Quantum Effects in Neural Networks

We develop the statistical mechanics of the Hopfield model in a transversefield to investigate how quantum fluctuations affect the macroscopic behaviorof neural networks. When the number of embedded patterns is finite, the Trotterdecomposition reduces the problem to that of a random Ising model. It turns outthat the effects of quantum fluctuations on macroscopic variables play the sameroles as those of thermal fluctuations. For an extensive number of embeddedpatterns, we apply the replica method to the Trotter-decomposed system. Theresult is summarized as a ground-state phase diagram drawn in terms of thenumber of patterns per site, $\alpha$, and the strength of the transversefield, $\Delta$. The phase diagram coincides very accurately with that of theconventional classical Hopfield model if we replace the temperature T in thelatter model by $\Delta$. Quantum fluctuations are thus concluded to be quitesimilar to thermal fluctuations in determination of the macroscopic behavior ofthe present model.Comment: 34 pages, LaTeX, 9 PS figures, uses jpsj.st