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

Abstract

Continuous-time stochastic processes pervade everyday experience, and the simulation of models of these processes is of great utility. Classical models of systems operating in continuous-time must typically track an unbounded amount 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 their optimal classical counterparts to simulate the future of discrete-time processes, and we demonstrate that this advantage extends to the continuous-time regime. Moreover, we show that this reduction in the memory requirement can be unboundedly large, allowing for arbitrary precision even with a finite quantum memory. We provide a systematic method for finding superior quantum constructions, and a protocol for analogue simulation of continuous-time renewal processes with a quantum machine.

Memory-efficient tracking of complex temporal and symbolic dynamics with quantum simulators

“…In figure 4(a) we show that with increasingly fine coarse-graining C μ diverges, while M q appears to converge towards a finite value. This suggests an unbounded advantage of the q-machine over the ε-machine, similar to that seen for renewal processes [34]. Inspection of the steady-state density matrix spectrum indicates that it falls off approximately as 1/n 2 (figure 4(b)).…”

“…Inspection of the steady-state density matrix spectrum indicates that it falls off approximately as 1/n 2 (figure 4(b)). When the spectrum has this dependence, the associated entropy is finite [34] providing further support that M q is bounded in the continuum limit.…”

“…In parallel, the field of quantum computational mechanics has emerged [26][27][28][29][30][31][32][33][34][35][36][37][38]. A central result arising from these works is that quantum models of stochastic processes can operate whilst tracking less information about the past than even the optimal classical models [26].…”

“…A central result arising from these works is that quantum models of stochastic processes can operate whilst tracking less information about the past than even the optimal classical models [26]. This quantum advantage can be immense, and the gap between quantum and optimal classical memory requirements can grow unbounded [32][33][34]. As with classical computational mechanics, the focus has largely been on discrete-time processes, and hitherto the quantum computational mechanics of continuous-time processes has been restricted to tracking only a limited set of temporal dynamics (renewal processes), where the times between consecutive emissions are all drawn from the same distribution [34].…”

“…This quantum advantage can be immense, and the gap between quantum and optimal classical memory requirements can grow unbounded [32][33][34]. As with classical computational mechanics, the focus has largely been on discrete-time processes, and hitherto the quantum computational mechanics of continuous-time processes has been restricted to tracking only a limited set of temporal dynamics (renewal processes), where the times between consecutive emissions are all drawn from the same distribution [34]. Nevertheless, it was found that quantum models of such processes can exhibit unbounded advantages, requiring only finite memory to predict processes that classically need infinite past information to model.…”