Kibble-Zurek mechanism in simulated annealing and quantum annealing

We study errors of quantum annealing and simulated annealing to highlight better performance of quantum annealing over simulated annealing. Quantum annealing and simulated annealing perform optimization through a quantum adiabatic evolution and a quasi-static evolution respectively. In both methods, dynamics across a phase transition plays a crucial role. The Kibble-Zurek mechanism is known as an underlying physics of defect formation during a time-evolution across a phase transition. We apply an argument for the Kibble-Zurek mechanism to the error generation of quantum annealing and simulated annealing. We show that, for the disordered Ising chain, the kink density and residual energy per spin of quantum annealing decay as (ln τ)-2 and (ln τ)-4 respectively, whereas those of simulated annealing decay as (ln τ)-1 and (ln τ)-2 with the annealing rate -1/τ. These results imply better performance of quantum annealing. We also develop our theory for a two-dimensional spin-glass model.