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Removing nodal and support-mismatch pathologies in Variational Monte Carlo via blurred sampling

Zhou-Quan Wan, Roeland Wiersema, Shiwei Zhang

arXiv:2603.18148v1cond-mat.str-elcond-mat.dis-nnphysics.comp-phquant-ph

TL;DR

VMC estimators can have infinite variance near continuum nodes or become biased under discrete support mismatch, undermining optimization and dynamics. The paper introduces blurred sampling, a local post-processing method with bounded reweighting factors, and reports stable, accurate simulations where conventional VMC breaks down while preserving existing sampling workflows.

  • Problem

    VMC suffers from heavy-tailed or divergent-variance estimators near nodes and biased estimators when discrete sampling misses Hamiltonian support.

  • Method

    Blurred sampling applies one local mixing step as post-processing to construct an implicit reference distribution while preserving the original sampling dynamics.

  • Results

    Blurred sampling restores stable and accurate simulations in regimes where conventional VMC breaks down, including finite-variance continuum estimation and unbiased discrete-force estimation.

  • Takeaways & Limitations

    The framework integrates with existing VMC workflows while resolving infinite-variance and support-mismatch pathologies with minimal overhead.

Abstract

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Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parameterized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators that form the backbone of the method can become unstable or biased due to the presence of nodes, a ubiquitous feature of quantum wave functions. In the continuum, this results in heavy-tailed estimators with potentially divergent variances, while in discrete Hilbert spaces the sampling distribution can miss parts of the support needed to form unbiased estimators. These statistical pathologies lead to unreliable optimization trajectories in stochastic reconfiguration or incorrect variational dynamics in time-dependent Variational Monte Carlo (t-VMC), and severely limit the power of the numerical simulations. We introduce blurred sampling to address these difficulties. The method has a number of rigorous properties that make it well-behaved, effective and efficient. Additionally it is a post-processing approach that can be used without modifying the underlying sampler and incurs only minimal overhead. We demonstrate its effectiveness on several representative examples where standard sampling approaches are known to fail, and apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.

I. INTRODUCTION

VMC estimators become unstable near wave-function nodes and biased when sampled and Hamiltonian supports differ. Blurred sampling addresses both pathologies through local post-processing that preserves standard sampling dynamics while restoring stable calculations.

  • I. INTRODUCTION: Blurred sampling locally mixes configurations as a post-processing step, implicitly regularizing nodal singularities without changing the underlying sampler.The method constructs an implicit reference measure rather than globally redefining the sampling distribution.
  • I. INTRODUCTION: Bounded reweighting factors give finite effective sample size and resolve both infinite-variance behavior and support-mismatch bias.The approach preserves standard VMC scaling and can often avoid additional wave-function evaluations in discrete spaces.
  • I. INTRODUCTION: VMC ratio-type estimators diverge near wave-function nodes because both local energies and logarithmic derivatives divide by ψθ(x).In continuum systems, nodal behavior can produce finite expectations but infinite estimator variance.
  • I. INTRODUCTION: In the continuum example, standard sampling has a heavy-tailed gradient estimator with α = 1.5 and nonconvergent variance, whereas blurred sampling has finite variance.The example uses two non-interacting spinless fermions on a ring.
  • I. INTRODUCTION: Support mismatch leaves discrete VMC estimators biased because configurations contributing to the exact force may lie outside supp(ψθ).The same ratio structure can also amplify fluctuations on rarely sampled configurations and affect the QGT.
  • I. INTRODUCTION: In the discrete single-spin example, standard sampling develops bias and can become trapped near θ →0, while blurred sampling follows the exact imaginary-time trajectory.The Hamiltonian is X and stochastic reconfiguration starts at θ = π/3.

C. Requirements for a Remedy

Importance-sampling remedies must simultaneously broaden support, control nodal singularities, avoid effective-sample-size collapse, and remain computationally efficient. Existing constructions face competing requirements because reference distributions and reweighting factors can become difficult to sample or statistically inefficient.

