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Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA)dynamics on a classical computer by a Quantum Monte Carlo whose parameters arechanged during the simulation, is a well established computational strategy tocope with the exponentially large Hilbert space. It has enjoyed some earlysuccesses but has also raised more recent criticisms. Here we investigate, onthe paradigmatic case of a one-dimensional transverse field Ising chain, twoissues related to SQA in its Path-Integral implementation: the question ofMonte Carlo vs physical (Schr\"odinger) dynamics and the issue of theimaginary-time continuum limit to eliminate the Trotter error. We show that,while a proper time-continuum limit is able to restitute the correctKibble-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 alarge number of Trotter time-slices, in the low-temperature ordered phase. Suchsampling problem, in turn, leads to SQA results which are apparently unrelatedto the coherent Schr\"odinger QA even at intermediate $\tau$.

On the dynamics of Simulated Quantum Annealing in random Ising chains