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arXiv:2311.12759 · Physical Review Research

Thermodynamic matrix exponentials and thermodynamic parallelism

Samuel Duffield, Maxwell Aifer, Gavin E. Crooks, Thomas D. Ahle, Patrick J. Coles

Published Feb 10, 2025• 9 citations
arXivHTMLPDF

Abstract

Thermodynamic computing exploits fluctuations and dissipation in physical systems to efficiently solve various mathematical problems. It was recently shown that certain linear algebra problems can be solved thermodynamically, leading to a speedup scaling with the matrix dimension. Here, we provide a thermodynamic algorithm for exponentiating a real matrix. We describe a simple electrical circuit involving coupled oscillators, which can implement our algorithm. We also show that this algorithm provides an asymptotic speedup that is linear in the dimension. Finally, we introduce the concept of thermodynamic parallelism to explain this speedup, stating that thermodynamic noise provides a resource leading to effective parallelization of computations.

DOI
10.1103/physrevresearch.7.013147

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Thermodynamic computing exploits fluctuations and dissipation in physical systems to efficiently solve various mathematical problems. It was recently shown that certain linear algebra problems can be solved thermodynamically, leading to a speedup scaling with the matrix dimension. Here, we provide a thermodynamic algorithm for exponentiating a real matrix. We describe a simple electrical circuit involving coupled oscillators, which can implement our algorithm. We also show that this algorithm provides an asymptotic speedup that is linear in the dimension. Finally, we introduce the concept of thermodynamic parallelism to explain this speedup, stating that thermodynamic noise provides a resource leading to effective parallelization of computations.
Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal "size" of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to 22002^{200}2200.
Abstract Linear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. 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. Here, 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 solving linear systems of equations, computing matrix inverses, and computing matrix determinants. 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 computers.
Thermodynamic computing has emerged as a promising paradigm for accelerating computation by harnessing the thermalization properties of physical systems. This work introduces a novel approach to solving quadratic programming problems using thermodynamic hardware. By incorporating a thermodynamic subroutine for solving linear systems into the interior-point method, we present a hybrid digital-analog algorithm that outperforms traditional digital algorithms in terms of speed. Notably, we achieve a polynomial asymptotic speedup compared to conventional digital approaches. Additionally, we simulate the algorithm for a support vector machine and predict substantial practical speedups with only minimal degradation in solution quality. Finally, we detail how our method can be applied to portfolio optimization and the simulation of nonlinear resistive networks.
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