Quantum accelerated approach to the thermal state of classical all-to-all connected spin systems with applications to pattern retrieval in the Hopfield neural network

We explore the question as to whether quantum effects can yield a speedup of the nonequilibrium evolution of fully connected quadratic spin models towards a classical thermal state. In our approach we exploit the fact that the thermal state of a spin system can be mapped onto a node-free quantum state whose coefficients are given by thermal weights. This perspective permits the construction of a dissipative yet quantum dynamics which encodes in its stationary state the thermal state of the original problem. We show for the case of an all-to-all connected Ising spin model that an appropriate transformation of this dissipative dynamics allows us to interpolate between a regime in which the order parameter obeys the classical equations of motion under Glauber dynamics and a quantum regime with an accelerated transient timescale before approaching stationarity. We show that this effect enables, in principle, a speedup of pattern retrieval in a Hopfield neural network.