{"id":"3d31bf20-e89c-4507-baf0-75dcbc3f2568","arxiv_id":"2411.10085","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A random sampling estimator for matrix permanents reduces the sample count for free-boson entanglement entropy from about 2^N to about 2^(0.2N), enabling dynamics for systems of order 100 sites.","lead":"This paper shows that a random sampling trick can shrink the cost of computing entanglement entropy in free boson systems from being exponential in the number of sites to a much gentler exponential, letting simulations reach around 100 sites. The work gives a practical tool for benchmarking entanglement growth in cold-atom experiments and for checking how entanglement scales with system size.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2^{0.2N} sampling-cost law rests on an extrapolated fit of c(Ns) from small sizes and a single time; no variance bound or large-N validation supports the Ns=100 entropy curves.","rationale":"The reader's conditional verdict correctly identifies the extrapolated alpha scaling as the weak point. The estimator identity is correct, the Ns=40 benchmark is genuine independent validation, and the dynamics are physically plausible. The only way the central claim fails is if the sample-count exponent grows with Ns beyond the fitted range or varies in time; this is exactly what a direct measurement at Ns=80/100 would expose. No reason to reject, but full acceptance requires that check or a rigorous variance bound.","tokens_in":16677,"tokens_out":13529,"duration_ms":144011,"concrete_test":"Measure c1D(Ns) directly at Ns=80 and Ns=100 at tJ=2Ns using 32 independent runs with Ntotal=2^{34}, and compare with the extrapolated fit 2^{0.219Ns - 8.8}; repeat at an intermediate time such as tJ=Ns. If the measured c deviates from the fit by more than a factor of 4, or if doubling Ntotal shifts S2/Ns by more than the stated bootstrap error, the 2^{0.2N} cost law and the Ns=100 curves are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise is that the variance constant c(Ns) fitted in Eqs. (35) and (40), c(Ns)=2^{alpha Ns - beta}, remains valid at Ns=100 and at all times, so that Ntotal=2^{0.2 Ns + 12} in Sec. III C suffices. This premise is used to justify the Ns=100 entropy curves. The fit is made at the single times tJ=2Ns (1D) and tJ=2Lx (2D), over Ns<=60 (1D) and Ns<=120 (2D), with a 2D exponent of only 0.20 +/- 0.08; no variance bound or larger-size check is given. Moreover, the estimator is rare-event dominated: for Ns=40 at tJ=2Ns, typical |Re p| ~ e^{-16} while permA ~ e^{-12}, so the mean is set by rare large samples; at Ns=100, the 2^{10}-block bootstrap may undersample those tails and underestimate the statistical error. The exact Ns=40 match validates the identity E[Gly_r(A)]=permA but not the extrapolated alpha.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17017,"tokens_out":5669,"duration_ms":58984,"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":[{"comment":"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.","section":"Sec. III B, Eqs. (35) and (40); Sec. III C"},{"comment":"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.","section":"Sec. III A, Figs. 2(b) and 3(b); Sec. III C"},{"comment":"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'.","section":"Sec. III C, Fig. 6(a)"}],"minor_comments":[{"comment":"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.","section":"Eq. (34) and Fig. 4 caption"},{"comment":"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)|]).","section":"Figs. 1 and 2"},{"comment":"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.","section":"Sec. II, sentence around Ref. [52]"},{"comment":"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'.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a useful, honestly-written methods paper, and the main numerical claim—that a Glynn random-sampling estimator evaluates the relevant permanent with cost ~2^{0.2N} rather than 2^N—is plausible but empirical, fitted over a narrow size window and extrapolated to Ns=100. The estimator identity in Eqs. (22)-(28) is clean, and the Ns=40 1D curve matches the exact permanent result, which is real evidence the implementation is correct. The dynamics results up to Ns=100 (1D) and 10x10 (2D) are a legitimate advance for benchmarking cold-atom experiments and checking volume-law scaling.\n\nThe soft spots are proportionate. The central cost claim c(Ns)=2^{alpha Ns - beta} is a fit over Ns>=40 (1D) and Ns>40 (2D) at a single time point, tJ=2Ns or tJ=2Lx. There is no variance bound, and the 2D exponent has a large error bar, alpha=0.20(8). The estimator is rare-event dominated in the long-time regime: for Ns=40, typical |Re p| ~ e^{-16} while permA ~ e^{-12}, so the mean is set by large, rare samples. The blocking/bootstrap with 2^10 blocks may undersample those tails at Ns=100, which could make the quoted error bars optimistic. The paper also says 'more than 100 sites' in the abstract and conclusion, but the largest system is exactly Ns=100; minor wording, but worth fixing.