{"id":"d780f929-53bf-4899-b1c4-4fb1e323f397","arxiv_id":"1908.10151","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Guiding wave functions do not change the asymptotic linear scaling of projective quantum Monte Carlo tunneling time with inverse energy gap in double wells, Ising chains, and the shamrock model.","lead":"This paper asks whether a guiding wave function, a standard trick in quantum Monte Carlo, changes how quickly simulations tunnel through energy barriers. The answer is no: the crossing time still grows as one over the energy gap, so the trick preserves the method's known speed advantage.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central Δ^-1 scaling claim rests on an unshown assertion that the tunneling-time stopping rule (threshold position, walker fraction p, population N_w) changes only the prefactor; no data or derivation supports this.","rationale":"The reader's weakest_assumption is exactly the load-bearing concern I identify: the threshold protocol is asserted to affect only the prefactor, but no data or derivation is shown. The manuscript's strongest claim—that all considered GWFs preserve the asymptotic Delta^{-1} scaling—depends on this because the fitted exponent b is extracted from a specific stopping rule. If different choices of p, x_th, or N_w changed the fitted exponent, the conclusion would be an artifact of the protocol. The semiclassical derivation does not fully close this gap: it computes the single-walker Kramers activation time, whereas the numerical protocol defines xi as a population-crossing time. For independent walkers with an asymptotically exponential first-passage distribution, the population-crossing time differs only by a prefactor log(1/(1-p)), which would support the paper's assertion, but this argument is not given and the branching/no-GWF case is not covered by it. The paper has genuine strengths: the double-well, Ising-chain, and shamrock results are mutually consistent; gaps are computed independently (finite difference, free fermions, exact diagonalization); and the fit-error analysis in Table II explicitly accounts for fitting-window variation. Those strengths make the central claim plausible, but not fully secured. A conditional verdict is therefore appropriate, with the requested check being a direct protocol-dependence scan. I do not see a more serious internal inconsistency, and I do not regard the WKB limitations as a flaw because the paper explicitly labels them as assumptions and invites a more general derivation.","tokens_in":15593,"tokens_out":4667,"duration_ms":57135,"concrete_test":"Repeat the double-well DMC tunneling-time measurement at a small-gap point (e.g., g such that 1/Delta~300) with the exact GWF and with no GWF, varying the stopping rule: p in {0.05,0.25,0.5}, x_th in {x_R/4, x_R/2, 3x_R/4}, and N_w in {10^3,10^4,10^5}. For each variant, fit xi=alpha*Delta^{-b} using the same finite-difference Delta values. If b remains consistent with 1 within statistical error for all variants, the protocol-independence assertion is confirmed and the concern is resolved. If b shifts with p or N_w, the reported linear scaling must be qualified as protocol-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that a guiding wave function changes only the prefactor, not the asymptotic exponent—requires that the measured 'tunneling time' xi be a trustworthy proxy for the physical 1/Delta scaling. In Sec. II B, xi is defined by a stopping rule: p=25% of the walker population must cross x_th=x_R/2. In Sec. III B, p=10% and M<0 define xi. The text states that 'a careful analysis shows that the asymptotic scaling of xi is independent of this specific choice up to a constant prefactor,' but no such analysis, data, or argument is presented. The semiclassical theory in Sec. II C derives instead the single-walker Kramers mean first-passage time tau_act, which is not obviously the same object as the time at which a prescribed fraction p of N_w walkers has crossed a threshold. If the population-level stopping time has a different dependence on p or N_w than the single-walker rate—for example through order-statistics/extreme-value effects or through population-control feedback when branching is present—then the fitted exponent b≈1 could be a property of the protocol rather than of the tunneling dynamics. The consistency across three models and the use of exact gaps from finite-difference/free-fermion/ED calculations are genuine supporting evidence, but they do not remove the need for an explicit check or derivation connecting xi to tau_act. This is the most load-bearing point because the headline claim is specifically about asymptotic scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper