  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) preferentially the wave
function of a quantum particle, which explores the configuration space of the
problem, 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 Schr\"odinger 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 feedbacks might be helpful in
solving a computationally hard optimization problem by a quantum reinforcement
algorithm.
