Quantum Hopfield Model

We find the free-energy in the thermodynamic limit of a one dimensional XY model associated to a system of N qubits. The coupling among the sigma_i^z is a long range two bodies random interaction. The randomness in the couplings is the typical interaction of the Hopfield model with p patterns (p<<N), with the patterns being p sequences of independent identically distributed (i.i.d.) random variables assuming values \pm 1 with probability 1/2. We show also that in the case p < alpha N the free-energy is asymptotically independent from the choice of the patterns, i.e. it is self-averaging. The Hamiltonian is the one used by (Neigovzen et al. 2009) in their experiment.