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International Journal of Modern Physics C

QUANTUM WAVE-PACKETS IN FUZZY AUTOMATA AND NEURAL ASSOCIATIVE MEMORIES

Gerasimos G. Rigatos

Published Oct 1, 2007• 14 citations
Publisher

Abstract

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.

DOI
10.1142/s012918310701156x

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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.Request access from your librarian to read this chapter's full text.
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.
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.