$\widetilde{O}(N^2)$ Representation of General Continuous Anti-symmetric Function

TL;DR

This paper introduces an efficient ${ ilde O}(N^2)$ ansatz for representing continuous anti-symmetric functions in quantum mechanics, enabling universal approximation and advancing neural network design for wave functions.

Contribution

It proposes a novel permutation-equivariant ansatz that can universally represent any continuous anti-symmetric function, addressing open problems in quantum wave function representation.

Findings

The ansatz can represent any continuous anti-symmetric function.

It can incorporate the determinant structure for universal representation.

The approach is efficient and generalizable for neural network design.

Abstract

In quantum mechanics, the wave function of fermion systems such as many-body electron systems are anti-symmetric (AS) and continuous, and it is crucial yet challenging to find an ansatz to represent them. This paper addresses this challenge by presenting an ${\widetilde O}(N^2)$ ansatz based on permutation-equivariant functions. We prove that our ansatz can represent any AS continuous functions, and can accommodate the determinant-based structure proposed by Hutter [14], solving the proposed open problems that ${O}(N)$ Slater determinants are sufficient to provide universal representation of AS continuous functions. Together, we offer a generalizable and efficient approach to representing AS continuous functions, shedding light on designing neural networks to learn wave functions.