Optimization by a quantum reinforcement algorithm

A reinforcement algorithm solves a classical optimization problem by introducing a feedback to the system, which slowly changes the energy landscape and converges the algorithm to an optimal solution in the configuration space. Here, we use this strategy to concentrate (localize) the wave function of a quantum particle, which explores the configuration space of the problem, preferentially on an optimal configuration. We examine the method by solving numerically the equations governing the evolution of the system, which are similar to the nonlinear Schrodinger equations, for small problem sizes. In particular, we observe that reinforcement increases the minimal energy gap of the system in a quantum annealing algorithm. Our numerical simulations and the latter observation show that such kind of quantum feedback might be helpful in solving a computationally hard optimization problem by a quantum reinforcement algorithm.