Dissipative Quantum Hopfield Network: A numerical analysis

Abstract

We present extensive simulations of a quantum version of the Hopfield Neural Network to explore its emergent behavior. The system is a network of $N$ qubits oscillating at a given $Ω$ frequency and which are coupled via Lindblad jump operators built with local fields $h_i$ depending on some given stored patterns. Our simulations show the emergence of pattern-antipattern oscillations of the overlaps with the stored patterns similar (for large $Ω$ and small temperature) to those reported within a recent mean-field description of such a system, and which are originated deterministically by the quantum term including $s_x^i$ qubit operators. However, in simulations we observe that such oscillations are stochastic due to the interplay between noise and the inherent metastability of the pattern attractors induced by quantum oscillations, and then are damped in finite systems when one averages over many quantum trajectories. In addition, we report the system behavior for large number of stored patterns at the lowest temperature we can reach in simulations (namely $T=0.005\, T_C$). Our study reveals that for small-size systems the quantum term of the Hamiltonian has a negative effect on storage capacity, decreasing the overlap with the starting memory pattern for increased values of $Ω$ and number of stored patterns. However, it also impedes the system to be trapped for long time in mixtures and spin-glass states. Interestingly, the system also presents a range of $Ω$ values for which, although the initial pattern is destabilized due to quantum oscillations, other patterns can be retrieved and remain stable even for many stored patterns, implying a quantum-dependent nonlinear relationship between the recall process and the number of stored patterns.