Wasserstein Quantum Monte Carlo: A Novel Approach for Solving the Quantum Many-Body Schrödinger Equation

Published: 21 Sept 2023, Last Modified: 16 Jan 2024NeurIPS 2023 spotlightEveryoneRevisionsBibTeX
Keywords: Quantum Monte Carlo, Schrödinger equation, Wasserstein Fisher-Rao gradient flow
TL;DR: We minimize the energy functional of quantum systems by following a projected Wasserstein gradient flow in the space of distributions, which results in faster convergence compared to the conventional Quantum Variational Monte Carlo.
Abstract: Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher--Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than the Fisher--Rao metric, and corresponds to *transporting* the probability mass, rather than *teleporting* it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.
Supplementary Material: zip
Submission Number: 13805