Theoretical framework for quantum associative memories

Associative memory refers to the ability to relate a memory with an input and targets the restoration of corrupted patterns. The prototypical example is the Hopfield neural network, where an attractor dynamics settles on stable solutions. Recently, several extensions to the quantum domain have been reported, focusing on specific tasks and dealing with classical-like patterns. Still, the potential of a quantum associative memory has not been established yet. In this work, we develop a comprehensive framework for a quantum associative memory based on open quantum system dynamics, which allows us to compare existing models, identify the theoretical prerequisites for performing associative memory tasks, and extend it in different forms. We explicitly derive a general form of dynamical map for a quantum associative memory that achieves an exponential increase in the number of stored patterns, giving a computational advantage and opening up new possibilities beyond those of classical associative memory systems. We establish the crucial role of symmetries and dissipation in the operation of quantum associative memory. 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. This opens new avenues for practical applications in quantum computing and machine learning, such as quantum error correction or quantum memories.