Quantum Advantage in Simulating Stochastic Processes

We investigate the problem of simulating classical stochastic processes through quantum dynamics and present three scenarios where memory or time quantum advantages arise. First, by introducing and analyzing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of space-time cost of a stochastic process P to the quantum domain, we prove an advantage of the quantum cost of simulating P over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.

Automated summary

The study shows that quantum dynamics can simulate classical processes that require memory, with advantages in cost, and that Markovian master equations with quantum controls can access more classical states compared to those with classical controls, potentially leading to advantages in cooling protocols.