nboot = 2^12
    Manual choice for bootstrap resampling count in Sec. III A.
assumptions (5)
  • domain assumption Renyi-2 entropy of the quenched CDW state equals -ln perm A with A in Eq. (14) constructed from Z in Eq. (12).
    Taken from the authors' previous work, Ref. [33]; used without re-derivation in this paper.
  • standard math Time evolution of free-boson operators is obtained by diagonalizing H0 and propagating single-particle wavefunctions, Eqs. (6)-(11).
    Standard free-boson second quantization; exact for U = 0.
  • standard math The Glynn estimator over i.i.d. unit-modulus complex random variables is an unbiased estimator of the permanent, Eqs. (22)-(28).
    Proved in the text from orthogonality of phase factors.
  • domain assumption Block averages in the bootstrap are effectively independent so the standard error estimate is reliable.
    No autocorrelation or block-length sensitivity analysis is provided; central to the reported error bars.
  • ad hoc to paper The fitted scaling c(Ns) = 2^(alpha Ns - beta) for Ns >= 40 (1D) and Ns > 40 (2D) continues to hold at larger Ns and at the times used for dynamics.
    This extrapolation underlies the Ns = 100 results; no proof or test against exact results above Ns = 40.

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Pith. "Pith review of Entanglement entropy dynamics of non-Gaussian states in free boson systems: Random sampling approach." pith.science (2026). https://pith.science/paper/WQJKQ22X

@misc{pith2026241110085,
  author       = {Pith},
  title        = {Pith review of: Entanglement entropy dynamics of non-Gaussian states in free boson systems: Random sampling approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQJKQ22X}},
  note         = {Machine review of arXiv:2411.10085}
}
abstract

We develop a random sampling method for calculating the time evolution of the R\'{e}nyi entanglement entropy after a quantum quench from an insulating state in free boson systems. Because of the non-Gaussian nature of the initial state, calculating the R\'{e}nyi entanglement entropy calls for the exponential cost of computing a matrix permanent. We numerically demonstrate that a simple random sampling method reduces the computational cost of a permanent; for an $N_{\mathrm{s}}\times N_{\mathrm{s}}$ matrix corresponding to $N_{\mathrm{s}}$ sites at half filling, the sampling cost becomes $\mathcal{O}(2^{\alpha N_{\mathrm{s}}})$ with a constant $\alpha\ll 1$, in contrast to the conventional algorithm with the $\mathcal{O}(2^{N_{\mathrm{s}}})$ number of summations requiring the exponential time cost. Although the computational cost is still exponential, this improvement allows us to obtain the entanglement entropy dynamics in free boson systems for more than $100$ sites. We present several examples of the entanglement entropy dynamics in low-dimensional free boson systems.

Figures

Figures reproduced from arXiv: 2411.10085 by the authors.

Figure 1
Figure 1. FIG. 1. Distribution of the value [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Distribution of the value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (a) Size dependence of the standard error of Rényi entan [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (2 more)
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Comparison with the exact result of the time evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Time evolution of the Rényi entanglement entropy density in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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Reference graph

Works this paper leans on

73 extracted references · 59 canonical work pages

  1. [52]

    Huh, Phys

    J. Huh, Phys. Rev. A111, 012418 (2025)

    work page 2025

  2. [1]

    Heyl, Rep

    M. Heyl, Rep. Prog. Phys.81, 054001 (2018)

    2018

  3. [2]

    K.Życzkowski,P.Horodecki,M.Horodecki,andR.Horodecki, Phys. Rev. A65, 012101 (2001)

    work page 2001

  4. [3]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2010)

    2010

  5. [4]

    M.Greiner,O.Mandel,T.Esslinger,T.W.Hänsch,andI.Bloch, Nature 415, 39 (2002)

    work page 2002

  6. [5]

    Calabrese and J

    P. Calabrese and J. Cardy, J. Stat. Mech.2005, P04010 (2005)

    2005

  7. [6]

    G.D.Chiara,S.Montangero,P.Calabrese,andR.Fazio,J.Stat. Mech. 2006, P03001 (2006)

    work page 2006

  8. [7]

    Cramer, C

    M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Phys. Rev. Lett.100, 030602 (2008)

    work page 2008

Show all 73 references
  1. [8]

    Cramer, A

    M. Cramer, A. Flesch, I. P. McCulloch, U. Schollwöck, and J. Eisert, Phys. Rev. Lett.101, 063001 (2008)

    2008

  2. [9]

    Flesch, M

    A. Flesch, M. Cramer, I. P. McCulloch, U. Schollwöck, and J. Eisert, Phys. Rev. A78, 033608 (2008)

    2008

  3. [10]

    Fagotti and P

    M. Fagotti and P. Calabrese, Phys. Rev. A78, 010306 (2008)

    2008

  4. [11]

    A. M. Läuchli and C. Kollath, J. Stat. Mech.2008, P05018 (2008)

    2008

  5. [12]

    82,277 10 (2010)

    J.Eisert,M.Cramer,andM.B.Plenio,Rev.Mod.Phys. 82,277 10 (2010)

