Heavy-tailed random error in quantum Monte Carlo

TL;DR

This paper demonstrates that the common assumption of the Central Limit Theorem's validity in quantum Monte Carlo methods often fails, leading to heavy-tailed errors that require a generalized approach for accurate error estimation.

Contribution

It shows that the Central Limit Theorem is often invalid in Variational Monte Carlo, necessitating the use of a Generalised Central Limit Theorem for error analysis.

Findings

The distribution of random errors can be heavy-tailed and non-Gaussian.

Standard error estimates may be unreliable in many cases.

The approach applies broadly to quantum Monte Carlo methods.

Abstract

The combination of continuum Many-Body Quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the Central Limit Theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the Variational Monte Carlo method, the Central Limit Theorem has limited validity, or is invalid and must be replaced by a Generalised Central Limit Theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the Central Limit Theorem is not valid. The approach used is generally applicable to characterising the random error of estimates, and to Quantum Monte Carlo methods beyond Variational Monte Carlo.