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

TL;DR

This paper introduces blurred sampling to improve the stability and reliability of Variational Monte Carlo methods, especially in the presence of wave function nodes, enabling more accurate quantum simulations.

Contribution

The authors propose a post-processing blurred sampling technique that addresses pathologies in VMC caused by nodes, without altering the original sampling process.

Findings

Enhanced stability of VMC estimators with blurred sampling

Successful application to large-scale spin dynamics problems

Robustness demonstrated on challenging quantum wave functions

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 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.