Non-linear biases, stochastically-sampled effective Hamiltonians and spectral functions in quantum Monte Carlo methods

TL;DR

This paper investigates systematic biases in quantum Monte Carlo methods caused by non-linear expectation values, analyzing their origins and proposing correction strategies within stochastic Hamiltonian frameworks.

Contribution

It introduces a detailed analysis of non-linear biases in QMC, especially in KP-FCIQMC, and demonstrates how basis adjustments can mitigate these errors.

Findings

Non-linear expectation values cause systematic biases in QMC.

Moving to an appropriate basis reduces these biases.

The approach extends to correlation function QMC methods.

Abstract

In this article we study examples of systematic biases that can occur in quantum Monte Carlo methods due to the accumulation of non-linear expectation values, and approaches by which these errors can be corrected. We begin with a study of the Krylov-projected FCIQMC (KP-FCIQMC) approach, which was recently introduced to allow efficient, stochastic calculation of dynamical properties. This requires the solution of a sampled effective Hamiltonian, resulting in a non-linear operation on these stochastic variables. We investigate the probability distribution of this eigenvalue problem to study both stochastic errors and systematic biases in the approach, and demonstrate that such errors can be significantly corrected by moving to a more appropriate basis. This is lastly expanded to include consideration of the correlation function QMC approach of Ceperley and Bernu, showing how such an approach can be taken in the FCIQMC framework.