Entropy, Precision, and the Hardness of Continuous-Time Computation

The rate of improvement in conventional computational methods has slowed in recent years. While alternative frameworks such as gate-based quantum computation are being explored, significant resources have shifted toward task-specific hardware and algorithms—GPUs for linear algebraic operations and quantum annealers for optimization. Beyond the discrete-time paradigm underlying digital and quantum gate models, continuous-time dynamical systems provide a distinct framework for computation. Research in this direction seeks to identify the fundamental resources that determine their performance on hard problems.

We examine how numerical or physical precision constrains the computational capacity of such systems and how it relates to invariant measures of chaos and information loss. Using quantities such as Lyapunov spectra, escape rates, and Kolmogorov–Sinai entropy, we characterize continuous-time solvers applied to combinatorial optimization problems such as (MAX)SAT and QAP. We derive an inequality linking attainable precision to chaos-theoretic invariants, enabling prediction of a threshold required for reliable computation. These findings suggest that precision acts as the exchange rate between two fundamental resources in computation: time and energy.