Simulated Quantum Annealing Can Be Exponentially Faster than Classical Simulated Annealing

TL;DR

This paper demonstrates that Simulated Quantum Annealing (SQA) can efficiently find the global minimum of a specific cost function in polynomial time, suggesting SQA inherits some quantum tunneling advantages over classical simulated annealing.

Contribution

The authors rigorously analyze a particular model to show SQA's polynomial-time performance, providing evidence that SQA benefits from quantum tunneling effects.

Findings

SQA efficiently samples the target distribution in polynomial time.

SQA inherits some advantages of quantum tunneling.

Classical simulated annealing takes exponential time on the same problem.

Abstract

Simulated Quantum Annealing (SQA) is a Markov Chain Monte-Carlo algorithm that samples the equilibrium thermal state of a Quantum Annealing (QA) Hamiltonian. In addition to simulating quantum systems, SQA has also been proposed as another physics-inspired classical algorithm for combinatorial optimization, alongside classical simulated annealing. However, in many cases it remains an open challenge to determine the performance of both QA and SQA. One piece of evidence for the strength of QA over classical simulated annealing comes from an example by Farhi, Goldstone and Gutmann . There a bit-symmetric cost function with a thin, high energy barrier was designed to show an exponential seperation between classical simulated annealing, for which thermal fluctuations take exponential time to climb the barrier, and quantum annealing which passes through the barrier and reaches the global minimum in poly time, arguably by taking advantage of quantum tunneling. In this work we apply a comparison method to rigorously show that the Markov chain underlying SQA efficiently samples the target distribution and finds the global minimum of this spike cost function in polynomial time. Our work provides evidence for the growing consensus that SQA inherits at least some of the advantages of tunneling in QA, and so QA is unlikely to achieve exponential speedups over classical computing solely by the use of quantum tunneling. Since we analyze only a particular model this evidence is not decisive. However, techniques applied here---including warm starts from the adiabatic path and the use of the quantum ground state probability distribution to understand the stationary distribution of SQA---may be valuable for future studies of the performance of SQA on cost functions for which QA is efficient.