Physical Capacity Regions of Computation

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.