Fokker-Planck equation governing the distribution of walkers in AFQMC

TL;DR

This paper derives the Fokker-Planck equation for AFQMC, revealing the sampled wavefunction's nature and opening pathways for systematic accuracy improvements in quantum Monte Carlo simulations.

Contribution

The paper introduces the first derivation of the Fokker-Planck equation for AFQMC and proposes a numerical scheme to analyze and improve the method.

Findings

Reveals the wavefunction sampled by AFQMC is not exact even with an exact guiding wavefunction.

Provides a new framework for understanding the distribution of walkers in AFQMC.

Suggests avenues for systematic accuracy improvements in AFQMC.

Abstract

Auxiliary-field quantum Monte Carlo (AFQMC) is typically formulated as an open-ended random walk in an overcomplete space of Slater determinants, implemented through a Langevin equation. However, the explicit form of the underlying Fokker-Planck equation governing the walker population distribution has remained unknown. In this paper, we derive the Fokker-Planck equation for AFQMC and propose a novel numerical scheme to solve it. The solution of the Fokker-Planck equation reveals the wavefunction actually sampled by the AFQMC algorithm. Interestingly, we find that even when the exact ground state is used as a guiding wavefunction in constrained path AFQMC, contrary to the common assumption, the wavefunction sampled by AFQMC is not exact. Beyond clarifying several fundamental aspects of AFQMC, the availability of a Fokker-Planck equation formulation opens new avenues for systematically improving its accuracy, which we outline in this paper.