Mathematical Foundation of Quantum Annealing

TL;DR

This paper reviews the mathematical and theoretical foundations of quantum annealing, including convergence theorems, error reduction strategies, and connections to classical simulated annealing, providing a rigorous basis for understanding its effectiveness in optimization.

Contribution

It presents new theorems for convergence conditions of quantum annealing and proposes methods to reduce errors through optimized annealing schedules.

Findings

Convergence to the optimal state is guaranteed under specific asymptotic control parameter behaviors.

Error reduction is achievable by carefully designing the annealing schedule.

The convergence condition for classical simulated annealing is derived from quantum adiabaticity principles.

Abstract

Quantum annealing is a generic name of quantum algorithms to use quantum-mechanical fluctuations to search for the solution of optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundation of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schroedinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence both for the Schroedinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.