Thermodynamic Linear Algebra

TL;DR

This paper introduces thermodynamic algorithms for linear algebra problems, leveraging classical thermodynamics principles to achieve asymptotic speedups over digital methods, potentially enabling near-term hardware acceleration.

Contribution

It establishes a novel connection between thermodynamics and linear algebra, providing simple algorithms for key problems with proven asymptotic speedups.

Findings

Thermodynamic algorithms achieve linear scaling speedups.

Algorithms solve linear systems, invert matrices, compute determinants, and Lyapunov equations.

The approach exploits ergodicity, entropy, and equilibration principles.

Abstract

Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities, so this approach remains long-term in timescale. Here we consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. In this work, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for (1) solving linear systems of equations, (2) computing matrix inverses, (3) computing matrix determinants, and (4) solving Lyapunov equations. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computing hardware.