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arXiv:2603.18148 · Physical Review X

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

Zhou-Quan Wan, Roeland Wiersema, Shiwei Zhang

cond-mat.str-elcond-mat.dis-nnphysics.comp-phquant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations• Revised Sep 25, 2026 (v2)
v1 · Mar 18, 2026v2 · Sep 25, 2026
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Abstract

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 computationally 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 in a broad range of problems -- including ground state optimization in both continuum and discrete settings, as well as in time-dependent variational Monte Carlo, where it resolves previously identified failure modes -- and further apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.

Journal reference
Phys. Rev. X 16, 031059 (2026)
DOI
10.1103/jrn5-gv19
Comments
28 pages, 12 figures

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Real-time dynamics after a local quench by a charged operator encodes the response functions measured in spectroscopic experiments, yet they have long posed a challenge for variational Monte Carlo calculations. The obstacle is a support mismatch: the projective action by a charged local operator forces an exponentially large number of configurations to vanish, but these configurations may still contribute to the dynamics, biasing the estimators and freezing the evolution at the very first step. This difficulty is an artifact of the chosen sampling basis, and the support mismatch generated by a charged local operator is itself local. We demonstrate that the missing support can be restored by a local rotation of the sampling basis, without changing the underlying variational dynamics. We propose a local basis-rotation sampling scheme that resolves the support-mismatch problem and can be readily incorporated into existing variational Monte Carlo algorithms. Benchmarks show that rotation sampling accurately captures long-time quantum dynamics, enabling variational Monte Carlo calculations of dynamical structure factors in one dimension and unbiased local-operator quench dynamics in two dimensions. We also show that this resolution of the support-mismatch problem extends beyond real-time dynamics, and may also be helpful for ground state variational Monte Carlo calculations.
Solving the ground state of quantum many-body systems remains a fundamental challenge in physics and chemistry. Recent advancements in quantum hardware have opened new avenues for addressing this challenge. Inspired by the quantum-enhanced Markov chain Monte Carlo (QeMCMC) algorithm, which was originally designed for sampling the Boltzmann distribution of classical spin models using quantum computers, we introduce a quantum-assisted variational Monte Carlo (QA-VMC) algorithm for solving the ground state of quantum many-body systems by adapting QeMCMC to sample the distribution of a (neural-network) wave function in VMC. The central question is whether such a quantum-assisted proposal can potentially offer a computational advantage over classical methods. Through numerical investigations for the Fermi-Hubbard model and molecular systems, we demonstrate that the quantum-assisted proposal exhibits larger absolute spectral gaps and reduced autocorrelation times compared to conventional classical proposals, leading to more efficient sampling and faster convergence to the ground state in VMC as well as a more accurate and precise estimation of physical observables. This advantage is especially pronounced for specific parameter ranges, where the ground-state configurations are more concentrated in some configurations separated by large Hamming distances. Our results underscore the potential of quantum-assisted algorithms to enhance classical variational methods for solving the ground state of quantum many-body systems.