What Is the Optimal Annealing Schedule in Quantum Annealing

Abstract

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 some 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 nonquantum 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).

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

“…For large g, which we define for all protocols as g ≡ τ a /τ I , the excitation number decays as a power law. For any g and N , our analytically solvable model (Protocol 1) always outperforms the other protocols, although all of them show scaling similar to 1/g for large g. In numerous other tests (not shown), we found that all non-powerlaw schedules, e.g., with g(t) decaying exponentially, had much worse performance for the same values of τ I , τ a , and T , in agreement with [22]. Figure 3 also shows the data that we used to extrapolate the results to larger N .…”

“…If a protocol is optimal, i.e., outperforms all other protocols at some imposed conditions on the QA schedule and for a certain class of H I , it must remain optimal after time-rescaling, t → λt, in the Schrödinger equation, because the latter merely means the change of timecounting procedure. It has been recently proved [22] that if such a protocol exists, it must correspond to a powerlaw decay of the coupling: g(t) ∼ t a . We will use this result because it strongly restricts the class of the schedules that should be tested in order to prove the optimality.…”

“…To test this hypothesis, we employ the result in [22] that allows us only to compare the performance of the solvable protocol with a family of the protocols with a power law decay of the coupling, g(t) ∼ 1/t α , and identical for each protocol fully random H I , as well as the annealing time τ a and T /τ a . First, we note that the protocols with α < 1 produce definitely worse than n ∼ 1/τ a scaling for the excitations if we set T /τ a = const.…”

Many Known Quantum Algorithms Are Optimal: Symmetry-Based Proofs

“…Researchers have found that empirically, the following two schedules work best: power law and exponential ones [7], [22], [23]. In this section, following [14], we describe the method and corresponding schedules in some detail, and prove that these two schedules are indeed optimal (in some reasonable sense).…”

“…In this paper, we summarize results of several papers of ours in which optimality was proven for specific empirically effective quantum algorithms. From this viewpoint, this paper can be viewed as an extended version of several of our previous published papers, in particular, our quantum annealing paper [14].…”