  The standard quantum annealing algorithm tries to approach the ground state
of a classical system by slowly decreasing the hopping rates of a quantum
random walk in the configuration space of the problem, where the on-site
energies are provided by the classical energy function. In a quantum
reinforcement algorithm, the annealing works instead by increasing gradually
the strength of the on-site energies according to the probability of finding
the walker on each site of the configuration space. Here, by using the
path-integral Monte Carlo simulations of the quantum algorithms, we show that
annealing via reinforcement can significantly enhance the success probability
of the quantum walker. More precisely, we implement a local version of the
quantum reinforcement algorithm, where the system wave function is replaced by
an approximate wave function using the local expectation values of the system.
We use this algorithm to find solutions to a prototypical constraint
satisfaction problem (XORSAT) close to the satisfiability to unsatisfiability
phase transition. The study is limited to small problem sizes (a few hundreds
of variables), nevertheless, the numerical results suggest that quantum
reinforcement may provide a useful strategy to deal with other computationally
hard problems and larger problem sizes even as a classical optimization
algorithm.
