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Integrated Computer-Aided Engineering

Neurodynamics and attractors in quantum associative memories

Gerasimos G. Rigatos, Spyros G. Tzafestaş

Published May 13, 2007• 24 citations
Publisher

Abstract

Quantum associative memories are connectionist structures that demonstrate the particle-wave nature of information and are compatible with quantum mechanics postulates. Following the solution of Schrödinger's diffusion equation, and using the Hopfiel
DOI
10.3233/ica-2007-14303

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In this paper the existence of quantum wave-packets in neural structures, such as automata and associative memories, is studied. The synaptic weights are considered to be stochastic variables, with probability density functions given by the solution of Schrödinger's equation. It is shown that the weights' update can be performed with the use of unitary operators, as indicated by the postulates of quantum mechanics. Moreover, it is proved that the number of attractors of quantum associative memories increases exponentially with respect to conventional associative memories. Finally, simulation tests are used to demonstrate the improved pattern storage capabilities of quantum associative memories.
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Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary importance for the realization of quantum computing machines. To this end, this chapter studies neural structures with weights that follow the model of the quantum harmonic oscillator. The proposed neural networks have stochastic weights which are calculated from the solution of Schrödingers equation under the assumption of a parabolic (harmonic) potential. These weights correspond to diffusing particles, which interact with each other as the theory of Brownian motion (Wiener process) predicts. The learning of the stochastic weights (convergence of the diffusing particles to an equilibrium) is analyzed. In the case of associative memories the proposed neural model results in an exponential increase of patterns storage capacity (number of attractors). It is also shown that conventional neural networks and learning algorithms based on error gradient can be conceived as a subset of the proposed quantum neural structures. Thus, the complementarity between classical and quantum physics is also validated in the field of neural computation.
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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.