We consider the minimal thermodynamic cost of an individual computation, where a single input $x$ is mapped to a single output $y$. In prior work, Zurek proposed that this cost was given by $K(x\vert y)$, the conditional Kolmogorov complexity of $x$ given $y$ (up to an additive constant which does not depend on $x$ or $y$). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that $K(x\vert y)$ is a minimal cost of mapping $x$ to $y$ that must be paid using some combination of heat, noise, and protocol complexity, implying a tradeoff between these three resources. Our result is a kind of "algorithmic fluctuation theorem" with implications for the relationship between the Second Law and the Physical Church-Turing thesis.

翻译：我们考虑单次计算的最小热力学成本，其中单个输入$x$映射到单个输出$y$。在先前的工作中，祖雷克提出该成本由$K(x\vert y)$（即给定$y$条件下$x$的条件柯尔莫哥洛夫复杂度）给出，结果精确至一个与$x$或$y$无关的加法常数。然而，该结论基于非严格论证，仅适用于确定性计算，且对协议选择存在任意依赖（通过加法常数体现）。本文利用随机热力学，从严格的哈密顿量表述出发推导出祖雷克界的广义版本。该界适用于所有量子与经典过程（无论含噪或确定性），并显式刻画了协议依赖关系。我们证明$K(x\vert y)$是将$x$映射到$y$的最小成本，该成本必须通过热量、噪声与协议复杂度的某种组合来支付，从而揭示这三类资源间的权衡关系。本研究可视为一种“算法涨落定理”，对理解热力学第二定律与物理丘奇-图灵论题之间的关系具有重要启示。