asks whether the asymptotic scaling relation between the tunneling time measured in projective quantum Monte Carlo (PQMC) simulations and the first energy gap, ξ ∝ Δ^{-1}, is affected by the use of a guiding wave function (GWF). The authors perform DMC/PQMC simulations on a continuous double-well potential, the ferromagnetic quantum Ising chain, and the frustrated shamrock model, using no GWF, a Boltzmann-type GWF, an exact ground-state GWF for the double well, and an unrestricted Boltzmann-machine GWF for the Ising chain. In every case they fit the large-barrier/large-system data to ξ = α Δ^{-b} and find b consistent with 1 (values between 0.96 and 1.04), concluding that the GWF changes only the prefactor. For the double well with exact GWF, they derive ξ ∝ Δ^{-1} from a WKB/Kramers analysis of the Fokker-Planck dynamics.","tokens_in":15887,"tokens_out":11739,"duration_ms":122425,"significance":"The result, if it holds, is significant: it extends the quadratic speedup of PQMC tunneling relative to incoherent quantum tunneling to guided simulations, which are the practically relevant ones for large many-body systems. The empirical core is well benchmarked: the gap is obtained from independent finite-difference, free-fermion, and exact-diagonalization calculations, so the fitted exponent is not a re-fit of a quantity derived from the tunneling data. The semiclassical derivation is a genuine parameter-free prediction for the exact-GWF case, and the numerical observation is a falsifiable scaling claim. The main weakness is that the protocol used to define ξ is asserted, not demonstrated, to affect only the prefactor; a direct sensitivity check is needed before the asymptotic claim can be considered fully established.","major_comments":[{"comment":"The measured quantity ξ is defined by a stopping rule: in the double well, the simulation stops when p = 25% of the walker population crosses x_th = x_R/2, and in the spin models when p = 10% reaches M < 0. The paper states that \"a careful analysis shows that the asymptotic scaling of ξ is independent of this specific choice up to a constant prefactor,\" but no such analysis, data, or derivation is presented, and no reference is given. Since the central claim concerns asymptotic scaling, this is a load-bearing assumption: if ξ's asymptotic behavior depended on p, x_th, or N_w, the fitted exponent b ≈ 1 could be an artifact of the protocol rather than a property of the tunneling dynamics. The consistency across three models and the independent gap benchmarks are reassuring, but they do not replace a direct check. I request a sensitivity analysis (varying p and the threshold in at least one model and showing b is unchanged) or an argument that the population-threshold time is asymptotically proportional to the single-walker escape rate.","section":"Sec. II B (and Sec. III B)"},{"comment":"The semiclassical theory equates ξ with the single-walker Kramers activation time τ_act for the effective potential Ṽ = -ln Ψ_G, and it is exact only when the branching term in Eq. (6) vanishes, i.e., when Ψ_G is the exact ground state. For the Boltzmann and uRBM GWFs the same scaling is observed numerically, but the theory does not cover those cases; the paper itself restricts the derivation to situations where branching may be neglected, but the abstract and conclusions could be read as claiming the theory explains all GWF choices. Please state this limitation more prominently and separate the explained exact-GWF case from the empirical approximate-GWF cases.","section":"Sec. II C and Sec. IV"}],"minor_comments":[{"comment":"The sentence \"This is a surprising results\" contains a typo; it should read \"This is a surprising result.\"","section":"Sec. II B"},{"comment":"The paragraph beginning \"The shamrock model\" is not given a distinct subsection number or heading, which makes the structure of Section III confusing; please format it as a proper subsection.","section":"Sec. III B"},{"comment":"The closed symbols are labelled \"α/Δ\" while the y-axis label is \"Tunneling time ξ\"; the caption should clarify that the closed symbols are α Δ^{-1} reference values obtained from the exact free-fermion gap, not directly measured tunneling times.","section":"Fig. 3"},{"comment":"In the introductory discussion of D-Wave devices, the phrase \"see, e.g., [1, 12–15]\" is followed by a misplaced period and then \"In particular,\"; please fix the punctuation and citation formatting.","section":"Sec. I"},{"comment":"The Conclusions state \"The proof we presented relies on the local validity of the semiclassical approximation,\" but the empirical results for approximate GWFs are not covered by that proof; please add a sentence