Pattern-recalling processes in quantum Hopfield networks far from saturation

TL;DR

This paper develops a quantum-mechanical extension of the Hopfield neural network model, deriving deterministic equations for pattern recall processes influenced by quantum noise, especially in the regime far from saturation.

Contribution

It introduces a quantum-mechanical variant of the Hopfield model and analytically derives macroscopic equations for pattern recall under quantum noise, extending classical thermal noise models.

Findings

Derived deterministic equations for order parameters in quantum Hopfield networks.

Analyzed pattern recall processes under quantum noise far from saturation.

Extended classical models to include quantum-mechanical effects in neural networks.

Abstract

As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of 'heat bath' that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the 'temperature'. Several studies concerning the pattern-recalling processes of the Hopfield model governed by the Glauber-dynamics at finite temperature were already reported. However, we might extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo method (the quantum MCMC), we analytically derive macroscopically deterministic equations of order parameters such as 'overlap' in a quantum-mechanical variant of the Hopfield neural networks (let us call "quantum Hopfield model" or "quantum Hopfield networks"). For the case in which non-extensive number $p$ of patterns are embedded via asymmetric Hebbian connections, namely, $p/N \to 0$ for the number of neuron $N \to \infty$ ('far from saturation'), we evaluate the recalling processes for one of the built-in patterns under the influence of quantum-mechanical noise.