FROM CLASSICAL NEURAL NETWORKS TO QUANTUM NEURAL NETWORKS

First I give a brief description of the classical Hopfield model introducing the fundamental concepts of patterns, retrieval, pattern recognition, neural dynamics, capacity and describe the fundamental results obtained in this field by Amit, Gutfreund and Sompolinsky,(1) using the non rigorous method of replica and the rigorous version given by Pastur, Shcherbina, Tirozzi(2) using the cavity method. Then I give a formulation of the theory of Quantum Neural Networks (QNN) in terms of the XY model with Hebbian interaction. The problem of retrieval and storage is discussed. The retrieval states are the states of the minimum energy. I apply the estimates found by Lieb(3) which give lower and upper bound of the free-energy and expectation of the observables of the quantum model. I discuss also some experiment and the search of ground state using Monte Carlo Dynamics applied to the equivalent classical two dimensional Ising model constructed by Suzuki et al.(6) At the end there is a list of open problems