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arXiv:2408.14272 · Quantum Science and Technology

Theoretical framework for quantum associative memories

Adrià Labay-Mora, Eliana Fiorelli, Roberta Zambrini, Gian Luca Giorgi

Published Jun 5, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Associative memory (AM) refers to the ability to relate a memory with an input and targets the restoration of corrupted patterns. It has been intensively studied in classical physical systems, as in neural networks where an attractor dynamics settles on stable solutions. Several extensions to the quantum domain have been recently reported, displaying different features. In this work, we develop a comprehensive framework for a quantum AM (QAM) based on open quantum system dynamics, which allows us to compare existing models, identify the theoretical prerequisites for performing AM tasks, and extend it in different forms. The map that achieves an exponential increase in the number of stored patterns with respect to classical systems is derived. We establish the crucial role of symmetries and dissipation in the operation of QAM. Our theoretical analysis demonstrates the feasibility of addressing both quantum and classical patterns, orthogonal and non-orthogonal memories, stationary and metastable operating regimes, and measurement-based outputs. Finally, this opens up new avenues for practical applications in quantum computing and machine learning, such as quantum error correction or quantum memories.

DOI
10.1088/2058-9565/ade184

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Abstract Associative memory (AM) refers to the ability to relate a memory with an input and targets the restoration of corrupted patterns. It has been intensively studied in classical physical systems, as in neural networks where an attractor dynamics settles on stable solutions. Several extensions to the quantum domain have been recently reported, displaying different features. In this work, we develop a comprehensive framework for a quantum AM (QAM) based on open quantum system dynamics, which allows us to compare existing models, identify the theoretical prerequisites for performing AM tasks, and extend it in different forms. The map that achieves an exponential increase in the number of stored patterns with respect to classical systems is derived. We establish the crucial role of symmetries and dissipation in the operation of QAM. Our theoretical analysis demonstrates the feasibility of addressing both quantum and classical patterns, orthogonal and non-orthogonal memories, stationary and metastable operating regimes, and measurement-based outputs. Finally, this opens up new avenues for practical applications in quantum computing and machine learning, such as quantum error correction or quantum memories.
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.
There is increasing interest in quantum computing, because of its enormous computing potential. A small number of powerful quantum algorithms have been proposed to date; however, the development of new quantum algorithms for practical use remains essential. Parallel computing with a neural network has successfully realized certain unique functions such as learning and recognition; therefore, the introduction of certain neural computing techniques into quantum computing to enlarge the quantum computing application field is worthwhile. In this paper, a novel quantum associative memory (QuAM) is proposed, which is achieved with a quantum neural network by employing adiabatic Hamiltonian evolution. The memorization and retrieval procedures are inspired by the concept of associative memory realized with an artificial neural network. To study the detailed dynamics of our QuAM, we examine two types of Hamiltonians for pattern memorization. The first is a Hamiltonian having diagonal elements, which is known as an Ising Hamiltonian and which is similar to the cost function of a Hopfield network. The second is a Hamiltonian having non-diagonal elements, which is known as a neuro-inspired Hamiltonian and which is based on interactions between qubits. Numerical simulations indicate that the proposed methods for pattern memorization and retrieval work well with both types of Hamiltonians. Further, both Hamiltonians yield almost identical performance, although their retrieval properties differ. The QuAM exhibits new and unique features, such as a large memory capacity, which differs from a conventional neural associative memory.
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.
Nowadays, the research field of Machine Learning (ML) is undergoing a rapid expansion. In addition, in the last decades, within the research area of quantum computation, several quantum protocols and algorithms have shown the ability to outperform their classical counterparts. In the light of these twofold advances, a frontier field investigating the potentiality of quantum implementations of ML architectures arose. Among the latter, many efforts are concentrated on Artificial Neural Networks (ANNs), which represent a highly successful ML architecture. This thesis focuses on characterizing and understanding the role of quantum effects on the behavior of an instance of ANN, the so-called Hopfield Neural Network (HNN). The HNN is an example of an associative memory model and it permits the recognition, or retrieval, of patterns, such as letters of an alphabet. Formally, it can be described in terms of a classical spin network with an all-to-all interaction, ruled by a stochastic dynamics. Its relatively intuitive behavior, and its physical description make the HNN being a good candidate for a theoretical investigation of its possible quantum generalizations. In the first part of our work we deal with the implementation of fully connected Ising models, such as the HNN one, in terms of quantum systems. To this end, we refer to the cases where the theory of Open Quantum System (OQS) is exploited for engineering steady states of out-of-equilibrium dynamics. This paradigm allows us to build a dissipative - yet quantum - dynamics which encodes in its stationary state the thermal state of the original (classical) problem. Such a construction gives us the chance to address the question as to whether the quantum description could lead to a speed-up in reaching the shared stationary state with respect to the classical counterpart. We show that a suitable choice of the parameters allows the system to switch between a classical regime and a quantum one, the latter being characterized by an accelerated approach towards the long-time limit state. In the second part of our work we focus on practically relevant implementations of quantum systems that can show analogies with the HNN model. To this end, we consider spin-boson models as described by the Dicke Hamiltonian. Indeed, at strong spin-boson interaction couplings, and when the latter are additionally characterized by disorder, the equilibrium Dicke model is able to reproduce the retrieval mechanism typical of HNNs. When dissipation is included, the theoretical description becomes more challenging. We analyze the open, disordered Dicke model by employing some perturbative techniques in order to deal with the out-of-equilibrium dynamics at strong coupling. As a result, we are able to highlight similarities and differences between the quantum model and HNNs.
Quantum associative memories (QAM) have storage capacity exponentially larger than classical associative memories. Given an input with n bits, the QAM can load all of 2n binary patterns. In this paper we propose a nonunitary quantum memory retrieval algorithm. The proposed method is used to recover incomplete input information and also in pattern classification task as an ensemble of memories. The advantages of the proposed method are the simplicity of recovery procedure in a unique iteration and the fact the circuit has linear complexity in function of the input length with deterministic output.
Abstract We propose a new framework to understand how quantum effects may impact on the dynamics of neural networks. We implement the dynamics of neural networks in terms of Markovian open quantum systems, which allows us to treat thermal and quantum coherent effects on the same footing. In particular, we propose an open quantum generalisation of the Hopfield neural network, the simplest toy model of associative memory. We determine its phase diagram and show that quantum fluctuations give rise to a qualitatively new non-equilibrium phase. This novel phase is characterised by limit cycles corresponding to high-dimensional stationary manifolds that may be regarded as a generalisation of storage patterns to the quantum domain.