  Quantum annealing aims to provide a faster method for finding the minima of
complicated functions, compared to classical computing, so there is an
increasing interest in the relaxation dynamics of quantum spin systems.
Moreover, it is known that problems in quantum annealing caused by first order
phase transitions can be reduced via appropriate temporal adjustment of control
parameters, aimed at steering the system away from local minima. To do this
optimally, it would be helpful to predict the evolution of the system at the
level of macroscopic observables. Solving the dynamics of a quantum ensemble is
nontrivial, as it requires modelling not just the quantum spin system itself
but also its interaction with the environment, with which it exchanges energy.
An interesting alternative approach to the dynamics of quantum spin systems was
proposed about a decade ago. It involves creating a 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.
