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Representational power of selected neural network quantum states in second quantization

Zhendong Li, Tong Zhao, Bohan Zhang

TL;DR

This work establishes the universal representational power of neuron product states (NPS) for fermionic wavefunctions in second quantization by deriving conditions on activation functions that guarantee arbitrary-state approximation. It extends RBM-based approaches by proving universality for NPS under mild assumptions and provides elementary proofs for the universal approximation capabilities of both feedforward neural networks (FNN) and neural network backflow (NNBF) within the same framework. The results connect NPS to existing variational ansätze such as CPS and clarify how long-range, simple nonlocal correlators can jointly approximate complex many-body states. Together, these findings offer a rigorous foundation for using neural-network architectures to represent and optimize quantum many-body states, with implications for variational Monte Carlo and beyond.

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.

Representational power of selected neural network quantum states in second quantization

TL;DR

This work establishes the universal representational power of neuron product states (NPS) for fermionic wavefunctions in second quantization by deriving conditions on activation functions that guarantee arbitrary-state approximation. It extends RBM-based approaches by proving universality for NPS under mild assumptions and provides elementary proofs for the universal approximation capabilities of both feedforward neural networks (FNN) and neural network backflow (NNBF) within the same framework. The results connect NPS to existing variational ansätze such as CPS and clarify how long-range, simple nonlocal correlators can jointly approximate complex many-body states. Together, these findings offer a rigorous foundation for using neural-network architectures to represent and optimize quantum many-body states, with implications for variational Monte Carlo and beyond.

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.

Paper Structure

This paper contains 10 sections, 8 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Feedforward neural network (FNN) states
  3. Neural network backflow (NNBF) states
  4. Neural network product states (NPS)
  5. Conclusion
  6. Representation of $\Psi$ by multilinear polynomials
  7. Representation of functions $f(b+\vec{w}^T\vec{z})$
  8. Introduction of intermediate functions $\tilde{\Psi}(\vec{n})$ and $\tilde{\Psi}_+(\vec{n})$
  9. Recursive approximation of $\tilde{\Psi}_+(\vec{n})$ by NPS
  10. Proof of Eq. \ref{['eq:approx']}

Key Result

Theorem 1

Given sufficiently large $N_h$, the FNN ansatz eq:FNN1 is universal for representing wavefunction in second quantization, denoted as a vector by $\vec{\Psi} \in \mathbb{R}^{D_K}$ in the occupation number representation, in the sense that for any wavefunction $\Psi:\mathcal{B}^K \rightarrow \mathbb{R

Figures (2)

  • Figure 1: Illustration of different neural-network wavefunction ansätze for $K=6$: (a) restricted Boltzmann machine (RBM) and neuron product states (NPS); (b) feedforward neural networks (FNN); (c) correlator product stateschanglani2009approximating (CPS). Each red circle in (a) and (b) represents a hidden neuron.
  • Figure 2: Hierarchical structure of neurons $\kappa(b_{I_{t,k}}+ \vec{\omega}^T_{I_{t,k}}\vec{z}_{I_{t,k}})$ used to eliminate the leading terms in $\mathcal{Z}_{I_{t,k}}$ recursively. Specifically, we can match the coefficients in Eq. \ref{['eq:gexpansion']} from the highest tier to the lowest tier recursively using neurons with supports shown in the figure.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Lemma 2: Gordan's lemma
  • Lemma 3
  • proof
  • ...and 3 more