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

## TL;DR

This paper demonstrates that quantum devices can simulate continuous-time stochastic processes with significantly less memory than classical models, enabling arbitrary precision with finite quantum memory.

## Contribution

It extends the known memory advantage of quantum simulations from discrete to continuous-time processes and provides systematic methods and protocols for such quantum simulations.

## Key findings

- Quantum models require less memory than classical models for continuous-time processes.
- Memory advantage can be unbounded, allowing arbitrary precision.
- A systematic method and protocol for quantum simulation of renewal processes are provided.

## 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.