Thermodynamics of classifiers

Abstract

Reducing computational accuracy can lower energy consumption, and this principle is widely used to improve energy efficiency in computing. This raises a fundamental question: what is the quantitative relationship between error and thermodynamic cost in information processing? In this study, we derive the error-cost trade-off in the binary classifier by considering classification based on Markov processes. We obtain the lower bounds on the Bayes error in terms of thermodynamic costs such as entropy production and dynamical activity. Our results show that when entropy production or dynamical activity vanishes, the Bayes error reaches $1/2$, equivalent to random guessing, while greater thermodynamic costs enable lower error. This establishes a fundamental trade-off between error and cost in information processing by thermodynamic systems. Because the Bayes error provides the lowest achievable error among all possible classifiers, the classification error cannot fall below the obtained bounds given the entropy production or dynamical activity. We also discuss the quantum generalization and show that the Bayes error of the quantum classifier is bounded from below by the variance of the Hamiltonian.