A space–time tradeoff for implementing a function with master equation dynamics

We consider the problem of constructing a physical system that evolves according to some specified conditional distribution. We restrict attention to physical systems that can be modeled as a time-inhomogeneous continuous-time Markov chain (CTMC) over a finite state space, which includes many of the systems considered in stochastic thermodynamics. Examples range from constructing a logical gate to be used in a digital circuit to constructing an entire digital computer. It is known that many conditional distributions over a space $X$ cannot be implemented by any CTMC, even approximately. This raises the question of how they can arise in the real world. Here we focus on the case where the conditional distribution is a (single-valued) function $f$. Any $f$ over a set of "visible" states $X$ can be implemented --- if the system has access to some additional "hidden" states not in $X$. Motivated by engineering considerations, we consider a natural decomposition of any such CTMC into a sequence of "hidden" timesteps, demarcated by changes in the set of allowed transitions between states. We demonstrate a tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given $f$ using a CTMC, analogous to space / time tradeoffs in theoretical computer science.

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