Stochastic differential equation approach to understanding the population control bias in full configuration interaction quantum Monte Carlo

TL;DR

This paper analyzes the population control bias in FCIQMC, showing how a stochastic differential equation model can predict and potentially mitigate this bias, especially in bosonic systems without the sign problem.

Contribution

It introduces an exactly solvable stochastic differential equation model to understand and predict the population control bias in FCIQMC, providing analytical insights and scaling laws.

Findings

Shift estimator provides an upper bound for ground state energy in sign-problem-free cases.

Population control bias scales as a power law with walker number in Bose-Hubbard models.

Analytical bias prediction for non-interacting Bose-Hubbard Hamiltonian.

Abstract

We investigate a systematic statistical bias found in full configuration quantum Monte Carlo (FCIQMC) that originates from controlling a walker population with a fluctuating shift parameter. This bias can become the dominant error when the sign problem is absent, e.g. in bosonic systems. FCIQMC is a powerful statistical method for obtaining information about the ground state of a sparse and abstract matrix. We show that, when the sign problem is absent, the shift estimator has the nice property of providing an upper bound for the exact ground state energy and all projected energy estimators, while a variational estimator is still an upper bound to the exact energy with substantially reduced bias. A scalar model of the general FCIQMC population dynamics leads to an exactly solvable It\^o stochastic differential equation. It provides further insights into the nature of the bias and gives accurate analytical predictions for delayed cross-covariance and auto-covariance functions of the shift energy estimator and the walker number. The model provides a toe-hold on finding a cure for the population control bias. We provide evidence for non-universal power-law scaling of the population control bias with walker number in the Bose-Hubbard model for various estimators of the ground state energy based on the shift or on projected energies. For the specific case of the non-interacting Bose-Hubbard Hamiltonian we obtain a full analytical prediction for the bias of the shift energy estimator.