Representational power of selected neural network quantum states in second quantization

TL;DR

This paper investigates the representational capabilities of neural network quantum states, particularly neuron product states, demonstrating their universal approximation power for many-body wavefunctions in second quantization.

Contribution

It introduces neuron product states (NPS) as a generalization of restricted Boltzmann machines for Fermions and proves their universal approximation capabilities, along with elementary proofs for FNN and NNBF.

Findings

NPS can approximate any wavefunction under certain conditions

Feedforward neural networks are universally approximating in second quantization

Neural network backflow also has universal approximation capabilities

Abstract

Neural network quantum states emerge as a promising tool for solving quantum many-body problems. However, its successes and limitations are still not well-understood in particular for Fermions with complex sign structures. Based on our recent work [J. Chem. Theory Comput. 21, 10252-10262 (2025)], we generalizes the restricted Boltzmann machine to a more general class of states for Fermions, formed by product of `neurons' and hence will be referred to as neuron product states (NPS). NPS builds correlation in a very different way, compared with the closely related correlator product states (CPS) [H. J. Changlani, et al. Phys. Rev. B, 80, 245116 (2009)], which use full-rank local correlators. In constrast, each correlator in NPS contains long-range correlations across all the sites, with its representational power constrained by the simple function form. We prove that products of such simple nonlocal correlators can approximate any wavefunction arbitrarily well under certain mild conditions on the form of activation functions. In addition, we also provide elementary proofs for the universal approximation capabilities of feedforward neural network (FNN) and neural network backflow (NNBF) in second quantization. Together, these results provide a deeper insight into the neural network representation of many-body wavefunctions in second quantization.