Digitally Optimized Initializations for Fast Thermodynamic Computing

TL;DR

This paper proposes a hybrid digital-thermodynamic method that uses optimized initializations to significantly speed up relaxation times in thermodynamic computing, enabling faster matrix operations.

Contribution

It introduces a novel initialization scheme inspired by the Mpemba effect that accelerates thermodynamic relaxation in physical systems performing computations.

Findings

Analytic expressions for speedup derived

Numerical analysis of matrix inversion and determinant computation

Spectral analysis of the Fokker-Planck operator

Abstract

Thermodynamic computing harnesses the relaxation dynamics of physical systems to perform matrix operations. A key limitation of such approaches is the often long thermalization time required for the system to approach equilibrium with sufficient accuracy. Here, we introduce a hybrid digital-thermodynamic algorithm that substantially accelerates relaxation through optimized initializations inspired by the Mpemba effect. In the proposed scheme, a classical digital processor efficiently computes an initialization that suppresses slow relaxation modes, after which the physical system performs the remaining computation through its intrinsic relaxation dynamics. We focus on overdamped Langevin dynamics for quadratic energy landscapes, analyzing the spectral structure of the associated Fokker-Planck operator and identifying the corresponding optimal initial covariances. This yields a predictable reduction in thermalization time, determined by the spectrum of the encoded matrix. We derive analytic expressions for the resulting speedups and numerically analyze thermodynamic implementations of matrix inversion and determinant computation as concrete examples. Our results show that optimized initialization protocols provide a simple and broadly applicable route to accelerating thermodynamic computations.