  • C. Requirements for a Remedy: Table I compares importance-sampling methods by effective-sample-size scaling and additional computational complexity, with blurred sampling adding zero overhead in discrete space and Nconn cost in the continuum.Additional complexity is measured beyond standard sampling, especially in wave-function evaluations.
  • C. Requirements for a Remedy: Reference sampling can restore missing contributions when supp(Hψθ) ∪ supp(ψθ) lies within supp(r), but constructing such an r is difficult.The analogous QGT support condition must cover derivative-product support as well.
  • C. Requirements for a Remedy: Reweighting-factor fluctuations strongly affect statistical efficiency because high-dimensional wave-function probabilities can span many orders of magnitude.This makes it difficult to broaden the distribution without substantially modifying sampling elsewhere.
  • C. Requirements for a Remedy: Overdispersed sampling broadens the distribution but can cause exponential effective-sample-size degradation as system size grows.Its reweighting factor scales as ω ∝|ψ|2−α for r(x) ∝ |ψ(x)|α with α < 2.
  • C. Requirements for a Remedy: A valid reference distribution must broaden support, regularize nodal singularities, avoid effective-sample-size collapse, and remain efficient to sample and evaluate.These requirements are generally in tension, and no existing reference distribution satisfies all of them generically.

III. BLURRED SAMPLING METHOD

Blurred sampling locally mixes sampled configurations to regularize nodal singularities and restore missing support without redefining the underlying sampler. Its bounded weights, unbiased estimators, and sparse post-processing preserve stability and computational scaling.

  • Construction: Blurred sampling applies a local mixing step to sampled configurations, inducing an implicit reference distribution that assigns weight near nodal regions.The method modifies configurations as post-processing rather than globally redefining the sampling distribution.
  • Construction: In continuum systems, sparse local displacements smear nodal singularities while keeping connectivity linear in system size.With q = 2/3 and ϵ = 0.2, the example eliminates the infinite-variance problem.
  • Construction: For discrete spaces, Hamiltonian-connected blurring expands support to include configurations contributing to the variational force, eliminating support-mismatch bias.The resulting estimator becomes unbiased because supp(ψθ) ∪ supp(Hψθ) is covered by the blurred distribution.
  • Statistical Guarantees and Computational Scaling: The reweighting factor has unit expectation and is bounded by 1/(1−q), yielding finite-sample unbiasedness and preventing exponential ESS degradation.These guarantees hold for fixed 0 < q < 1, with stability demonstrated across a broad range of q values.
  • Statistical Guarantees and Computational Scaling: Sparse post-processing requires no additional wave-function evaluations in discrete spaces and preserves standard VMC computational scaling.The discrete reweighting calculation uses amplitudes already computed for the local energy, while sparse connectivity keeps costs polynomial.

C. Generalizations

Blurred sampling generalizes to randomized sparse kernels that broaden accessible support while retaining conditional weighting and computational efficiency. In benchmark t-VMC examples, it restores correct dynamics where standard sampling fails.

  • C. Generalizations: Randomized blurred sampling draws a sparse kernel for each sample, broadening effective support while preserving finite-sample unbiasedness.Conditional weighting keeps computational cost tied to each realized kernel’s sparsity rather than the averaged connectivity.
  • A. Illustration with Previously Identified Difficult Examples: In the single-spin and 2 × 2 Heisenberg benchmarks, standard t-VMC breaks down from support-mismatch bias, whereas blurred sampling recovers the correct dynamics.The demonstrations use a Hamiltonian-induced kernel with q = 0.5.

B. Solution of the Parity Mixing Problem

Parity-mixing quenches expose support mismatch because the Hamiltonian transfers weight from an initially even sector to an odd sector absent from standard samples. Blurred sampling restores that access and accurately tracks parity dynamics across initializations, system sizes, and quench rates.

