Quantum walk in a reinforced free-energy landscape: Quantum annealing with reinforcement

Providing an optimal path to a quantum annealing algorithm is key to finding good approximate solutions to computationally hard optimization problems. Reinforcement is one of the strategies that can be used to circumvent the exponentially small energy gaps of the system in the annealing process. Here a time-dependent reinforcement term is added to the Hamiltonian in order to give lower energies to the most probable states of the evolving system. In this study, we take a local entropy in the configuration space for the reinforcement and apply the algorithm to a number of easy and hard optimization problems. The reinforced algorithm performs better than the standard quantum annealing algorithm in the quantum search problem, where the optimal parameters behave very differently depending on the number of solutions. Moreover, the reinforcements can change the discontinuous phase transitions of the mean-field p-spin model ($p>2$) to a continuous transition. The algorithm's performance in the binary perceptron problem is also superior to that of the standard quantum annealing algorithm, which already works better than a classical simulated annealing algorithm.Comment: 28 pages, 12 figure