Representational power of selected neural network quantum states in second quantization

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 [Z. Wu et al., J. Chem. Theory Comput. 21, 10252-10262 (2025)], we generalize the restricted Boltzmann machine Ansatz to a more general class of states for fermions, which is formed by the 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 [H. J. Changlani et al., Phys. Rev. B 80, 245116 (2009)], which use full-rank local correlators. In contrast, 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 networks and neural network backflow in second quantization. Together, these results provide a deeper insight into the neural network representation of many-body wavefunctions in second quantization.