Topological field theory and computing with instantons

TL;DR

This paper links topological sigma models, a class of topological field theories, to digital memcomputing machines, revealing how instantons and topological supersymmetry breaking enable efficient problem-solving in these dynamical systems.

Contribution

It introduces a novel connection between topological sigma models and digital memcomputing machines, explaining their efficiency through instantons and supersymmetry breaking.

Findings

DMMs are described by topological sigma models.

Instantons cause supersymmetry breaking, leading to long-range order.

DMMs can efficiently solve complex problems like prime factorization.

Abstract

Chern-Simons topological field theories TFTs are the only TFTs that have already found application in the description of some exotic strongly-correlated electron systems and the corresponding concept of topological quantum computing. Here, we show that TFTs of another type, specifically the gauge-field-less Witten-type TFTs known as topological sigma models, describe the recently proposed digital memcomputing machines (DMMs) - engineered dynamical systems with point attractors being the solutions of the corresponding logic circuit that solves a specific task. This result derives from the recent finding that any stochastic differential equation possesses a topological supersymmetry, and the realization that the solution search by a DMM proceeds via an instantonic phase. Certain TFT correlators in DMMs then reveal the presence of a transient long-range order both in space and time, associated with the effective breakdown of the topological supersymmetry by instantons. The ensuing non-locality and the low dimensionality of instantons are the physical reasons why DMMs can solve complex problems efficiently, despite their non-quantum character. We exemplify these results with the solution of prime factorization.