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 advantages arise. First, by introducing analysing a version embeddability for matrices, we show that memoryless dynamics can simulate necessarily require memory. Second, extending notion space-time cost process $P$ to domain, prove an advantage over cost. Third, demonstrate set states accessible via Markovian master equations with controls is larger than those controls, leading, e.g., potential in cooling protocols.