Theoretical framework for quantum associative memories

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.