Thermodynamic Matrix Exponentials and Thermodynamic Parallelism

TL;DR

This work provides a thermodynamic algorithm for exponentiating a real matrix and introduces the concept of thermodynamic parallelism to explain this speedup, stating that thermodynamic noise provides a resource leading to effective parallelization of computations.

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.

Figures (5)

• Figure 1: Analog and thermodynamic approaches. (A) The target matrix $e^{-A\tau}$ has $d^2$ elements that must be experimentally characterized. An individual experiment can be viewed as sampling these elements from a distribution, due to inherent randomness in the experiment (either unintentional in the analog case or intentional in the thermodynamic case). (B) A naive analog approach would sample $d$ elements at a time, while (C) The thermodynamic approach samples all $d^2$ elements at once by using noise as a resource, an advantage we call thermodynamic parallelism.
• Figure 2: Possible hardware architecture for matrix exponentiation. The overall system is composed of two subunits. One subunit (bottom) outputs, as a vector of voltages, Gaussian noise with zero mean and covariance matrix proportional to $A$. The other subunit (top) takes this output as its noise source and directly simulates the stochastic differential equation in \ref{['eq:specific_OU']}.
• Figure 3: Error as a function of analog integration time for varying matrix dimension. The error is the Frobenius norm of the sampled matrix minus the true matrix exponential. (A) Matrices are positive-definite and drawn from a Wishart distribution (with $2d$ degrees of freedom). (B) Matrices are asymmetric, with elements drawn from the Haar distribution over orthogonal matrices. To make the eigenvalues have positive real part, a term $c\mathbb{I}$ is added as described in the text, with $c=1.1$. Vertical lines represent the times $t_C$ when the error falls below a threshold. Inset: Crossing time $t_C$ versus matrix dimension $d$.
• Figure 4: Relative error as a function of analog integration time for varying matrix dimension. The error is the relative Frobenius norm of the sampled matrix from the true matrix exponential. (A) Matrices are positive-definite and drawn from a Wishart distribution (with $2d$ degrees of freedom). (B) Matrices are asymmetric, with elements drawn from the Haar distribution over orthogonal matrices. To make the eigenvalues have positive real part, a term $c\mathbb{I}$ is added as described in the text, with $c=1.1$. Vertical lines represent the times $t_C$ when the error falls below a threshold. Inset: Crossing time $t_C$ versus matrix dimension $d$, with the dashed line showing $t_C = d$.
• Figure 5: Potential circuit diagram for matrix exponentiation hardware. The lower subunit produces Gaussian noise, at the output of the integrators, with zero mean and a covariance matrix proportional to $A$. The top subunit directly simulates the desired stochastic differential equation in Eq. \ref{['eq:specific_OU']}.