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Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

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Pith's one-line read Projective quantum Monte Carlo simulations guided by approximate wave functions still tunnel at a rate set linearly by the energy gap.

desk verdict 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. read the letter →

arxiv 1908.10151 v2 pith:WSQPVRXI submitted 2019-08-27 quant-ph cond-mat.stat-mech

T. Parolini , E. M. Inack , G. Giudici , S. Pilati This is my paper · ORCID
classification quant-phcond-mat.stat-mech PACS 02.70.Ss03.65.Xp
keywords quantumMonteCarlotunnelingrateguidingwavefunctionenergygapscalingannealingDiffusionunrestrictedBoltzmannmachineshamrockmodel
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 5 minor

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.

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 (2)
  1. [Sec. II B (and Sec. III B)] 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.
  2. [Sec. II C and Sec. IV] 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.
minor comments (5)
  1. [Sec. II B] The sentence "This is a surprising results" contains a typo; it should read "This is a surprising result."
  2. [Sec. III B] 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.
  3. [Fig. 3] 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.
  4. [Sec. I] 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.
  5. [Sec. IV] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Δ^{-1} scaling is tested against independently computed gaps and derived from WKB/Kramers theory, not from the fitted tunneling data.

full rationale

The paper's central claim is that the asymptotic PQMC tunneling exponent is the same with and without guiding wave functions, with the GWF affecting only the prefactor. The load-bearing comparison is made against energy gaps obtained from independent sources: finite-difference diagonalization for the continuous double well, the exact free-fermion formula for the ferromagnetic Ising chain, and exact diagonalization for the shamrock model. The reported exponents b come from fitting ξ(Δ)=αΔ^{-b} to these independently computed gaps, so the fitted exponent is not statistically forced to equal 1 by construction. For the exact-GWF double-well case, the semiclassical derivation is self-contained: Eq. (10) reduces the stochastic dynamics to a Fokker-Planck equation, Eq. (11) is the standard Kramers activation time, and the WKB estimates for Ψ0(0) and the gap Δ, Eqs. (3) and (12), combine to give ξ∼τ_act∼1/Δ in Eq. (13). That derivation does not feed the numerical tunneling times back into its inputs. The paper does rely on same-author prior work, Ref. [28] for the no-GWF linear scaling and Ref. [37] for the uRBM guiding wave function, but those are used as previously published, externally falsifiable baselines and algorithmic tools rather than as premises that by definition imply the new GWF result. The unsupported statement in Sec. II B that 'a careful analysis shows that the asymptotic scaling of ξ is independent of this specific choice' is a missing-evidence concern about the stopping-rule protocol, not a circular reduction: the threshold and walker-fraction choices are not the quantity being predicted, and the central scaling is verified against independently computed gaps across three distinct models. Overall, the derivation and numerical protocol are not circular.

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

No new physical entities are introduced. The free parameters are either variational parameters of the guiding wave functions or fit parameters in the power-law analysis. The explanatory theory rests on standard Kramers and WKB results, with the mapping from DMC escape to classical activation being the most fragile step.

free parameters (5)
  • Prefactor alpha in scaling fits = 0.28(5) to 112(8), depending on protocol and model
    Appears in the fit xi = alpha Delta^{-b} for each dataset. It is not predicted by the theory and absorbs protocol- and model-dependent constants.
  • Exponent b in scaling fits = 0.96(3) to 1.04(3)
    Fitted via power-law regression to test the claimed linear scaling. The central claim is that b equals 1, so the exponent is fitted, not imposed.
  • Boltzmann guiding wave function inverse temperature beta = Not quoted; variationally minimized
    Free variational parameter in the ansatz exp(-beta V(x)). It is optimized against the energy, not against the tunneling-time data.
  • uRBM couplings K1, K2, K3 = Not quoted; optimized via stochastic reconfiguration
    Variational parameters of the neural-network guiding wave function. They are tuned for ground-state accuracy and are not fitted to the tunneling scaling.
  • DMC time step tau and walker population N_w = tau = 0.007 for 1/Delta > 70; N_w = 5000 to 10000
    Chosen to make systematic bias negligible. The paper states these values are sufficient but does not show a systematic convergence study.
assumptions (6)
  • domain assumption The ground-state wave functions Psi_0(x) are real and nonnegative for the stoquastic Hamiltonians considered.
    Invoked in Sec. II A to justify DMC walker sampling and the absence of the negative-sign problem, with Refs. [42,43] as support.
  • domain assumption The linear short-time Trotter approximation, plus zero-time-step extrapolation, correctly describes the DMC and PQMC dynamics.
    Used in Secs. II A and III A. It introduces a finite-tau bias that the authors control numerically but do not display.
  • domain assumption The Kramers escape-rate formula Eq. (11) applies to the Fokker-Planck dynamics Eq. (10) and gives the DMC tunneling time to exponential accuracy.
    Sec. II C. This mapping from stochastic population escape to classical activation is the theoretical bridge between DMC tunneling time and Delta^{-1}.
  • standard math WKB semiclassical approximation is valid for the ground-state wave function near the barrier, giving Psi_0(0) ~ exp(-g/3), Delta ~ exp(-2g/3), and Delta proportional to Psi_R(0) Psi_R'(0).
    Used in Sec. II C. The paper explicitly assumes local WKB validity and that turning points do not approach the origin in the large-barrier limit.
  • domain assumption For generic double wells, the single-well function Psi_R(x) is asymptotically localized so that the integral of [Psi_R(x) Psi_R(-x) - Psi_R(x)^2] over the negative half-line is small.
    Stated in Sec. II C as the condition for the generic gap formula Delta proportional to Psi_R(0) Psi_R'(0).
  • domain assumption The ferromagnetic quantum Ising chain and the shamrock model can be described by an effective double-well potential in the magnetization, with the gap controlled by tunneling between the two polarized minima.
    Sec. III. This standard mapping is used to interpret the measured tunneling times in the spin models.

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Pith. "Pith review of Tunneling in projective quantum Monte Carlo simulations with guiding wave functions." pith.science (2026). https://pith.science/paper/WSQPVRXI

@misc{pith2026190810151,
  author       = {Pith},
  title        = {Pith review of: Tunneling in projective quantum Monte Carlo simulations with guiding wave functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WSQPVRXI}},
  note         = {Machine review of arXiv:1908.10151}
}
read the original abstract

Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function --- which is essential to make many-body PQMC simulations computationally feasible --- affects the tunneling rate. The chosen testbeds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically-exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.

Figures

Figures reproduced from arXiv: 1908.10151 by the authors.

Figure 1
Figure 1. FIG. 1. (color online). Profile of the quartic double-well po [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (color online). DMC tunneling time [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. displays the tunneling time ξ obtained with the Boltzmann GWF, as a function of the number of spins N, for different transverse field intensities Γ. In the large system-size regime, where the energy gap ∆ is small, the exponential growth of ξ closely matches the scaling of the inverse energy gap α∆−1 , where α is an appropriate prefactor. The energy gap values are computed using the exact formula obtained from the f… view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (color online). The shamrock, a model of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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