Convergence theorems for quantum annealing

We prove several theorems to give sufficient conditions for convergence of quantum annealing, which is a protocol to solve generic optimization problems by quantum dynamics. In particular, the property of strong ergodicity is proved for the path-integral Monte Carlo implementation of quantum annealing for the transverse Ising model under a power decay of the transverse field. This result is to be compared with the much slower inverse-log decay of temperature in the conventional simulated annealing. Similar results are proved for the Green's function Monte Carlo approach. Optimization problems in continuous space of particle configurations are also discussed.