Quantum Effects in Neural Networks

TL;DR

This paper explores how quantum fluctuations influence neural network behavior by analyzing a quantum Hopfield model, revealing that quantum effects mimic thermal fluctuations in determining macroscopic states.

Contribution

It introduces a quantum statistical mechanics framework for the Hopfield model, showing quantum fluctuations have effects similar to thermal fluctuations in neural networks.

Findings

Quantum fluctuations act like thermal fluctuations in the model.

The phase diagram with quantum effects matches the classical one when temperature is replaced by quantum strength.

Quantum effects do not fundamentally alter the macroscopic behavior of the network.

Abstract

We develop the statistical mechanics of the Hopfield model in a transverse field to investigate how quantum fluctuations affect the macroscopic behavior of neural networks. When the number of embedded patterns is finite, the Trotter decomposition reduces the problem to that of a random Ising model. It turns out that the effects of quantum fluctuations on macroscopic variables play the same roles as those of thermal fluctuations. For an extensive number of embedded patterns, we apply the replica method to the Trotter-decomposed system. The result is summarized as a ground-state phase diagram drawn in terms of the number of patterns per site, $\alpha$, and the strength of the transverse field, $\Delta$. The phase diagram coincides very accurately with that of the conventional classical Hopfield model if we replace the temperature T in the latter model by $\Delta$. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.