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.

中文翻译：

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模拟随机过程的量子优势

我们研究了通过量子动力学模拟经典随机过程的问题，并提出了出现内存或时间量子优势的三种情况。首先，通过介绍和分析随机矩阵可嵌入性问题的量子形式，我们证明了无记忆量子动力学可以模拟经典过程，而经典过程必然需要记忆。其次，通过扩展随机过程的时空成本概念$P$ 到量子域，我们证明了模拟的量子成本的优势 $P$超过经典的成本。第三，我们证明了通过具有量子控制的马尔可夫主方程可访问的经典状态的集合要大于通过经典控制可访问的经典状态的集合，从而例如导致了冷却协议的潜在优势。