Alternative sampling for variational quantum Monte Carlo

TL;DR

This paper introduces a residual sampling strategy for variational quantum Monte Carlo that restores the validity of the Central Limit Theorem, enabling better control of random errors in energy estimates.

Contribution

The paper presents a novel residual sampling method that ensures the Central Limit Theorem applies, improving error control in variational quantum Monte Carlo calculations.

Findings

Residual sampling reestablishes the Central Limit Theorem in VMC.

Improved control over random errors in energy and variance estimates.

Method applicable to other operators and quantum Monte Carlo variants.

Abstract

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within Many-Body Quantum mechanics, and these multi-dimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum Quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the Central Limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative Generalised Central Limit theorem and non-Normal random error. In both cases the random error is not controlled. Here we consider a new `residual sampling strategy' that reintroduces the Central Limit Theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within Variational Monte Carlo are considered in detail, and the approach presented may be generalised to expectation values of other operators, and to other variants of the Quantum Monte Carlo method.