  • B. Solution of the Parity Mixing Problem: Standard sampling freezes parity dynamics at t = 0 because it samples only the initially occupied even-parity sector.The TDVP trajectory is incorrectly frozen when the Hamiltonian couples even configurations to odd ones.
  • B. Solution of the Parity Mixing Problem: Even a small initial odd-parity component does not rescue standard sampling, whose tiny statistical weight produces biased, noisy force estimates and later trajectory errors.The standard trajectory initially follows the exact solution but deviates at later times.
  • B. Solution of the Parity Mixing Problem: Blurred sampling restores missing parity-sector access and reproduces exact real-time dynamics over long times for q ∈ [0.1, 0.9].The results are reported as robust to changes in the blur strength.
  • B. Solution of the Parity Mixing Problem: For n = 64 spins, blurred sampling accurately reproduces parity mixing for both slow h = 1/8 and fast h = 1 quenches, while standard t-VMC fails.A Gaussian-state ansatz is used to separate estimator bias from insufficient RBM expressiveness.
  • C. Unbiased Spin Relaxation Dynamics: Randomized blurring can enlarge support beyond Hamiltonian-induced transitions, addressing additional QGT support mismatch in localized spin-relaxation dynamics.The framework combines Hamiltonian-induced blurring with sampled spin-flip masks controlled by q1 and q2.

V. DISCUSSION AND CONCLUDING REMARKS

Blurred sampling provides a general framework for regularizing VMC pathologies while preserving existing sampling dynamics and computational scaling. The discussion highlights successful spin-dynamics applications and directions for broader use and refinement.

  • Blurred sampling regularizes infinite-variance and support-mismatch pathologies through one-step local configuration mixing, without requiring an explicit global reference distribution.The method is presented as a flexible framework for neural-network variational calculations across ground-state, finite-temperature, and real- and imaginary-time settings.
  • The post-processing design preserves the original sampling dynamics and autocorrelation structure, enabling direct integration with existing VMC workflows.This contrasts with approaches requiring modified sampling distributions or additional Markov chains.
  • Figure 6 shows randomized blurred sampling recovering spin relaxation for both uncorrelated and correlated parameterizations, whereas standard t-VMC remains trapped and Hamiltonian-only blur fails in the correlated case.The simulation uses 32 TFIM spins and compares magnetization dynamics with exact Majorana-fermion results.
  • Future work includes adaptive or problem-specific blur kernels and studying blurred sampling for ground-state optimization, projector methods, finite-temperature simulations, and classical Monte Carlo.The blur strength q can interpolate between conventional and fully blurred estimates, while tailored local choices may improve ergodicity and sampling efficiency.

Appendix A: Detailed derivation and properties of blurred sampling

The appendix establishes unbiased finite-sample estimators under blurred sampling and shows that bounded reweighting factors control variance-related pathologies. It also explains why self-normalized covariance estimates can still yield unchanged SR and TDVP updates.

  • Blurred sampling constructs estimators under a post-processed distribution r while retaining unbiased recovery of expectation values under the original distribution p.Samples x come from p, configurations x′ are generated by a Markov kernel, and the reweighting factor is ω(x′)=p(x′)/r(x′).
  • The finite-sample covariance estimator remains unbiased when constructed from the reweighted moments under r.The derivation uses the unit expectation of ω and the standard unbiased covariance estimator under the blurred distribution.
  • Self-normalized covariance estimates are biased by a common multiplicative factor, but that factor cancels from SR and TDVP parameter-update ratios.Consequently, the resulting time evolution is unaffected by this overall factor.
  • The reweighting factors are bounded, preventing infinite-variance behavior and supporting finite effective sample size for fixed blur strength.The appendix derives the bound from the blurred distribution and nonnegative p(x′).

4. Randomized blur construction

Randomized blurred sampling averages over sparse, randomly selected blur kernels and evaluates weights conditionally on each realized kernel. Its modified QGT remains well-behaved and yields exact projected TDVP dynamics when the ansatz is sufficiently expressive.

  • Randomized blurred sampling independently draws a sparse blur kernel parameter λ for each sample, preserving finite-sample unbiasedness when support mismatch is absent.The formulation can randomize continuum displacement magnitudes and coordinate directions, producing an effective mixed kernel.
  • Conditional reweighting keeps computational cost tied to each sparse kernel rather than the potentially difficult mixed kernel.This avoids requiring an auxiliary Monte Carlo calculation for the mixed distribution.
  • The randomized estimator generally deviates from the exact QGT when the averaged indicator weight differs from one.The deviation reflects the conditional evaluation of reweighting factors under each realized kernel.
  • Despite modifying the QGT, randomized blurred sampling induces a positive deformation of the Hilbert-space metric rather than invalidating projected dynamics.The modified norm remains strictly positive definite because the averaged indicator is positive for every configuration.
  • When the variational ansatz is sufficiently expressive, randomized blurred sampling produces the exact projected TDVP parameter dynamics.The same parameter velocity minimizes both the ordinary and modified tangent-space distances.

