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Communications in Theoretical Physics

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

Xi Qin, Yu‐qiang Ma

Published Sep 15, 2000• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/34/2/217

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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.
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.
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.
Phase diagrams and magnetization curves of a diluted spin-1 transverse Ising model in a random field on honeycomb, square, and simple-cubic lattices, respectively, are investigated by the use of an effective-field theory with correlations. The tricritical point is found in the system, in contrast to the corresponding spin-1/2 Ising counterpart. The behavior of the tricritical point is also examined as a function of applied transverse field. The possible reentrant phenomena displayed by the system due to the competition effects that occur for appropriate ranges of the random and transverse fields are investigated.