Optimization by Quantum Annealing: Lessons from Simple Cases

This paper investigates the basic behavior and performance of simulated quantum annealing (QA) in comparison with classical annealing (CA). Three simple one dimensional case study systems are considered, namely a parabolic well, a double well, and a curved washboard. The time dependent Schr\"odinger evolution in either real or imaginary time describing QA is contrasted with the Fokker Planck evolution of CA. The asymptotic decrease of excess energy with annealing time is studied in each case, and the reasons for differences are examined and discussed. The Huse-Fisher classical power law of double well CA is replaced with a different power law in QA. The multi-well washboard problem studied in CA by Shinomoto and Kabashima and leading classically to a logarithmic annealing even in the absence of disorder, turns to a power law behavior when annealed with QA. The crucial role of disorder and localization is briefly discussed.