Appendix B: Details of the parity mixing problem in TFIM

The appendix defines the TFIM parity-mixing benchmark, its observable, and the RBM ansatz used to initialize the chosen parity-sector state. It also specifies the parameter initialization and numerical perturbation.

  • The benchmark begins from an equal-weight superposition of computational-basis states with even Z-parity.The computational-basis spin variables satisfy x_i∈{±1}.
  • The evolving observable is the Z-parity expectation under periodic-boundary TFIM dynamics.The appendix defines P_Z(t) using the time-evolved parity operator and the initial state.
  • A complex-valued RBM serves as the variational ansatz, with ansatz size α=M/N determined by the number of hidden units.The parameters a_i, b_j, and W_ij are complex.
  • Specific RBM parameters exactly reproduce the target parity-sector amplitudes, while a perturbation of order 10^-3 keeps logarithmic derivatives well defined.The initialization sets most parameters to zero and assigns specified imaginary values to the remaining parameters.

2. Exact solution via mapping to Majorana fermions

The transverse-field Ising model is mapped to Majorana fermions, enabling analytic evaluation of its parity dynamics through Gaussian-state techniques. This provides exact dynamics for system sizes beyond exact diagonalization.

  • The transverse-field Ising model is represented using 2N Majorana fermions, with spin operators mapped into fermionic operators.
  • The initial state lies in a fixed X-parity sector, allowing the Hamiltonian and parity observables to be treated within that symmetry sector.
  • Although the initial state is non-Gaussian, parity symmetry decomposes it into Gaussian-state contributions whose evolution remains Gaussian.
  • Pfaffian methods evaluate Gaussian-state overlaps and parity expectations, making the time-dependent Z-parity analytically tractable.
  • The resulting analytic Z-parity dynamics reaches system sizes far beyond exact diagonalization.

3. Gaussian ansatz and its initialization

A Gaussian ansatz is introduced to separate t-VMC estimator effects from the expressiveness limits of the RBM. Its symmetry-aware construction enables efficient amplitude evaluation and can represent the exact evolved state.

  • The Gaussian ansatz is used for larger systems because the RBM is not sufficiently expressive for accurate long-time evolution.
  • The ansatz can represent the exact time-evolved state at all times while permitting efficient wavefunction-amplitude evaluation.
  • Its parameters are two 2N × 2N antisymmetric real matrices, each containing N(2N − 1) independent parameters.
  • Amplitudes are evaluated through overlaps with symmetric combinations of basis states, which are Gaussian and preserve X-parity.
  • A perturbation of order 10^-3 is added at initialization so logarithmic derivatives remain well defined.
  • The benchmark studies real-time total-Z-magnetization dynamics under the transverse-field Ising Hamiltonian from a computational-basis product state.

2. Dressed RBM ansatz and its initialization

The dressed RBM augments a translation-symmetric RBM with sector-dependent amplitudes and optional cross-sector correlations. The resulting construction exposes support mismatch in t-VMC and enables comparison of blurred-sampling behaviors.

  • The dressed RBM combines a translation-symmetric complex RBM with a sector-dependent factor parameterized by n + 1 complex amplitudes.
  • Sector amplitudes can selectively enhance or suppress entire spin sectors, including support restricted to the fully polarized all-down configuration.
  • A correlated parameterization couples the single-spin-up and another sector, making the amplitude of the s = 3 sector depend on ϕ1.
  • The resulting state exhibits a pronounced support-mismatch problem in both variational-force and QGT estimators.
  • For an uncorrelated ansatz, Hamiltonian-induced blur yields limited QGT support, while force components concentrate on one mode with other supports below 10^-8.
  • For the correlated ansatz, the force has broader support and the dynamics diverges around the corresponding point in Fig. 6(b).
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