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International Joint Conference on Neural Networks (IJCNN)

Energy spectrum of quantum associative memories

Gerasimos G. Rigatos

Published Jan 1, 2006• 4 citations
Publisher

Abstract

Quantum associative memories are derived from the Hopfleld memory model under the assumption that the elements of the correlation weight matrix W are stochastic variables. The probability density function of each weight is given as a solution of Schrodinger's diffusion equation. Spectral analysis of quantum associative memories follows previous studies on the wavelets' energy spectrum. Spectral analysis shows that (i) the basis functions of the stochastic weights express the distribution of energy with respect to the weights' values, (ii) the stochastic weights satisfy the principle of uncertainty.

DOI
10.1109/ijcnn.2006.1716092

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Quantum associative memories are derived from the Hopfleld memory model under the assumption that the elements of the correlation weight matrix W are stochastic variables. The probability density function of each weight is given as a solution of Schrodinger's diffusion equation. Spectral analysis of quantum associative memories follows previous studies on the wavelets' energy spectrum. Spectral analysis shows that (i) the basis functions of the stochastic weights express the distribution of energy with respect to the weights' values, (ii) the stochastic weights satisfy the principle of uncertainty.
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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.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.