Analysis of Discrete Modern Hopfield Networks in Open Quantum System

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.