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

TL;DR

The paper derives a rigorous, Hamiltonian-based bound on the thermodynamic cost of an individual computation, unifying classical and quantum, deterministic and noisy processes. It shows the cost is bounded by a combination of heat $Q$, noise via $-\log_{2}p(y|x)$, and a protocol-complexity term $K(\mathcal{P})$, with a refined bound using $K(x|y,\mathcal{P}^*)$ that ties the minimal algorithmic information $x$ must carry given $y$ to physical resources. A generalized Zurek bound $\frac{\beta}{\ln 2}Q(x\shortrightarrow y)+\log_{2}\frac{1}{p(y|x)}+K(\mathcal{P})+\gamma_{\mathfrak{C}}\ge K(x|y)$ formalizes a tradeoff among heat, noise, and protocol description length, and the approach yields an algorithmic fluctuation theorem flavor by linking symmetry breaking in the forward map to thermodynamic cost. The results are illustrated with erasure and randomization tasks, and the discussion connects to the Second Law, Landauer's principle, and the Physical Church-Turing thesis, suggesting new avenues for fully quantum formulations and algorithmic thermodynamics.

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.

Figures (2)

• Figure 1: We analyze the cost of mapping a single input $x$ to a single output $y$, given a system coupled to a heat bath and a work reservoir that drives protocol $\mathcal{P}$. We show that the conditional Kolmogorov complexity $K(x\vert y)$ is a fundamental minimal cost, which bounds accessible combinations of heat, noise, and complexity of the protocol.
• Figure 2: Illustration of our results using the erasure task, where an input string $x$ is mapped to an output $y$ consisting of all 0s, possibly in a noisy fashion. Four different processes that achieve our bound (\ref{['eq:resInt2Simpler']}): a) Landauer erasure, where all possible inputs are mapped to the target $y$. b) Free expansion, where all inputs are allowed to thermalize to a uniform equilibrium distribution. c) Swap, where a single input and a single output are swapped, while all other states are left in place. d) Controlled expansion, where a single input is allowed to thermalize to a uniform equilibrium distribution.