  We propose a message-passing algorithm to compute the Hamiltonian expectation
with respect to an appropriate class of trial wave functions for an interacting
system of fermions. To this end, we connect the quantum expectations to average
quantities in a classical system with both local and global interactions, which
are related to the variational parameters and use the Bethe approximation to
estimate the average energy within the replica-symmetric approximation. The
global interactions, which are needed to obtain a good estimation of the
average fermion sign, make the average energy a nonlocal function of the
variational parameters. We use some heuristic minimization algorithms to find
approximate ground states of the Hubbard model on random regular graphs and
observe significant qualitative improvements with respect to the mean-field
approximation.
