Hopfield networks are a variant of associative memory that recall patterns stored in the couplings of an Ising model. Stored memories are conventionally accessed as fixed points in the network dynamics that correspond to energetic minima of the spin state. We show that memories stored in a Hopfield network may also be recalled by energy minimization using adiabatic quantum optimization (AQO). Numerical simulations of the underlying quantum dynamics allow us to quantify AQO recall accuracy with respect to the number of stored memories and noise in the input key. We investigate AQO performance with respect to how memories are stored in the Ising model according to different learning rules. Our results demonstrate that AQO recall accuracy varies strongly with learning rule, a behavior that is attributed to differences in energy landscapes. Consequently, learning rules offer a family of methods for programming adiabatic quantum optimization that we expect to be useful for characterizing AQO performance.
Hopfield networks are a variant of associative memory that recall patterns stored in the couplings of an Ising model. Stored memories are conventionally accessed as fixed points in the network dynamics that correspond to energetic minima of the spin state. We show that memories stored in a Hopfield network may also be recalled by energy minimization using adiabatic quantum optimization (AQO). Numerical simulations of the underlying quantum dynamics allow us to quantify AQO recall accuracy with respect to the number of stored memories and noise in the input key. We investigate AQO performance with respect to how memories are stored in the Ising model according to different learning rules. Our results demonstrate that AQO recall accuracy varies strongly with learning rule, a behavior that is attributed to differences in energy landscapes. Consequently, learning rules offer a family of methods for programming adiabatic quantum optimization that we expect to be useful for characterizing AQO performance.
A content-addressable memory (CAM) stores key-value associations such that the key is recalled by providing its associated value. While CAM recall is traditionally performed using recurrent neural network models, we show how to solve this problem using adiabatic quantum optimization. Our approach maps the recurrent neural network to a commercially available quantum processing unit by taking advantage of the common underlying Ising spin model. We then assess the accuracy of the quantum processor to store key-value associations by quantifying recall performance against an ensemble of problem sets. We observe that different learning rules from the neural network community influence recall accuracy but performance appears to be limited by potential noise in the processor. The strong connection established between quantum processors and neural network problems supports the growing intersection of these two ideas.
Associative memory models, in theoretical neuro- and computer sciences, can generally store at most a linear number of memories. Recalling memories in these models can be understood as retrieval of the energy minimizing configuration of classical Ising spins, closest in Hamming distance to an imperfect input memory, where the energy landscape is determined by the set of stored memories. We present an Ising formulation for associative memory models and consider the problem of memory recall using quantum annealing. We show that allowing for input-dependent energy landscapes allows storage of up to an exponential number of memories (in terms of the number of neurons). Further, we show how quantum annealing may naturally be used for recall tasks in such input-dependent energy landscapes, although the recall time may increase with the number of stored memories. Theoretically, we obtain the radius of attractor basins R(N) and the capacity C(N) of such a scheme and their tradeoffs. Our calculations establish that for randomly chosen memories the capacity of our model using the Hebbian learning rule as a function of problem size can be expressed as C(N)=O(eC1N), ${C}_{1}\ensuremath{\ge}0$, and succeeds on randomly chosen memory sets with a probability of $(1\ensuremath{-}{e}^{\ensuremath{-}{C}_{2}N})$, ${C}_{2}\ensuremath{\ge}0$ with ${C}_{1}+{C}_{2}={(0.5\ensuremath{-}f)}^{2}/(1\ensuremath{-}f)$, where f=R(N)/N, $0\ensuremath{\le}f\ensuremath{\le}0.5$, is the radius of attraction in terms of the Hamming distance of an input probe from a stored memory as a fraction of the problem size. We demonstrate the application of this scheme on a programmable quantum annealing device, the D-wave processor.
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?
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.
Adiabatic quantum computing is successfully used to solve discrete optimization problems. The classical Hopfield neural network can also be applied to such problems but is primarily used for pattern recognition and reconstruction. Since its dynamics are deterministic, this can adversely affect its performance: the network sometimes gets trapped in local energy minima. This paper proposes a novel approach in which the Hopfield neural network learning rule (Hebbian rule) is combined with forward and reverse quantum annealing. This enables the use of adiabatic quantum annealers for pattern recognition, providing an alternative to traditional approaches.