making this distinction explicit.","section":"Sec. IV"}],"recommendation":"major_revision","confidential_remarks":"I recommend major revision rather than rejection. The missing stopping-rule analysis is readily supplied, and the rest of the evidence—especially the external gap benchmarks—is solid. If the authors provide the requested sensitivity check and clarify the scope of the semiclassical theory, I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a solid within-subfield advance. The genuinely new content is the demonstration that three different guiding wave functions — a Boltzmann ansatz, the exact ground state, and an unrestricted Boltzmann machine — leave the Δ^-1 tunneling-time scaling of PQMC intact across a continuous double well, the ferromagnetic Ising chain, and the shamrock model. That is a useful result for practitioners who need importance sampling to make PQMC feasible. The paper does it well: gaps are computed independently (finite differences, free-fermion formula, exact diagonalization), so the scaling benchmark is not circular. Fitted exponents are all consistent with 1, and the WKB/Kramers semiclassical derivation for the exact-GWF case is a genuine explanatory addition, even though it explicitly leans on semiclassical assumptions.\n\nThe main soft spot is the one-sentence assertion in Sec. II B that the tunneling-time definition is independent of the stopping rule up to a prefactor. No data or derivation is shown. The stress-test concern is fair: the population-level stopping time (p=25% crossing x_R/2, or p=10% reaching negative magnetization) is not obviously the same object as the single-walker Kramers activation time, and in principle the fitted exponent could be affected by the protocol. The consistency across three models and two different protocols is reassuring, but it is not a substitute for a direct check. A referee should ask for a supplementary figure varying p and x_th, or an explicit argument.\n\nOther soft spots are minor: no code or data release, and the \"competitive quantum-inspired optimizer\" framing remains an outlook, since total computational cost is not analyzed. The self-citations to Ref. [28] are appropriate — that is the prior result being extended.\n\nOverall, the central claim holds up reasonably well. This paper is for QMC practitioners and people benchmarking quantum annealers, and it deserves a serious refereeing. The protocol-independence check is the main thing I would want before accepting.","headline":"A careful numerical study showing that guiding wave functions preserve the Δ^-1 PQMC tunneling scaling, with one unbacked protocol-independence claim that a referee should push on.","tokens_in":16456,"tokens_out":1651,"would_cite":true,"duration_ms":18177,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["02.70.Ss","03.65.Xp"],"model":"deepseek-v4-flash","headline":"Projective quantum Monte Carlo simulations guided by approximate wave functions still tunnel at a rate set linearly by the energy gap.","keywords":["quantum Monte Carlo","tunneling rate","guiding wave function","energy gap scaling","quantum annealing","Diffusion Monte Carlo","unrestricted Boltzmann machine","shamrock model"],"falsifier":"Repeat the double-well and Ising-chain runs with very different stopping rules (for example p = 1% versus p = 75%, and thresholds at the barrier top versus deep in the target well), then fit $\\xi = \\alpha \\Delta^{-b}$ in the small-gap regime; if the fitted b moves away from 1 as the rule changes, the claimed linear law is an artifact of the measurement protocol.","tokens_in":15362,"feed_emoji":"⚛️","tokens_out":9802,"duration_ms":81302,"temperature":0.7,"pith_summary":"Projective quantum Monte Carlo (PQMC) methods can cross energy barriers by quantum tunneling, and in double-well-type problems their tunneling rate was previously found to scale linearly with the first energy gap—a quadratic speedup over the gap-squared rate of real quantum annealers. Since practical PQMC simulations need a guiding wave function (GWF) to keep costs manageable, the paper asks whether the GWF changes this scaling. Testing a continuous double well, the ferromagnetic Ising chain, and the frustrated shamrock model, with Boltzmann, exact, and neural-network GWFs, it finds that the asymptotic linear scaling $\\xi \\propto \\Delta^{-1}$ survives in every case; only the prefactor changes. A semiclassical WKB argument explains the linear law for the double well with an exact GWF. If the result holds generally, guided PQMC can serve as both a benchmark for quantum annealers and a competitive quantum-inspired optimizer.","feed_headline":"Guiding wave functions don't change QMC tunneling scaling","feed_subtitle":"Tested