Quantum Effects in Neural Networks

Hidetoshi Nishimori, Yoshihiko Nonomura

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


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, α, and the strength of the transverse field, Δ. 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 Δ. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.

Original languageEnglish
Pages (from-to)3780-3796
Number of pages17
Journaljournal of the physical society of japan
Issue number12
Publication statusPublished - 1996
Externally publishedYes


  • Hopfield model
  • Macrovariables
  • Neural networks
  • Phase diagram
  • Quantum effects

ASJC Scopus subject areas

  • Physics and Astronomy(all)


Dive into the research topics of 'Quantum Effects in Neural Networks'. Together they form a unique fingerprint.

Cite this