Quantum and classical annealing in a continuous space with multiple local minima

TL;DR

This paper demonstrates that quantum annealing significantly outperforms classical simulated annealing in a continuous space optimization problem with multiple local minima, showing exponential speedup through numerical analysis.

Contribution

It provides the first detailed numerical comparison showing quantum annealing's power-law convergence and the role of tunneling, surpassing classical methods in a continuous domain.

Findings

Quantum annealing achieves power-law convergence, faster than simulated annealing.

Introducing quasi-global searches improves classical convergence but remains slower than quantum.

Quantum tunneling and diabatic dynamics effectively guide the system to the global minimum.

Abstract

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasi-global searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasi-global search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.