Dynamical Analysis of Quantum Annealing

Quantum annealing aims to provide a faster method than classical computing for finding the minima of complicated functions, and it has created increasing interest in the relaxation dynamics of quantum spin systems. Moreover, problems in quantum annealing caused by first-order phase transitions can be reduced via appropriate temporal adjustment of control parameters, and in order to do this optimally, it is helpful to predict the evolution of the system at the level of macroscopic observables. Solving the dynamics of quantum ensembles is nontrivial, requiring modeling of both the quantum spin system and its interaction with the environment with which it exchanges energy. An alternative approach to the dynamics of quantum spin systems was proposed about a decade ago. It involves creating stochastic proxy dynamics via the Suzuki-Trotter mapping of the quantum ensemble to a classical one (the quantum Monte Carlo method), and deriving from this new dynamics closed macroscopic equations for macroscopic observables using the dynamical replica method. In this chapter, we give an introduction to this approach, focusing on the ideas and assumptions behind the derivations, and on its potential and limitations.