A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation

TL;DR

This paper introduces a theoretically grounded normalizing flow framework that enables efficient sampling of wavefunctions for solving the Electronic Schrödinger Equation, improving over traditional MCMC methods.

Contribution

It proposes a novel normalizing flow-based ansatz with specific properties for quantum wavefunctions, including equivariance and efficient sampling, supported by rigorous proofs.

Findings

Normalizing flow ansatz satisfies quantum properties

Efficient sampling reduces computational cost

Framework captures wavefunction cusps and generalizes across molecules

Abstract

A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.