on double wells, Ising chains, and the shamrock model: the gap scaling stays linear, only prefactors move.","key_machinery":"The load-bearing object is the importance-sampled projective QMC evolution of the product $\\rho(x,t)=\\Psi(x,t)\\Psi_G(x)$, which obeys a Fokker–Planck equation with drift, diffusion, and branching. When the GWF is the exact ground state $\\Psi_G=\\Psi_0$, the local energy is constant, branching is suppressed, and the dynamics reduces to classical Kramers activation over the effective potential $\\tilde V(x)=-\\ln \\Psi_G(x)$. The semiclassical mechanism then uses WKB theory in two places: the ground-state amplitude at the barrier scales as $\\Psi_0(0)\\sim e^{-g/3}$, and the gap in a double well obeys $\\Delta \\propto \\Psi_R(0)^2 \\propto \\Psi_0(0)^2$; combining these yields $\\xi \\sim 1/\\Delta$. This identity—tunneling time set by the squared ground-state amplitude at the barrier—is what carries the linear-scaling result.","core_discovery":"The central claim is that a guiding wave function—even a very accurate one such as the exact ground state or a neural-network state—does not alter the leading exponential scaling of the PQMC tunneling time with the energy gap. In all three models, the measured tunneling time $\\xi$ approaches $\\xi \\propto \\Delta^{-1}$ as the gap $\\Delta$ becomes small, with fitted exponents $b \\simeq 1$ for the double well, the Ising chain, and the shamrock model. The GWF only rescales the prefactor $\\alpha$ in $\\xi = \\alpha \\Delta^{-b}$. For the double well with the exact ground state as GWF, the branching term disappears and the algorithm becomes classical activation over the effective potential $\\tilde{V} = -\\ln \\Psi_G$; Kramers' activation formula combined with WKB estimates gives $\\xi \\sim \\Psi_0(0)^{-2} \\sim \\Delta^{-1}$, matching the numerics and extending to generic double wells when WKB holds near the barrier.","pith_inferences":["A testable extension, not claimed by the paper, would vary the stopping protocol (walker fraction p and threshold position) over a wide range; the paper asserts but does not show that only the prefactor changes, so a protocol-dependent exponent would undercut the universality claim.","The semiclassical argument points to the ground-state amplitude at the barrier as the controlling quantity, so a GWF that is energy-accurate but wrong near the barrier could in principle break the linear scaling; constructing such an ansatz would be a sharp test of the mechanism.","If the linear law survives in higher-dimensional Ising spin glasses, guided PQMC could become a practical classical heuristic for optimization problems where annealing gaps close exponentially, but the paper only tests one-dimensional and small frustrated models.","The authors themselves leave open a derivation that does not rely on WKB; finding a counterexample where the $\\Delta^{-1}$ scaling fails would clarify exactly which feature of double-well tunneling is essential."],"forward_implications":["Guided PQMC retains the quadratic speedup over incoherent quantum tunneling (rate $\\propto \\Delta^2$) even when the guiding ansatz is approximate or neural-network based.","Because an accurate GWF also makes equilibrium PQMC costs polynomial in system size, the same guided algorithm can efficiently access both ground-state properties and tunneling dynamics.","The Boltzmann-machine-guided PQMC result on the shamrock model extends the linear gap scaling to a frustrated setting where finite-temperature path-integral Monte Carlo slows down exponentially.","The WKB-based derivation implies the $\\xi \\propto \\Delta^{-1}$ law is generic for double-well potentials whose ground-state wave function is semiclassically accurate at the barrier, not just for the quartic well.","The practical message for quantum annealing benchmarks is that adding a guiding wave function is safe: it improves equilibrium sampling without spoiling the favorable tunneling scaling."],"supporting_citations":[{"why":"Established the baseline result that PQMC without GWF tunnels at rate $\\xi \\propto \\Delta^{-1}$; this paper extends it to guided simulations.","marker":"[28]"},{"why":"Supplies the importance-sampling/GWF formalism that defines the guided PQMC algorithm studied here.","marker":"[17]"},{"why":"Introduced the unrestricted Boltzmann machine GWF and showed it reduces PQMC cost to polynomial; that GWF is one of the ansatzes tested.","marker":"[37]"},{"why":"Reported the $\\Delta^{-2}$ PIMC tunneling scaling that the linear PQMC result is contrasted against.","marker":"[23]"},{"why":"Provided the plateau-potential variant and the measurement protocol adapted here for the width-dependence test.","marker":"[24]"},{"why":"Defined