Mathematical foundation of quantum annealing
Abstract
Quantum annealing is a generic name of quantum algorithms that use quantummechanical 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 infinitetime 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 classicalquantum mapping.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 December 2008
 DOI:
 10.1063/1.2995837
 arXiv:
 arXiv:0806.1859
 Bibcode:
 2008JMP....49l5210M
 Keywords:

 Quantum Physics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 51pages, 8 figures