Quantum Hopfield Model

TL;DR

This paper calculates the free-energy of a quantum spin system with Hopfield-like randomness, revealing self-averaging properties and connecting to experimental models of qubits.

Contribution

It introduces a quantum Hopfield model with long-range random interactions and analyzes its free-energy in the thermodynamic limit, highlighting self-averaging behavior.

Findings

Free-energy derived for the quantum Hopfield model.

Self-averaging property when p < alpha N.

Connection to experimental quantum systems.

Abstract

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.