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arXiv · 2510.19914

Fokker-Planck equation governing the distribution of walkers in AFQMC

Alfred Li·Ankit Mahajan·Sandeep Sharma
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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.

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Alfred Li, Ankit Mahajan, Sandeep Sharma. 2025-10-22. Fokker-Planck equation governing the distribution of walkers in AFQMC. https://arxiv.org/abs/2510.19914

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