\n\nCredit where due: the same sampling procedure is explicitly credited to Ref. [52], the authors acknowledge the empirical nature of the scaling, and the conclusion discusses bounds and future variance-reduction ideas. No code or data is released, which would strengthen the reproducibility but is not a fatal omission.\n\nBottom line: this deserves a serious referee. The method is simple, the validation at Ns=40 is reassuring, and the dynamics results are useful. The referee should ask for either a theoretical variance argument or a more extensive empirical check (larger sizes, multiple times), and for the 'more than 100' phrasing to be corrected. I would accept it with revisions. I'd cite it as the first demonstration that this estimator works for these permanent matrices at this scale.","headline":"Useful empirical methods paper; the 2^{0.2N} cost claim is real but extrapolated, and the paper slightly oversells the system-size reach.","tokens_in":17498,"tokens_out":2316,"would_cite":true,"duration_ms":23330,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Mn","02.70.Uu","05.30.Jp"],"model":"deepseek-v4-flash","headline":"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.","keywords":["matrix permanent","Rényi entanglement entropy","free bosons","random sampling","quantum quench","charge-density-wave state","entanglement entropy dynamics","computational cost"],"falsifier":"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.","tokens_in":16485,"feed_emoji":"⚛️","tokens_out":14847,"duration_ms":131816,"temperature":0.7,"pith_summary":"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.","feed_headline":"Random sampling reaches 100-site boson entanglement dynamics","feed_subtitle":"A random-phase permanent estimator makes 100-site free-boson entropy dynamics practical on a single CPU core.","key_machinery":"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 $2^{N_{\\mathrm{s}}}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the permanent formula for Rényi entropy and the exact 40-site benchmark that the random sampling result reproduces.","marker":"[33]"},{"why":"It gives the Glynn/BBFG permanent formula on which the random sampling estimator is built.","marker":"[47]"},{"why":"It proposed the equivalent random-phase sampling procedure, which this paper adopts and analyzes for efficiency.","marker":"[52]"},{"why":"It provides the random-phase permanent expectation identity used to justify replacing the sum with a sample mean.","marker":"[54]"},{"why":"It supplies the permanent expectation inequalities used in the paper's variance and bound discussion.","marker":"[55]"},{"why":"It gives the uniform random-phase implementation used to generate the sample vectors.","marker":"[56]"},{"why":"It provides the Gaussian-initial-state entanglement-growth predictions used to compare the two-dimensional dynamics.","marker":"[22, 23]"},{"why":"It frames the exponential sample requirement as a negative-sign-problem-like cancellation, motivating the cost analysis.","marker":"[57]"}],"fun_headline_variants":["Random sampling cracks 100-site boson entropy dynamics","Permanent speedup enables 100-site boson entropy dynamics","Random-phase estimator tames 100-site boson entropy dynamics","Exponential speedup for boson entropy: 100 sites now feasible","Sampling beats permanent: 100-site boson entropy dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Random sampling cracks 100-site boson entropy dynamics","Permanent speedup enables 100-site boson entropy dynamics","Random-phase estimator tames 100-site boson entropy dynamics","Exponential speedup for boson entropy: 100 sites now feasible","Sampling beats permanent: 100-site boson entropy dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000382,"raw_usage":{"total_tokens":2052,"prompt_tokens":997,"completion_tokens":1055,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":969}},"tokens_in":613,"tokens_out":1055,"duration_ms":8210,"temperature":1.0,"reasoning_tokens":969,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:58:57.640410+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Kagamihara, R","cited_arxiv_id":null,"evidence_quote":"It supplies the permanent formula for Rényi entropy and the exact 40-site benchmark that the random sampling result reproduces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the Glynn/BBFG permanent formula on which the random sampling estimator is built."},{"cited_title":"Huh, Phys","cited_arxiv_id":null,"evidence_quote":"It proposed the equivalent random-phase sampling procedure, which this paper adopts and analyzes for efficiency."},{"cited_title":"Aaronson and T","cited_arxiv_id":null,"evidence_quote":"It provides the random-phase permanent expectation identity used to justify replacing the sum with a sample mean."},{"cited_title":"Berkowitz and P","cited_arxiv_id":null,"evidence_quote":"It supplies the permanent expectation inequalities used in the paper's variance and bound discussion."},{"cited_title":"Kocharovsky, V","cited_arxiv_id":null,"evidence_quote":"It gives the uniform random-phase implementation used to generate the sample vectors."}],"review_version":1}