Fermi Sets: Universal and interpretable neural architectures for fermions

TL;DR

Fermi Sets introduces a neural architecture for fermionic wavefunctions that is both universal and physically interpretable, with provably minimal basis requirements and successful application to complex quantum systems.

Contribution

The paper presents Fermi Sets, a novel neural architecture that efficiently approximates fermionic wavefunctions using a small number of basis functions and permutation-invariant networks, improving interpretability and scalability.

Findings

Fermi Sets can approximate any continuous fermionic wavefunction with minimal basis functions.

The architecture achieves universal approximation with basis size K=1 in 1D, K=2 in 2D, and linear growth in higher dimensions.

Applied to metallic solid hydrogen, Fermi Sets outperform diffusion Monte Carlo benchmarks.

Abstract

We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability. As a numerical illustration, a single Fermi Sets model applied to metallic solid hydrogen in three dimensions, trained simultaneously across multiple nuclear geometries, surpasses all diffusion Monte Carlo benchmarks.