Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations 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 Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger 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.
Skip Nav Destination
Article navigation
December 2008
Research Article|
December 15 2008
Mathematical foundation of quantum annealing
Satoshi Morita;
Satoshi Morita
a)
1
International School for Advanced Studies (SISSA)
, Via Beirut 2-4, I-34014 Trieste, Italy
Search for other works by this author on:
Hidetoshi Nishimori
Hidetoshi Nishimori
2Department of Physics,
Tokyo Institute of Technology
, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
Search for other works by this author on:
a)
Electronic mail: smorita@sissa.it.
J. Math. Phys. 49, 125210 (2008)
Article history
Received:
June 04 2008
Accepted:
September 10 2008
Citation
Satoshi Morita, Hidetoshi Nishimori; Mathematical foundation of quantum annealing. J. Math. Phys. 1 December 2008; 49 (12): 125210. https://doi.org/10.1063/1.2995837
Download citation file:
Sign in
Don't already have an account? Register
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Sign in via your Institution
Sign in via your InstitutionPay-Per-View Access
$40.00