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Physical Capacity Regions of Computation

Michael Rey

Published Sep 1, 2025• 0 citations
Publisher

Abstract

We establish a unified framework for understanding the fundamental physical limits of computation by developing a geometric theory of resource trade-offs. Our approach treats computation as a physical process constrained by thermodynamics, quantum mechanics, and relativity, leading to a multidimensional capacity region in resource space. We prove that any algorithm achieving error ε on problem class P must satisfy joint constraints across five dimensions: space S, time T, bandwidth H, energy E, and coherence C. The capacity region is characterized by fundamental physical constants: Landauer’s principle (kBT ln 2 per bit erasure), the Margolus-Levitin quantum speed limit (2E/(π~) operations per second), the Bekenstein bound (maximum information density), and statistical sampling limits. We provide explicit constructions showing these bounds are achievable within constant factors for canonical problems including matrix multiplication, covariance estimation, and Gaussian process inference. This framework unifies classical complexity theory with quantum information and thermodynamics, providing the first general theory of multi-resource computational limits with applications to algorithm design, hardware optimization, and fundamental physics.

DOI
10.20944/preprints202509.0010.v1

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We introduce a unified physical framework for algorithmic tradeoffs based on Space (S), Time (T), Energy (E), and Entropy (H) --- abbreviated as STEH. This framework integrates classical and quantum perspectives, and yields rigorous lower bounds on computational processes by linking memory, runtime, thermodynamic cost, and uncertainty. We derive a set of inequalities governing the achievable resource combinations and illustrate how they sharpen classical results such as Landauer's principle and quantum speed limits. The STEH tradeoff formalism provides a systematic approach to study computation as a constrained physical process, highlighting resource complementarity and impossibility results that shape the limits of algorithms.
Phase III of the Qubit Delusion series. Examines fundamental physical limits on computation: Landauer bound, Margolus-Levitin theorem, Bremermann limit, Bekenstein bound. Shows that quantum error-correction overhead (1e2-1e3 physical operations per logical operation) multiplies thermodynamic cost beyond what any polynomial algorithmic speedup can overcome. Proposes a falsifiable criterion: joules per solution on commercially relevant problems. Assesses all major quantum platforms against this criterion. Identifies reversible computing as the only paradigm that can approach the true physical limits. Full text and updates: https://papers.qnfo.org/papers/paper-physics-of-computation/
We study the maximum speed of quantum computation and how it is affected by limitations on physical resources. We show how the resulting concepts generalize to a broader class of physical models of computation within dynamical systems and introduce a specific algebraic structure representing these speed limits. We derive a family of quantum speed limit results in resource-constrained quantum systems with pure states and a finite dimensional state space, by using a geometric method based on right invariant action functionals on [Formula: see text]. We show that when the action functional is bi-invariant, the minimum time for implementing any quantum gate using a potentially time-dependent Hamiltonian is equal to the minimum time when using a constant Hamiltonian, thus constant Hamiltonians are time optimal for these constraints. We give an explicit formula for the time in these cases, in terms of the resource constraint. We show how our method produces a rich family of speed limit results, of which the generalized Margolus–Levitin theorem and the Mandelstam–Tamm inequality are special cases. We discuss the broader context of geometric approaches to speed limits in physical computation, including the way geometric approaches to quantum speed limits are a model for physical speed limits to computation arising from a limited resource.
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