  Stochastic modelling of complex systems plays an essential, yet often
computationally intensive role across the quantitative sciences. Recent
advances in quantum information processing have elucidated the potential for
quantum simulators to exhibit memory advantages for such tasks. Heretofore, the
focus has been on lossless memory compression, wherein the advantage is
typically in terms of lessening the amount of information tracked by the model,
while -- arguably more practical -- reductions in memory dimension are not
always possible. Here we address the case of lossy compression for quantum
stochastic modelling of continuous-time processes, introducing a method for
coarse-graining in quantum state space that drastically reduces the requisite
memory dimension for modelling temporal dynamics whilst retaining near-exact
statistics. In contrast to classical coarse-graining, this compression is not
based on sacrificing temporal resolution, and brings memory-efficient,
high-fidelity stochastic modelling within reach of present quantum
technologies.