    2010

  6. [13]

    A 85, 053625 (2012)

    P.Barmettler,D.Poletti,M.Cheneau,andC.Kollath,Phys.Rev. A 85, 053625 (2012)

    2012

  7. [14]

    J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012)

    2012

  8. [15]

    Cheneau, P

    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T.Fukuhara,C.Gross,I.Bloch,C.Kollath,andS.Kuhr,Nature 481, 484 (2012)

    2012

  9. [16]

    Trotzky, Y.-A

    S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Scholl- wöck, J. Eisert, and I. Bloch, Nat. Phys.8, 325 (2012)

    2012

  10. [17]

    Carleo, F

    G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M. Fabrizio, Phys. Rev. A89, 031602 (2014)

    2014

  11. [18]

    A.Bauer,F.Dorfner,andF.Heidrich-Meisner,Phys.Rev.A 91, 053628 (2015)

    2015

  12. [19]

    Frérot and T

    I. Frérot and T. Roscilde, Phys. Rev. B92, 115129 (2015)

    2015

  13. [20]

    Frérot and T

    I. Frérot and T. Roscilde, Phys. Rev. Lett.116, 190401 (2016)

    2016

  14. [21]

    Laflorencie, Phys

    N. Laflorencie, Phys. Rep.646, 1 (2016)

    2016

  15. [22]

    Alba and P

    V. Alba and P. Calabrese, Proc. Natl. Acad. Sci. U.S.A.114, 7947 (2017)

    2017

  16. [23]

    Alba and P

    V. Alba and P. Calabrese, SciPost Phys.4, 017 (2018)

    2018

  17. [24]

    Goto and I

    S. Goto and I. Danshita, Phys. Rev. B99, 054307 (2019)

    2019

  18. [25]

    K.Nagao,M.Kunimi,Y.Takasu,Y.Takahashi,andI.Danshita, Phys. Rev. A99, 023622 (2019)

    2019

  19. [26]

    Yao and J

    R. Yao and J. Zakrzewski, Phys. Rev. B102, 104203 (2020)

    2020

  20. [27]

    Takasu, T

    Y. Takasu, T. Yagami, H. Asaka, Y. Fukushima, K. Nagao, S. Goto, I. Danshita, and Y. Takahashi, Sci. Adv.6, eaba9255 (2020)

    2020

  21. [28]

    Kunimi and I

    M. Kunimi and I. Danshita, Phys. Rev. A104, 043322 (2021)

    2021

  22. [29]

    Nagao, Y

    K. Nagao, Y. Takasu, Y. Takahashi, and I. Danshita, Phys. Rev. Res. 3, 043091 (2021)

    2021

  23. [30]

    91, 054601 (2022)

    R.Yoshii,S.Yamashika,andS.Tsuchiya,J.Phys.Soc.Jpn. 91, 054601 (2022)

    2022

  24. [31]

    Rylands, B

    C. Rylands, B. Bertini, and P. Calabrese, J. Stat. Mech.2022, 103103 (2022)

    2022

  25. [32]

    Kaneko and I

    R. Kaneko and I. Danshita, Commun. Phys.5, 65 (2022)

    2022

  26. [33]

    Kagamihara, R

    D. Kagamihara, R. Kaneko, S. Yamashika, K. Sugiyama, R.Yoshii,S.Tsuchiya,andI.Danshita,Phys.Rev.A 107,033305 (2023)

    2023

  27. [34]

    S.Yamashika,D.Kagamihara,R.Yoshii,andS.Tsuchiya,Phys. Rev. Res.5, 043102 (2023)

    2023

  28. [35]

    Yamashika, P

    S. Yamashika, P. Calabrese, and F. Ares, arXiv:2410.14299

    arXiv

  29. [36]

    D. A. Abanin and E. Demler, Phys. Rev. Lett.109, 020504 (2012)

    2012

  30. [37]

    A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Phys. Rev. Lett.109, 020505 (2012)

    2012

  31. [38]

    A.Elben,B.Vermersch,M.Dalmonte,J.I.Cirac,andP.Zoller, Phys. Rev. Lett.120, 050406 (2018)

    2018

  32. [39]

    Greiner, Nature528, 77 (2015)

    R.Islam,R.Ma,P.M.Preiss,M.EricTai,A.Lukin,M.Rispoli, and M. Greiner, Nature528, 77 (2015)

    2015

  33. [40]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science353, 794 (2016)

    2016

  34. [41]

    Lanyon, P

    T.Brydges,A.Elben,P.Jurcevic,B.Vermersch,C.Maier,B.P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Science364, 260 (2019)

    2019

  35. [42]

    Watrous, Quantum computational complexity, inEncyclope- dia of Complexity and Systems Science, edited by R

    J. Watrous, Quantum computational complexity, inEncyclope- dia of Complexity and Systems Science, edited by R. A. Meyers (Springer New York, New York, NY, 2009) pp. 7174–7201

    2009

  36. [43]

    M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett.104, 157201 (2010)

    2010

  37. [44]

