  A local and distributive algorithm is proposed to find an optimal trial
wave-function minimizing the Hamiltonian expectation in a quantum system. To
this end, the quantum state of the system is connected to the Gibbs state of a
classical system with the set of couplings playing the role of variational
parameters. The average energy is written within the replica-symmetric
approximation and the optimal parameters are obtained by a heuristic message
passing algorithm based on the Bethe approximation. The performance of this
approximate algorithm depends on the structure and quality of the trial
wave-functions; starting from a classical system of isolated elements, i.e.
mean-field approximation, and improving on that by considering the higher order
many-body interactions. The method is applied to some disordered quantum Ising
models in transverse fields and the results are compared with the exact ones
for small systems.
