Thermodynamic Matrix Exponentials and Thermodynamic Parallelism

TL;DR

This paper introduces a thermodynamic algorithm for matrix exponentiation using coupled oscillators, demonstrating linear asymptotic speedup and proposing thermodynamic parallelism as a resource for computational efficiency.

Contribution

It presents a novel thermodynamic algorithm for matrix exponentiation and introduces the concept of thermodynamic parallelism to explain speedup mechanisms.

Findings

The algorithm achieves linear asymptotic speedup with matrix dimension.

A simple electrical circuit can implement the thermodynamic matrix exponential.

Thermodynamic noise acts as a resource for effective parallelization.

Abstract

Thermodynamic computing exploits fluctuations and dissipation in physical systems to efficiently solve various mathematical problems. For example, it was recently shown that certain linear algebra problems can be solved thermodynamically, leading to an asymptotic speedup scaling with the matrix dimension. The origin of this "thermodynamic advantage" has not yet been fully explained, and it is not clear what other problems might benefit from it. Here we provide a new thermodynamic algorithm for exponentiating a real matrix, with applications in simulating linear dynamical systems. We describe a simple electrical circuit involving coupled oscillators, whose thermal equilibration can implement our algorithm. We also show that this algorithm also 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, and we hypothesize this as a mechanism to explain thermodynamic advantage more generally.