  We investigate a stochastic approach to non-equilibrium quantum spin systems
based on recent insights linking quantum and classical dynamics. Exploiting a
sequence of exact transformations, quantum expectation values can be recast as
averages over classical stochastic processes. We illustrate this approach for
the quantum Ising model by extracting the Loschmidt amplitude and the
magnetization dynamics from the numerical solution of stochastic differential
equations. We show that dynamical quantum phase transitions are accompanied by
clear signatures in the associated classical distribution functions, including
the presence of enhanced fluctuations. We demonstrate that the method is
capable of handling integrable and non-integrable problems in a unified
framework, including those in higher dimensions.
