  Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA)
dynamics on a classical computer by a Quantum Monte Carlo whose parameters are
changed during the simulation, is a well established computational strategy to
cope with the exponentially large Hilbert space. It has enjoyed some early
successes but has also raised more recent criticisms. Here we investigate, on
the paradigmatic case of a one-dimensional transverse field Ising chain, two
issues related to SQA in its Path-Integral implementation: the question of
Monte Carlo vs physical (Schr\"odinger) dynamics and the issue of the
imaginary-time continuum limit to eliminate the Trotter error. We show that,
while a proper time-continuum limit is able to restitute the correct
Kibble-Zurek scaling of the residual energy $\varepsilon_\mathrm{res}(\tau)\sim
\tau^{-1/2}$ for the ordered case --- $\tau$ being the total annealing time
---, the presence of disorder leads to a characteristic sampling crisis for a
large number of Trotter time-slices, in the low-temperature ordered phase. Such
sampling problem, in turn, leads to SQA results which are apparently unrelated
to the coherent Schr\"odinger QA even at intermediate $\tau$.
