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arXiv:1103.3787 · Journal of Physics Conference Series

Pattern-recalling processes in quantum Hopfield networks far from saturation

Jun-ichi Inoue

Published May 1, 2011• 14 citations
arXivHTMLPDF

Abstract

As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of 'heat bath' that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the 'temperature'. Several studies concerning the pattern-recalling processes of the Hopfield model governed by the Glauber-dynamics at finite temperature were already reported. However, we might extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo method (the quantum MCMC), we analytically derive macroscopically deterministic equations of order parameters such as 'overlap' in a quantum-mechanical variant of the Hopfield neural networks (let us call quantum Hopfield model or quantum Hopfield networks ). For the case in which non-extensive number p of patterns are embedded via asymmetric Hebbian connections, namely, p/N → 0 for the number of neuron N → ∞ ('far from saturation'), we evaluate the recalling processes for one of the built-in patterns under the influence of quantum-mechanical noise.

DOI
10.1088/1742-6596/297/1/012012

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cond-mat.stat-mechCross-listedarXiv:2608.08226•Aug 11, 2026
As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of 'heat bath' that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the 'temperature'. Several studies concerning the pattern-recalling processes of the Hopfield model governed by the Glauber-dynamics at finite temperature were already reported. However, we might extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo method (the quantum MCMC), we analytically derive macroscopically deterministic equations of order parameters such as 'overlap' in a quantum-mechanical variant of the Hopfield neural networks (let us call quantum Hopfield model or quantum Hopfield networks ). For the case in which non-extensive number p of patterns are embedded via asymmetric Hebbian connections, namely, p/N → 0 for the number of neuron N → ∞ ('far from saturation'), we evaluate the recalling processes for one of the built-in patterns under the influence of quantum-mechanical noise.
As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of ‘heat bath’ that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the ‘temperature’. Several studies concerning the pattern-recalling processes of the Hopfield model governed by the Glauber-dynamics at finite temperature were already reported. However, we might extend the ‘thermal noise’ to the quantum-mechanical variant. In this paper, in terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo method (the quantum MCMC), we analytically derive macroscopically deterministic equations of order parameters such as ‘overlap’ in a quantum-mechanical variant of the Hopfield neural networks (let us call quantum Hop eld model or quantum Hop eld networks). For the case in which non-extensive number p of patterns are embedded via asymmetric Hebbian connections, namely, p/N → 0 for the number of neuron N → ∞ (‘far from saturation’), we evaluate the recalling processes for one of the built-in patterns under the influence of quantummechanical noise.
Abstract We present extensive simulations of a quantum version of the Hopfield neural network to explore its emergent behavior. The system is a network of N qubits oscillating at a given Ω frequency and which are coupled via Lindblad jump operators built with local fields hi depending on some given stored patterns. Our simulations show the emergence of pattern-antipattern oscillations of the overlaps with the stored patterns similar (for large Ω and small temperature) to those reported within a recent mean-field description of such a system, and which are originated deterministically by the quantum term including s x i qubit operators. However, in simulations we observe that such oscillations are stochastic due to the interplay between noise and the inherent metastability of the pattern attractors induced by quantum oscillations, and then are damped in finite systems when one averages over many quantum trajectories. In addition, we report the system behavior for large number of stored patterns at the lowest temperature we can reach in simulations (namely T = 0.005 T C ). Our study reveals that for small-size systems the quantum term of the Hamiltonian has a negative effect on storage capacity, decreasing the overlap with the starting memory pattern for increased values of Ω and number of stored patterns. However, it also impedes the system to be trapped for long time in mixtures and spin-glass states. Interestingly, the system also presents a range of Ω values for which, although the initial pattern is destabilized due to quantum oscillations, other patterns can be retrieved and remain stable even for many stored patterns, implying a quantum-dependent nonlinear relationship between the recall process and the number of stored patterns.
Associative memory, the ability to reconstruct complete information from partial or noisy cues, is a defining feature of human cognition and an enduring objective in artificial intelligence. Classical neural models such as Hopfield networks have long provided insight into distributed storage and retrieval mechanisms, yet they remain constrained by limited storage capacity and susceptibility to noise. Recent developments in quantum computing have motivated investigation of hybrid quantum-classical architectures as alternative approaches to memory representation and retrieval. This study examines how effectively hybrid quantum-classical models can reproduce associative memory functions compared to Hopfield networks, with a particular focus on recall accuracy under varying noise levels, storage capacity, and computational efficiency. Through small-scale experiments using binary patterns and hybrid quantum circuits, we evaluate potential quantum advantages and practical trade-offs. The findings contribute to ongoing discourse in quantum machine learning and cognitive modeling, underscoring both the promise and current limitations of quantum-inspired associative memory systems. Our guiding research question is: To what extent can hybrid quantum–classical models replicate associative memory functions more effectively than classical neural memory models, particularly in terms of recall accuracy under noise and storage capacity?
Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here quantum generalizations of classical networks realizing associative memories---capable of retrieving patterns, or memories, from corrupted initial states---have been proposed. It is a challenging open problem to analyze quantum associative memories with an extensive number of patterns and to determine the maximal number of patterns the quantum networks can reliably store, i.e., their storage capacity. In this work, we propose and explore a general method for evaluating the maximal storage capacity of quantum neural network models. By generalizing what is known as Gardner's approach in the classical realm, we exploit the theory of classical spin glasses for deriving the optimal storage capacity of quantum networks with quenched pattern variables. As an example, we apply our method to an open-system quantum associative memory formed of interacting spin-1/2 particles realizing coupled artificial neurons. The system undergoes a Markovian time evolution resulting from a dissipative retrieval dynamics that competes with a coherent quantum dynamics. We map out the nonequilibrium phase diagram and study the effect of temperature and Hamiltonian dynamics on the storage capacity. Our method opens an avenue for a systematic characterization of the storage capacity of quantum associative memories.
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.
Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.