What Is the Optimal Annealing Schedule in Quantum Annealing

In many real-life situations in engineering (and in other disciplines), we need to solve an optimization problem: we want an optimal design, we want an optimal control, etc. One of the main problems in optimization is avoiding local maxima (or minima). One of the techniques that helps with solving this problem is annealing: whenever we find ourselves in a possibly local maximum, we jump out with sonic probability and continue search for the true optimum. A natural way to organize such a probabilistic perturbation of the deterministic optimization is to use quantum effects. It turns out that often, quantum annealing works much better than non-quantum one. Quantum annealing is the main technique behind the only commercially available computational devices that use quantum effects - D-Wave computers. The efficiency of quantum annealing depends on the proper selection of the annealing schedule, i.e., schedule that describes how the perturbations decrease with time. Empirically, it has been found that two schedules work best: power law and exponential ones. In this paper, we provide a theoretical explanation for these empirical successes, by proving that these two schedules are indeed optimal (in some reasonable sense).