the shamrock model and showed PIMC tunneling there scales as $2^K \\Delta^{-2}$, the slowdown the PQMC result surpasses.","marker":"[29]"},{"why":"Kramers' activation formula is the core of the semiclassical tunneling-time calculation.","marker":"[46]"},{"why":"Gives the WKB tunnel-splitting formula for one-dimensional wells used to connect $\\Delta$ to $\\Psi_0(0)^2$.","marker":"[40]"},{"why":"Introduced the diffusion Monte Carlo random-walk algorithm that underlies the continuous-space simulations.","marker":"[41]"},{"why":"Supplies the incoherent-tunneling $\\Delta^2$ rate that defines the quadratic speedup claimed for PQMC.","marker":"[25]"}],"fun_headline_variants":["Guiding waves don't change QMC tunneling scaling","Tunneling rate stays gap-linear with guiding ansatz","PQMC tunneling linear in gap regardless of GWF","Guiding wave functions keep tunneling scaling linear","QMC tunneling rate unaffected by guiding wave functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported scaling rests on the assumption that the precise stopping rule used to clock a tunneling event—what fraction of walkers must reach which point in the opposite well—changes only the overall prefactor, not the asymptotic gap scaling.","fun_headline_variants_meta":{"raw":{"variants":["Guiding waves don't change QMC tunneling scaling","Tunneling rate stays gap-linear with guiding ansatz","PQMC tunneling linear in gap regardless of GWF","Guiding wave functions keep tunneling scaling linear","QMC tunneling rate unaffected by guiding wave functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":1273,"prompt_tokens":995,"completion_tokens":278,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":204}},"tokens_in":611,"tokens_out":278,"duration_ms":3502,"temperature":1.0,"reasoning_tokens":204,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:52:11.679312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the double-well and Ising-chain runs with very different stopping rules (for example p = 1% versus p = 75%, and thresholds at the barrier top versus deep in the target well), then fit $\\xi = \\alpha \\Delta^{-b}$ in the small-gap regime; if the fitted b moves away from 1 as the rule changes, the claimed linear law is an artifact of the measurement protocol.","supporting_citations":[{"cited_title":"Understanding quantum tunneling using diffu- sion Monte Carlo simulations,","cited_arxiv_id":null,"evidence_quote":"Established the baseline result that PQMC without GWF tunnels at rate $\\xi \\propto \\Delta^{-1}$; this paper extends it to guided simulations."},{"cited_title":"Quantum Monte Carlo simulations of solids,","cited_arxiv_id":null,"evidence_quote":"Supplies the importance-sampling/GWF formalism that defines the guided PQMC algorithm studied here."},{"cited_title":"Projective quantum Monte Carlo simulations guided by unrestricted neural network states,","cited_arxiv_id":null,"evidence_quote":"Introduced the unrestricted Boltzmann machine GWF and showed it reduces PQMC cost to polynomial; that GWF is one of the ansatzes tested."},{"cited_title":"Understanding quantum tunneling through quantum Monte Carlo sim- ulations,","cited_arxiv_id":null,"evidence_quote":"Reported the $\\Delta^{-2}$ PIMC tunneling scaling that the linear PQMC result is contrasted against."},{"cited_title":"Quan- tum Monte Carlo tunneling from quantum chemistry to quantum annealing,","cited_arxiv_id":null,"evidence_quote":"Provided the plateau-potential variant and the measurement protocol adapted here for the width-dependence test."},{"cited_title":"Can quantum Monte Carlo simulate quantum annealing?","cited_arxiv_id":"1703.09277","evidence_quote":"Defined the shamrock model and showed PIMC tunneling there scales as $2^K \\Delta^{-2}$, the slowdown the PQMC result surpasses."},{"cited_title":"Brownian motion in a field of force and the diffusion model of chemical reactions,","cited_arxiv_id":null,"evidence_quote":"Kramers' activation formula is the core of the semiclassical tunneling-time calculation."},{"cited_title":"Tunnel splittings for one-dimensional potential wells revisited,","cited_arxiv_id":null,"evidence_quote":"Gives the WKB tunnel-splitting formula for one-dimensional wells used to connect $\\Delta$ to $\\Psi_0(0)^2$."},{"cited_title":"A random-walk simulation of the Schr¨ odinger equation: H+ 3 ,","cited_arxiv_id":null,"evidence_quote":"Introduced the diffusion Monte Carlo random-walk algorithm that underlies the continuous-space simulations."},{"cited_title":"Incoherent tunneling in a double well,","cited_arxiv_id":null,"evidence_quote":"Supplies the incoherent-tunneling $\\Delta^2$ rate that defines the quadratic speedup claimed for PQMC."}],"review_version":1}