Quantum advantage in simulating stochastic processes.

2020 
We investigate the problem of simulating classical stochastic processes through quantum dynamics, and present three scenarios where memory/time quantum advantages arise. First, by introducing and analysing 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 exponential 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 an advantage in cooling protocols. To achieve this last point, we develop the notion of continuous thermo-majorisation, which strengthens the so-called "second laws" by including the typical constraint that the dynamics is (effectively) Markovian.
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