Comparative Study of the Performance of Quantum Annealing and Simulated Annealing

TL;DR

This paper explores the theoretical relationship between quantum annealing and simulated annealing, showing their spectral equivalence and discussing implications for efficiency and implementation in solving Ising model problems.

Contribution

It establishes a spectral mapping between classical Markovian dynamics and quantum Hamiltonians, revealing differences in efficiency and providing insights into their comparative performance.

Findings

Quantum and classical operators share eigenvalues.

Simulated annealing can be efficiently simulated by quantum annealing.

Quantum annealing is easier to implement and more flexible.

Abstract

Relations of simulated annealing and quantum annealing are studied by a mapping from the transition matrix of classical Markovian dynamics of the Ising model to a quantum Hamiltonian and vice versa. It is shown that these two operators, the transition matrix and the Hamiltonian, share the eigenvalue spectrum. Thus, if simulated annealing with slow temperature change does not encounter a difficulty caused by an exponentially long relaxation time at a first-order phase transition, the same is true for the corresponding process of quantum annealing in the adiabatic limit. One of the important differences between the classical-to-quantum mapping and the converse quantum-to-classical mapping is that the Markovian dynamics of a short-range Ising model is mapped to a short-range quantum system, but the converse mapping from a short-range quantum system to a classical one results in long-range interactions. This leads to a difference in efficiencies that simulated annealing can be efficiently simulated by quantum annealing but the converse is not necessarily true. We conclude that quantum annealing is easier to implement and is more flexible than simulated annealing. We also point out that the present mapping can be extended to accommodate explicit time dependence of temperature, which is used to justify the quantum-mechanical analysis of simulated annealing by Somma, Batista, and Ortiz. Additionally, an alternative method to solve the non-equilibrium dynamics of the one-dimensional Ising model is provided through the classical-to-quantum mapping.