    Zhao, B.-B

    J. Zhao, B.-B. Chen, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, npj Quantum Materials7, 69 (2022)

    2022

  38. [45]

    H. J. Ryser,Combinatorial mathematics, Vol. 14 (American Mathematical Soc., 1963)

    1963

  39. [46]

    R. A. Brualdi and H. J. Ryser,Combinatorial matrix theory (Cambridge University Press, 1991)

    1991

  40. [47]

    D. G. Glynn, Eur. J. Combinat.31, 1887 (2010)

    2010

  41. [48]

    thesis, Indian Statistical Institute-Kolkata (1980)

    K.Balasubramanian, Combinatoricsanddiagonalsofmatrices , Ph.D. thesis, Indian Statistical Institute-Kolkata (1980)

    1980

  42. [49]

    Bax and J

    E. Bax and J. Franklin, CalTech-CS-TR-96-04 (1996)

    1996

  43. [50]

    E. T. Bax,Finite-difference algorithms for counting problems (California Institute of Technology, 1998)

    1998

  44. [51]

    D. G. Glynn, Des. Codes Cryptogr.68, 39 (2013)

    2013

  45. [53]

    Gurvits, inMathematical Foundations of Computer Science 2005, edited by J

    L. Gurvits, inMathematical Foundations of Computer Science 2005, edited by J. Je ¸drzejowicz and A. Szepietowski (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 447–458

    2005

  46. [54]

    Aaronson and T

    S. Aaronson and T. Hance, Quantum Inf. Comput.14, 541 (2014)

    2014

  47. [55]

    Berkowitz and P

    R. Berkowitz and P. Devlin, Isr. J. Math.224, 437 (2018)

    2018

  48. [56]

    Kocharovsky, V

    V. Kocharovsky, V. Kocharovsky, and S. Tarasov, Entropy22, 322 (2020)

    2020

  49. [57]

    E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Phys. Rev. B41, 9301 (1990)

    1990

  50. [58]

    Huber and J

    M. Huber and J. Law, inProceedings of the Nineteenth Annual ACM-SIAMSymposiumonDiscreteAlgorithms ,SODA’08(So- ciety for Industrial and Applied Mathematics, USA, 2008) pp. 681–689

    2008

  51. [59]

    J. Kuck, T. Dao, H. Rezatofighi, A. Sabharwal, and S. Ermon, inAdvancesinNeuralInformationProcessingSystems ,Vol.32, editedbyH.Wallach,H.Larochelle,A.Beygelzimer,F.d 'Alché- Buc, E. Fox, and R. Garnett (Curran Associates, Inc., 2019)

    2019

  52. [60]

    Beygelzimer, Y

    J.Harviainen,A.Röyskö,andM.Koivisto,in AdvancesinNeu- ralInformationProcessingSystems ,Vol.34,editedbyM.Ran- zato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan (Curran Associates, Inc., 2021) pp. 213–224

    2021

  53. [61]

    Jerrum, A

    M. Jerrum, A. Sinclair, and E. Vigoda, J. ACM51, 671 (2004)

    2004

  54. [62]

    Bezáková, D

    I. Bezáková, D. Štefankovič, V. V. Vazirani, and E. Vigoda, SIAM J. Comput.37, 1429 (2008)

    2008

  55. [63]

    S. C. Kou and P. McCullagh, Biometrika96, 635 (2009)

    2009

  56. [64]

    Bianchi, L

    E. Bianchi, L. Hackl, and N. Yokomizo, J. High Energ. Phys. 2018(03), 1

    2018

  57. [65]

    L.Hackl,E.Bianchi,R.Modak,andM.Rigol,Phys.Rev.A 97, 032321 (2018)

    2018

  58. [66]

    J. S. Liu, R. Chen, and W. H. Wong, J. Am. Stat. Assoc.93, 1022 (1998)

    1998

  59. [67]

    Aaronson and A

    S. Aaronson and A. Arkhipov, inProceedings of the forty-third annual ACM symposium on Theory of computing(2011) pp. 333–342

    2011

  60. [68]

    Qi, and R

    N.Quesada,R.S.Chadwick,B.A.Bell,J.M.Arrazola,T.Vin- cent, H. Qi, and R. García−Patrón, PRX Quantum3, 010306 (2022)

    2022

  61. [69]

    Clifford and R

    P. Clifford and R. Clifford, Phys. Scr.99, 065121 (2024)

    2024

  62. [70]

    Montanaro, and A

    A.Neville,C.Sparrow,R.Clifford,E.Johnston,P.M.Birchall, A. Montanaro, and A. Laing, Nat. Phys.13, 1153 (2017)

    2017

  63. [71]

    J.Wu,Y.Liu,B.Zhang,X.Jin,Y.Wang,H.Wang,andX.Yang, Natl. Sci. Rev.5, 715 (2018)

    2018

  64. [72]

    455,110990 (2022)

    P.-H.LundowandK.Markström,J.Comput.Phys. 455,110990 (2022)

    2022

  65. [73]

    Kuwahara, T

    T. Kuwahara, T. V. Vu, and K. Saito, Nat. Commun.15, 2520 (2024)

    2024

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