Engines of Parsimony: Part I; Limits on Computational Rates in Physical Systems

TL;DR

This paper establishes universal upper bounds on the maximum sustained computation rate in physical systems, highlighting the importance of reversible computation and geometric constraints, with implications for classical, quantum, and relativistic regimes.

Contribution

It derives universal bounds on computational rates based on geometric and physical constraints, extending to quantum, classical, and relativistic systems, and suggests practical implementation approaches.

Findings

Universal upper bound scales as √AV for classical and quantum systems.

Reversible computation is necessary to reach the maximum bounds.

Relativistic effects impose more restrictive bounds for large regions.

Abstract

We analyse the maximum achievable rate of sustained computation for a given convex region of three dimensional space subject to geometric constraints on power delivery and heat dissipation. We find a universal upper bound across both quantum and classical systems, scaling as $\sqrt{AV}$ where $V$ is the region volume and $A$ its area. Attaining this bound requires the use of reversible computation, else it falls to scaling as $A$. By specialising our analysis to the case of Brownian classical systems, we also give a semi-constructive proof suggestive of an implementation attaining these bounds by means of molecular computers. For regions of astronomical size, general relativistic effects become significant and more restrictive bounds proportional to $\sqrt{AR}$ and $R$ are found to apply, where $R$ is its radius. It is also shown that inhomogeneity in computational structure is generally to be avoided. These results are depicted graphically in Figure 1.