Cavity approach to variational quantum mechanics

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.