Simplicity of mean-field theories in neural quantum states

Abstract

The utilization of artificial neural networks to represent quantum many-body wave functions has garnered significant attention, with enormous recent progress for both ground states and nonequilibrium dynamics. However, quantifying state complexity within this neural quantum states framework remains elusive. In this study, we address this key open question from a complementary point of view: Which states are simple to represent with neural quantum states? Concretely, we show on a general level that ground states of mean-field theories with permutation symmetry require only a limited number of independent neural network parameters. We analytically establish that, in the thermodynamic limit, convergence to the ground state of the fully connected transverse-field Ising model (TFIM), the mean-field Ising model, can be achieved with just one single parameter. Expanding our analysis, we explore the behavior of the one-parameter ansatz under breaking of the permutation symmetry. For that purpose, we consider the TFIM with tunable long-range interactions, characterized by an interaction exponent α. We show analytically that the one-parameter ansatz for the neural quantum state still accurately captures the ground state for a whole range of values for 0≤α≤1, implying a mean-field description of the model in this regime. Published by the American Physical Society 2024