Statistical Analysis of Quantum Annealing

TL;DR

This paper compares classical and quantum annealing methods for solving optimization problems, establishing their mathematical equivalence under certain conditions, deriving success probability bounds, and analyzing differences through numerical simulations.

Contribution

It provides a theoretical and numerical comparison between classical and quantum annealing, including a lower bound on success probability and a classical MCMC implementation of quantum annealing.

Findings

Classical and quantum annealing are mathematically equivalent for the same Ising models.

Derived a lower bound on the success probability of quantum annealing.

Identified discrepancies between MCMC-based classical and quantum annealing in solving optimization problems.

Abstract

Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.