Analysis of discrete modern Hopfield networks in open quantum system

TL;DR

This paper introduces a quantum model for discrete modern Hopfield networks, combining quantum spin systems with classical and quantum effects, and analyzes their phase behavior.

Contribution

It generalizes the open quantum Hopfield network to include higher-order interactions and quantum effects, providing new insights into their phase structure.

Findings

Distinct phase behaviors from previous models

Additional stable fixed points in ferromagnetic and limit cycle phases

Analytical and numerical characterization of fixed points

Abstract

The modern Hopfield network, proposed by Krotov and Hopfield, is a mathematical generalization of the Hopfield network, which is a basic model of associative memory that employs higher-order interactions. This study introduces an open quantum model for discrete modern Hopfield networks that generalizes the open quantum Hopfield network. Our model integrates dissipative quantum spin systems, governed by quantum master equations, with classical hopping terms and additional quantum effects through a transverse field. We analytically examined the behavior of the stable fixed points and numerically determined the phase diagram. The results demonstrated qualitatively distinct behaviors from the open quantum Hopfield network, showing that the ferromagnetic and limit cycle phases have additional stable fixed points.