Mathematical aspects of quantum annealing

Sufficient conditions for convergence of quantum annealing are derived for optimization problems represented by the Ising model. Three different types of time evolution are considered; the real-time Schroedinger equation, the path-integral Monte Carlo method and the Green's function Monte Carlo method. It is proved that the system under each dynamics reaches the target solution in the limit of infinite time if the transverse field, representing the strength of quantum fluctuations, decreases inversely proportionally to the power of time in the asymptotic region