  The numerical simulation of dynamical phenomena in interacting quantum
systems is a notoriously hard problem. Although a number of promising numerical
methods exist, they often have limited applicability due to the growth of
entanglement or the presence of the so-called sign problem. In this work, we
develop an importance sampling scheme for the simulation of quantum spin
dynamics, building on a recent approach mapping quantum spin systems to
classical stochastic processes. The importance sampling scheme is based on
identifying the classical trajectory that yields the largest contribution to a
given quantum observable. An exact transformation is then carried out to
preferentially sample trajectories that are close to the dominant one. We
demonstrate that this approach is capable of reducing the temporal growth of
fluctuations in the stochastic quantities, thus extending the range of
accessible times and system sizes compared to direct sampling. We discuss
advantages and limitations of the proposed approach, outlining directions for
further developments.
