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

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 autocovariance functions of the shift energy estimator and the walker number. The model provides a toehold on finding a cure for the population control bias. We provide evidence for nonuniversal 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 noninteracting Bose-Hubbard Hamiltonian we obtain a full analytical prediction for the bias of the shift energy estimator.