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Continuous-time stochastic processes pervade everyday experience, and thesimulation of models of these processes is of great utility. Classical modelsof systems operating in continuous-time must typically track an unboundedamount of information about past behaviour, even for relatively simple models,enforcing limits on precision due to the finite memory of the machine. However,quantum machines can require less information about the past than even theiroptimal classical counterparts to simulate the future of discrete-timeprocesses, and we demonstrate that this advantage extends to thecontinuous-time regime. Moreover, we show that this reduction in the memoryrequirement can be unboundedly large, allowing for arbitrary precision evenwith a finite quantum memory. We provide a systematic method for findingsuperior quantum constructions, and a protocol for analogue simulation ofcontinuous-time renewal processes with a quantum machine.

Superior memory efficiency of quantum devices for the simulation of continuous-time stochastic processes