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arXiv:2509.12202 · Science

High-capacity associative memory in a quantum-optical spin glass

Brendan Marsh, David Atri Schuller, Yunpeng Ji, Henry S. Hunt, Surya Ganguli, Sarang Gopalakrishnan, Jonathan Keeling, Benjamin Lev

Published Sep 3, 2026• 0 citations
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Abstract

The Hopfield neural network stores memories using all-to-all-coupled spins and recalls those memories through equilibrium dynamics. Storing too many hampers recall because frustration causes an exponential number of spurious patterns to arise as the network becomes a spin glass. Despite this, we show that memory recall can be restored, and even enhanced, under quantum-optical nonequilibrium dynamics that convert spurious patterns into reliable memories. We experimentally observe associative memory with high storage capacity in a driven-dissipative spin glass made of atoms and photons. The capacity surpasses that of the Hopfield model under Hebbian learning by up to sevenfold in a 16-spin network. Atomic motion boosts capacity by dynamically modifying connectivity akin to short-term synaptic plasticity in neural networks, realizing a precursor to learning in a quantum-optical system.

DOI
10.1126/science.aec3917

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•Sep 14, 2026•0 citations
The Hopfield neural network stores memories using all-to-all-coupled spins and recalls those memories through equilibrium dynamics. Storing too many hampers recall because frustration causes an exponential number of spurious patterns to arise as the network becomes a spin glass. Despite this, we show that memory recall can be restored, and even enhanced, under quantum-optical nonequilibrium dynamics that convert spurious patterns into reliable memories. We experimentally observe associative memory with high storage capacity in a driven-dissipative spin glass made of atoms and photons. The capacity surpasses that of the Hopfield model under Hebbian learning by up to sevenfold in a 16-spin network. Atomic motion boosts capacity by dynamically modifying connectivity akin to short-term synaptic plasticity in neural networks, realizing a precursor to learning in a quantum-optical system.
We introduce a near-term experimental platform for realizing an associative memory.It can simultaneously store many memories by using spinful bosons coupled to a degenerate multimode optical cavity.The associative memory is realized by a confocal cavity QED neural network, with the modes serving as the synapses, connecting a network of superradiant atomic spin ensembles,which serve as the neurons.Memories are encoded in the connectivity matrix between the spins and can be accessed through the input and output of patterns of light.Each aspect of the scheme is based on recently demonstrated technology using a confocal cavity and Bose-condensed atoms.Our scheme has two conceptually novel elements.First, it introduces a new form of random spin system that interpolates between a ferromagnetic and a spin glass regime as a physical parameter is tuned-the positions of ensembles within the cavity.Second, and more importantly, the spins relax via deterministic steepest-descent dynamics rather than Glauber dynamics.We show that this nonequilibrium quantum-optical scheme has significant advantages for associative memory over Glauber dynamics: These dynamics can enhance the network's ability to store and recall memories beyond that of the standard Hopfield model.Surprisingly, the cavity QED dynamics can retrieve memories even when the system is in the spin glass phase.Thus, the experimental platform provides a novel physical instantiation of associative memories and spin glasses as well as provides an unusual form of relaxational dynamics that is conducive to memory recall even in regimes where it was thought to be impossible.
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.
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.
Algorithms for associative memory typically rely on a network of many connected units. The prototypical example is the Hopfield model, whose generalizations to the quantum realm are mainly based on open quantum Ising models. We propose a realization of associative memory with a single driven-dissipative quantum oscillator exploiting its infinite degrees of freedom in phase space. The model can improve the storage capacity of discrete neuron-based systems in a large regime and we prove successful state discrimination between n coherent states, which represent the stored patterns of the system. These can be tuned continuously by modifying the driving strength, constituting a modified learning rule. We show that the associative-memory capability is inherently related to the existence of a spectral separation in the Liouvillian superoperator, which results in a long timescale separation in the dynamics corresponding to a metastable phase.
We review our models of quantum associative memories that represent the “quantization” of fully coupled neural networks like the Hopfield model. The idea is to replace the classical irreversible attractor dynamics driven by an Ising model with pattern-dependent weights by the reversible rotation of an input quantum state onto an output quantum state consisting of a linear superposition with probability amplitudes peaked on the stored pattern closest to the input in Hamming distance, resulting in a high probability of measuring a memory pattern very similar to the input. The unitary operator implementing this transformation can be formulated as a sequence of one-qubit and two-qubit elementary quantum gates and is thus the exponential of an ordered quantum Ising model with sequential operations and with pattern-dependent interactions, exactly as in the classical case. Probabilistic quantum memories, that make use of postselection of the measurement result of control qubits, overcome the famed linear storage limitation of their classical counterparts because they permit to completely eliminate crosstalk and spurious memories. The number of control qubits plays the role of an inverse fictitious temperature. The accuracy of pattern retrieval can be tuned by lowering the fictitious temperature under a critical value for quantum content association while the complexity of the retrieval algorithm remains polynomial for any number of patterns polynomial in the number of qubits. These models thus solve the capacity shortage problem of classical associative memories, providing a polynomial improvement in capacity. The price to pay is the probabilistic nature of information retrieval.
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?