Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9512142

Quantum Hopfield Model

Yoshihiko Nonomura, Hidetoshi Nishimori

Published Dec 21, 1995• 0 citations
arXivHTMLPDF

Abstract

The Hopfield model in a transverse field is investigated in order to clarify how quantum fluctuations affect the macroscopic behavior of neural networks. Using the Trotter decomposition and the replica method, we find that the $α$ (the ratio of the number of stored patterns to the system size)-$Δ$ (the strength of the transverse field) phase diagram of this model in the ground state resembles the $α$-$T$ phase diagram of the Hopfield model quantitatively, within the replica-symmetric and static approximations. This fact suggests that quantum fluctuations play quite similar roles to thermal fluctuations in neural networks as long as macroscopic properties are concerned.

DOI
10.48550/arxiv.cond-mat/9512142

Related papers

Quantum Effects in Neural Networks

Hidetoshi Nishimori, Yoshihiko Nonomura

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.

arXiv:cond-mat/9603105•Journal of the Physical Society of Japan•Dec 15, 1996•43 citations

Statistical mechanics of a Hopfield neural-network model in a transverse field

Yu‐qiang Ma, Yueming Zhang, Y. G. +1 more

We construct a soluble model to describe the retrieval properties of the quantum transverse Hopfield model in neural networks. A set of equations for the order parameters describing retrieval and spin-glass phases of the Hopfield model in a transverse field are obtained. Based on these equations, phase diagrams are examined and the memory storage capacity of the networks is analyzed as a function of the temperature and transverse field.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1993•10 citations

Statistical mechanics of a multiconnected Hopfield neural-network model in a transverse field

Yu‐qiang Ma, Chang-De Gong

The Hopfield neural-network model with p-spin interactions in the presence of a transverse field is introduced and solved exactly in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. In the phase diagrams drawn as a function of the temperature, the important results such as reentrance are found, and the effects of the quantum fluctuations on the phase transitions, the retrieval phase, and the storage ratio \ensuremath{\alpha} are examined.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1995•5 citations

Exact Replica Symmetric Solution for Transverse Field Hopfield Model under Finite Trotter Size

Koki Okajima, Yoshiyuki Kabashima

We analyze the quantum Hopfield model in which an extensive number of patterns are embedded in the presence of a uniform transverse field. This analysis employs the replica method under the replica symmetric ansatz on the Suzuki-Trotter representation of the model, while keeping the number of Trotter slices $M$ finite. The statistical properties of the quantum Hopfield model in imaginary time are reduced to an effective $M$-spin long-range classical Ising model, which can be extensively studied using a dedicated Monte Carlo algorithm. This approach contrasts with the commonly applied static approximation, which ignores the imaginary time dependency of the order parameters, but allows $M \to \infty$ to be taken analytically. During the analysis, we introduce an exact but fundamentally weaker static relation, referred to as the quasi-static relation. We present the phase diagram of the model with respect to the transverse field strength and the number of embedded patterns, indicating a small but quantitative difference from previous results obtained using the static approximation.

arXiv:2411.02012•Journal of the Physical Society of Japan•Mar 12, 2025•1 citation

Tricritical Points and Reentry in the Quantum Hopfield Neural-Network Model

Xi Qin, Yu‐qiang Ma

We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.

Communications in Theoretical Physics•Sep 15, 2000•4 citations

Phase diagram of quantum generalized Potts-Hopfield neural networks

Eliana Fiorelli, Igor Lesanovsky, Markus Müller

Abstract We introduce and analyze an open quantum generalization of the q-state Potts-Hopfield neural network (NN), which is an associative memory model based on multi-level classical spins. The dynamics of this many-body system is formulated in terms of a Markovian master equation of Lindblad type, which allows to incorporate both probabilistic classical and coherent quantum processes on an equal footing. By employing a mean field description we investigate how classical fluctuations due to temperature and quantum fluctuations effectuated by coherent spin rotations affect the ability of the network to retrieve stored memory patterns. We construct the corresponding phase diagram, which in the low temperature regime displays pattern retrieval in analogy to the classical Potts-Hopfield NN. When increasing quantum fluctuations, however, a limit cycle phase emerges, which has no classical counterpart. This shows that quantum effects can qualitatively alter the structure of the stationary state manifold with respect to the classical model, and potentially allow one to encode and retrieve novel types of patterns.

arXiv:2109.10140•New Journal of Physics•Feb 11, 2022•17 citations
The Hopfield model in a transverse field is investigated in order to clarify how quantum fluctuations affect the macroscopic behavior of neural networks. Using the Trotter decomposition and the replica method, we find that the ααα (the ratio of the number of stored patterns to the system size)-ΔΔΔ (the strength of the transverse field) phase diagram of this model in the ground state resembles the ααα-TTT phase diagram of the Hopfield model quantitatively, within the replica-symmetric and static approximations. This fact suggests that quantum fluctuations play quite similar roles to thermal fluctuations in neural networks as long as macroscopic properties are concerned.
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, ααα, and the strength of the transverse field, ΔΔΔ. 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 ΔΔΔ. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.
We analyze the quantum Hopfield model in which an extensive number of patterns are embedded in the presence of a uniform transverse field. This analysis employs the replica method under the replica symmetric ansatz on the Suzuki-Trotter representation of the model, while keeping the number of Trotter slices MMM finite. The statistical properties of the quantum Hopfield model in imaginary time are reduced to an effective MMM-spin long-range classical Ising model, which can be extensively studied using a dedicated Monte Carlo algorithm. This approach contrasts with the commonly applied static approximation, which ignores the imaginary time dependency of the order parameters, but allows M→∞M \to \inftyM→∞ to be taken analytically. During the analysis, we introduce an exact but fundamentally weaker static relation, referred to as the quasi-static relation. We present the phase diagram of the model with respect to the transverse field strength and the number of embedded patterns, indicating a small but quantitative difference from previous results obtained using the static approximation.
We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.
Abstract We introduce and analyze an open quantum generalization of the q-state Potts-Hopfield neural network (NN), which is an associative memory model based on multi-level classical spins. The dynamics of this many-body system is formulated in terms of a Markovian master equation of Lindblad type, which allows to incorporate both probabilistic classical and coherent quantum processes on an equal footing. By employing a mean field description we investigate how classical fluctuations due to temperature and quantum fluctuations effectuated by coherent spin rotations affect the ability of the network to retrieve stored memory patterns. We construct the corresponding phase diagram, which in the low temperature regime displays pattern retrieval in analogy to the classical Potts-Hopfield NN. When increasing quantum fluctuations, however, a limit cycle phase emerges, which has no classical counterpart. This shows that quantum effects can qualitatively alter the structure of the stationary state manifold with respect to the classical model, and potentially allow one to encode and retrieve novel types of patterns.