Complex stochastic processes describe a rich and diverse range of phenomena: natural systems such as weather, geophysical, and biological processes; and societal constructs such as financial markets and traffic networks. The essential nature of these systems necessitates our ability to track and forecast their behaviour, a task for which we generally turn to large-scale simulations. Such simulations are resource intensive, requiring extreme amounts of memory, limiting the precision to which these complex systems can be studied. In particular, tracking time in such systems is invariably limited to a finite precision by the amount of memory available.
We show that this limitation is no longer the case when one has access to a quantum simulator. Our results demonstrate that with only a finite amount of quantum memory it is possible to track time in a complex system to arbitrarily fine precision. This is done by leveraging quantum effects to store time in a quantum superposition, allowing us to overcome the classical accuracy/storage trade-off, due to the ability to more efficiently isolate predictive features beyond what is possible with classical information processing. The presence of such extreme and unbounded reductions in resource requirements indicate the potential for emerging quantum technologies to revolutionise how we simulate, study, and understand complex systems. We discuss recent developments prescribing systematic construction methods for such efficient quantum simulators of structured complex processes operating in continuous-time.