  Restricted Boltzmann machines (RBMs) are a class of neural networks that have
been successfully employed as a variational ansatz for quantum many-body wave
functions. Here, we develop an analytic method to study quantum many-body spin
states encoded by random RBMs with independent and identically distributed
complex Gaussian weights. By mapping the computation of ensemble-averaged
quantities to statistical mechanics models, we are able to investigate the
parameter space of the RBM ensemble in the thermodynamic limit. We discover
qualitatively distinct wave functions by varying RBM parameters, which
correspond to distinct phases in the equivalent statistical mechanics model.
Notably, there is a regime in which the typical RBM states have near-maximal
entanglement entropy in the thermodynamic limit, similar to that of Haar-random
states. However, these states generically exhibit nonergodic behavior in the
Ising basis, and do not form quantum state designs, making them distinguishable
from Haar-random states.
