Efficient anti-symmetrization of a neural network layer by taming the sign problem

TL;DR

This paper presents an efficient method for antisymmetrizing neural network layers, addressing the sign problem and enabling their use in quantum physics applications, with the effectiveness depending on activation function choice.

Contribution

It introduces a practical approach to antisymmetrize neural networks efficiently, overcoming factorial complexity and controlling the sign problem through activation function selection.

Findings

Efficient evaluation of antisymmetric projection for two-layer networks.

Effectiveness depends on activation function choice and initialization.

Smooth activation functions require weight re-scaling for proper antisymmetrization.

Abstract

Explicit antisymmetrization of a neural network is a potential candidate for a universal function approximator for generic antisymmetric functions, which are ubiquitous in quantum physics. However, this procedure is a priori factorially costly to implement, making it impractical for large numbers of particles. The strategy also suffers from a sign problem. Namely, due to near-exact cancellation of positive and negative contributions, the magnitude of the antisymmetrized function may be significantly smaller than before anti-symmetrization. We show that the anti-symmetric projection of a two-layer neural network can be evaluated efficiently, opening the door to using a generic antisymmetric layer as a building block in anti-symmetric neural network Ansatzes. This approximation is effective when the sign problem is controlled, and we show that this property depends crucially the choice of activation function under standard Xavier/He initialization methods. As a consequence, using a smooth activation function requires re-scaling of the neural network weights compared to standard initializations.