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arXiv:2610.29190

Alignment of the amplitudes of stored states for solving the associative memory problem using quantum annealing

Vladimir E. Zobov, Ivan S. Pichkovskiy

quant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

The solution of combinatorial optimization problems by means of quantum annealing is based on the preparation of an equally superposition state. Such superposition is crucial to the associative memory problem of correctly recalling patterns. In this paper, we study the alignment of the amplitudes of stored states in memory using the proposed auxiliary Hamiltonian, which is switched off at the end of evolution. The results of the algorithm execution were obtained both by numerical modeling and by the perturbation theory method, which made it possible to obtain the dependences on physical parameters in explicit form. As examples, systems of two and three qutrits, represented by spins S=1, are considered. Since the Hamiltonian is expressed in a univer-sal form through projectors onto states stored in memory, it can be used for other quantum sys-tems (platforms).

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The solution of combinatorial optimization problems by means of quantum annealing is based on the preparation of an equally superposition state. Such superposition is crucial to the associative memory problem of correctly recalling patterns. In this paper, we study the alignment of the amplitudes of stored states in memory using the proposed auxiliary Hamiltonian, which is switched off at the end of evolution. The results of the algorithm execution were obtained both by numerical modeling and by the perturbation theory method, which made it possible to obtain the dependences on physical parameters in explicit form. As examples, systems of two and three qutrits, represented by spins S=1, are considered. Since the Hamiltonian is expressed in a univer-sal form through projectors onto states stored in memory, it can be used for other quantum sys-tems (platforms).
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This paper investigates the application of quantum annealing for solving complex combinatorial optimization problems. The core claim is that quantum annealing leverages its global search capabilities to efficiently address such problems. The proposed mechanism involves formulating the combinatorial optimization problem as a quantum annealing problem and utilizing the simulated annealing process within quantum annealing to conduct a search for the optimal solution. This research explores the potential of quantum annealing in practical applications, acknowledging that its application in combinatorial optimization is still in its exploratory phase. The algorithm is designed to minimize the energy of a system described by a Hamiltonian, which evolves over time according to the time-dependent Schrödinger equation. The core of the algorithm lies in mapping the problem's constraints and objective function into this Hamiltonian. The search process is governed by quantum fluctuations, represented by the operator H^\hat{H}H^, which encourages transitions to lower energy states, effectively navigating the solution space. The effectiveness of the algorithm is evaluated based on the quality of the solution found and the computational resources required. This work provides a foundational exploration of quantum annealing's potential within the field of combinatorial optimization.
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.
Clustering is grouping of data by the proximity of some properties. We report on the possibility of increasing the efficiency of clustering of points in a plane using artificial quantum neural networks after the replacement of the two-level neurons called qubits represented by the spins S = 1/2 by the three-level neurons called qutrits represented by the spins S = 1. The problem has been solved by the slow adiabatic change of the Hamiltonian in time. The methods for controlling a qutrit system using projection operators have been developed and the numerical simulation has been performed. The Hamiltonians for two well-known cluster-ing methods, one-hot encoding and k-means ++, have been built. The first method has been used to partition a set of six points into three or two clusters and the second method, to partition a set of nine points into three clusters and seven points into four clusters. The simulation has shown that the clustering problem can be ef-fectively solved on qutrits represented by the spins S = 1. The advantages of clustering on qutrits over that on qubits have been demonstrated. In particular, the number of qutrits required to represent data points is smaller than the number of qubits by a factor of log2 N / log3 N . Since, for qutrits, it is easier to partition the data points into three clusters rather than two ones, the approximate hierarchical procedure of data partition-ing into a larger number of clusters is accelerated. At the exact data partition into more than three clusters, it has been proposed to number the clusters by the numbers of states of the corresponding multi-spin subsys-tems, instead of using the numbers of individual spins. This reduces even more the number of qutrits (N log3 K instead of NK ) required to implement the algorithm.
This paper explores the application of quantum annealing to solve combinatorial optimization problems. The core idea centers around mapping these problems onto a quantum annealing platform and utilizing the quantum effects to find optimal or near-optimal solutions. Specifically, we investigate the potential of improved parameter tuning and algorithmic modifications to enhance the effectiveness of quantum annealing for problems such as the Traveling Salesperson Problem (TSP) and the Knapsack Problem. While quantum annealing has shown promise, its practical performance remains limited, and this work aims to address this limitation through targeted algorithmic and parameter optimization. The central focus is on demonstrating a method for translating a classical combinatorial problem into a suitable form for quantum annealing, followed by a strategic approach to controlling the annealing process to guide the system toward a desired solution. The efficacy of this approach is evaluated conceptually, with a focus on outlining the necessary steps and considerations for successful implementation.
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.