  We explore the dynamics of quantum spin systems in two and three dimensions
using an exact mapping to classical stochastic processes. In recent work we
explored the effectiveness of sampling around the mean field evolution as
determined by a stochastically averaged Weiss field. Here, we show that this
approach can be significantly extended by sampling around the instantaneous
Weiss field associated with each stochastic trajectory taken separately. This
trajectory-resolved approach incorporates sample to sample fluctuations and
allows for longer simulation times. We demonstrate the utility of this approach
for quenches in the two-dimensional and three-dimensional quantum Ising model.
We show that the method is particularly advantageous in situations where the
average Weiss-field vanishes, but the trajectory-resolved Weiss fields are
non-zero. We discuss the connection to the gauge-P phase space approach, where
the trajectory-resolved Weiss field can be interpreted as a gauge degree of
freedom.
