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arXiv:2504.00111 · Physical Review Letters

Multiphoton Quantum Simulation of the Generalized Hopfield Memory Model

Gennaro Zanfardino, Stefano Paesani, Luca Leuzzi, Raffaele Santagati, Fabio Sciarrino, Fabrizio Illuminati, Giancarlo Ruocco, Marco Leonetti

Published Jan 9, 2026• 0 citations
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Abstract

In the present Letter, we introduce, develop, and investigate a connection between multiphoton quantum interference, a core element of emerging photonic quantum technologies, and Hopfield-like Hamiltonians of classical neural networks, the paradigmatic models for associative memory and machine learning in systems of artificial intelligence. Specifically, we show that combining a system composed of N_{ph} indistinguishable photons in superposition over M field modes, a controlled array of M binary phase shifters, and a linear-optical interferometer, yields output photon statistics described by means of a p-body Hopfield Hamiltonian of M Ising-like neurons ±1, with p=2N_{ph}. We investigate in detail the generalized four-body Hopfield model obtained through this procedure, undergoing a transition from a memory retrieval to a memory black-out regime, i.e., a spin-glass phase, as the amount of stored memory increases. The mapping enables novel routes to the realization and investigation of disordered and complex classical systems via efficient photonic quantum simulators and describes aspects of structured photonic systems in terms of classical spin Hamiltonians.

DOI
10.1103/945c-11wt

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In the present Letter, we introduce, develop, and investigate a connection between multiphoton quantum interference, a core element of emerging photonic quantum technologies, and Hopfield-like Hamiltonians of classical neural networks, the paradigmatic models for associative memory and machine learning in systems of artificial intelligence. Specifically, we show that combining a system composed of N_{ph} indistinguishable photons in superposition over M field modes, a controlled array of M binary phase shifters, and a linear-optical interferometer, yields output photon statistics described by means of a p-body Hopfield Hamiltonian of M Ising-like neurons ±1, with p=2N_{ph}. We investigate in detail the generalized four-body Hopfield model obtained through this procedure, undergoing a transition from a memory retrieval to a memory black-out regime, i.e., a spin-glass phase, as the amount of stored memory increases. The mapping enables novel routes to the realization and investigation of disordered and complex classical systems via efficient photonic quantum simulators and describes aspects of structured photonic systems in terms of classical spin Hamiltonians.
Models of interacting complex systems provide the fundamental statistical physics reference frame for the study and the understanding of associative memories, machine learning, and the dynamics of neural networks. On the other hand, simulating complex multi-synaptic interactions on a classical hardware is computationally demanding due to the super-linear scaling of the system complexity. Photonic quantum technologies provide a promising solution to these limitations by leveraging on their inherent speed and parallel processing ability in order to simulate complex networks. Recently, a connection between multiphoton processes and generalized ppp-body Hopfield models has been theoretically established. Here, we design and demonstrate an experimental platform that exploits single photons distributed across a set of optical modes, in which controlled arrays of binary phase shifters act as Ising-like neurons. We focus specifically on a fully connected Hopfield Hamiltonian with four-body local interaction terms, realized via two-photon processes. Through quantum simulations on programmable photonic processors, the study identifies three distinct regimes: a memory retrieval phase, a spin-glass memory "black-out" phase, and a paramagnetic phase. Experimental results confirm successful memory retrieval at low storage capacities and temperatures, where the system consistently relaxes to fixed points with high memory overlap, effectively reconstructing the stored patterns. Future research will extend the platform design to investigate networks with local or dilute interactions, while advances in the realization of scalable photonic circuits will enable architectures that encompass very large numbers of interacting spins.
Quantum simulation of the associative memory in Hopfield neural networks is an interesting crossover between quantum information and machine learning. The $q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}m$ $s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c$ $w\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}k$ has been proposed for such simulations, yet not realized experimentally. The authors successfully map this scheme to a three-dimensional photonic chip, and achieve quantum stochastic walk evolution. A good match rate between the experimental quantum scheme and the expected result for a Hopfield neural net is attained. This proof of principle, combined with the scalability of low-loss integrated chips and straightforward Hamiltonian engineering, is a primary step toward photonic artificial intelligence.
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.
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.
This article explores the groundbreaking intersection of quantum optics and neural network models, focusing on how multiphoton quantum interference in photonic circuits can simulate associative memory akin to the human brain. Drawing from recent research, we delve into the technical mechanisms, memory limits, and phase transitions observed in these systems. We employ geometric interpretations to elucidate the underlying ideas, simulate a model demonstrating latent memory in photon-like states for AI applications, and propose avenues for future research. The discussion covers historical context, experimental implications, energy efficiency advantages, potential challenges such as scalability and noise, and broader impacts on artificial intelligence, physics simulations, and neuromorphic hardware. Mamun, SM (2026)………………………1 Quantum Photonic Associative Memory: Multiphoton Interference as a Pathway to Brain-Inspired Computing (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132157923/download_file?s=portfolio
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?