Generalized Zurek's bound on the cost of an individual classical or quantum computation

TL;DR

This paper generalizes Zurek's thermodynamic cost bound for individual classical and quantum computations, establishing a rigorous, protocol-dependent relation involving heat, noise, and complexity, applicable to noisy or deterministic processes.

Contribution

It provides a rigorous Hamiltonian derivation of a generalized Zurek's bound that applies to all quantum and classical processes, capturing protocol dependence and resource tradeoffs.

Findings

The bound applies to noisy and deterministic processes.

It reveals a tradeoff between heat, noise, and protocol complexity.

The result links thermodynamics with computational complexity and the Church-Turing thesis.

Abstract

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.