Basic Theory of Quantum Annealing

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Simulated annealing is a widely used method to approximately solve various combinatorial optimization problems. It is necessary to control the temperature appropriately in simulated annealing to achieve the optimal state. In the present research project, we have established the method to use quantum fluctuations instead of thermal fluctuations for transitions between states, and applied this technique the traveling salesman problem. We have also introduced a new point of view to image restoration by using quantum fluctuations.Properties of the Ising model in a transverse field at the ground state is in many respects similar to those at finite temperatures without transverse field. It is thus expected that the transverse field plays similar roles as the temperature not just in equilibrium aspects but in dynamical properties as well. Indeed we have confirmed this conjecture by numerically calculating the ground state of the Ising model with time-dependent transverse field; we have shown that the quantum annealing gives a larger probability for the system to reach the true ground state than the thermal agitations. We have applied this idea to a typical optimization problem, the traveling salesman problem, and stochastic image restoration and have obtained very satisfactory results. In particular, in the latter application, we have found that appropriate combinations of thermal and quantum fluctuations yield restoration protocol robust against uncertainties in hyperparameter estimation.