Fermionic Neural Networks through the lens of Group Theory

TL;DR

This paper explores how group theory can be used to design neural network wave functions that respect physical symmetries in quantum many-body problems, revealing why determinant-based approaches are particularly effective.

Contribution

It demonstrates that determinants can be viewed as group convolutions, providing a theoretical foundation for symmetry-respecting neural network ans"atze in quantum physics.

Findings

Determinants are a form of group convolution.

Group representation theory explains the efficiency of determinant-based methods.

Symmetry incorporation via group theory enhances neural quantum state modeling.

Abstract

We present an overview of the method of Neural Quantum States applied to the many-body problem of atomic nuclei. Through the lens of group representation theory, we focus on the problem of constructing neural-network ans\"atze that respect physical symmetries. We explicitly prove that determinants, which are among the most common methods to build antisymmetric neural-network wave functions, can be understood as the result of a group convolution. We also identify the reason why this construction is so efficient in practice compared to other group convolutional operations. We conclude that group representation theory is a promising avenue to incorporate explicitly symmetries in Neural Quantum States.