Improved energies and wave function accuracy with Weighted Variational Monte Carlo

TL;DR

This paper introduces a weighted VMC approach that improves wave function accuracy, especially in probability tails, by optimizing the wave function in different regions of state space, demonstrated on a spin chain model.

Contribution

It presents a new theoretical interpretation of VMC as a gradient flow with projection, enabling tailored accuracy in different probability regions.

Findings

Reduces ground state energy error by a factor of 2

Decreases local energy errors by factors of 10^2 to 10^4

Demonstrates improved accuracy on the Heisenberg model

Abstract

Neural network parametrizations have increasingly been used to represent the ground and excited states in variational Monte Carlo (VMC) with promising results. However, traditional VMC methods only optimize the wave function in regions of peak probability. The wave function is uncontrolled in the tails of the probability distribution, which can limit the accuracy of the trained wavefunction approximation. To improve the approximation accuracy in the probability tails, this paper interprets VMC as a gradient flow in the space of wave functions, followed by a projection step. From this perspective, arbitrary probability distributions can be used in the projection step, allowing the user to prioritize accuracy in different regions of state space. Motivated by this theoretical perspective, the paper tests a new weighted VMC method on the antiferromagnetic Heisenberg model for a periodic spin chain. Compared to traditional VMC, weighted VMC reduces the error in the ground state energy by a factor of 2 and it reduces the errors in the local energies away from the mode by large factors of $10^2$--$10^4$.