Eliana Fiorelli, Open Quantum Hopfield-type networks: phase diagram and storage capacity

Wednesday 9 Nov 2022, 16:15 → 17:00 Europe/Stockholm

Albano 3: 6228 - Mega (22 seats) (Albano Building 3)

In this work we focus on Hopfield-type NNs, which are network of interconnected binary units representing the simplest instances of associative memories, permitting the recognition, or retrieval, of patterns. The latter are encoded in a chosen learning prescription via the neurons connections. The retrieval of information is described by means of a classical non-equilibrium dynamics, where memories are stored as long-time solutions of a stochastic dynamics. Such a classical out-of-equilibrium dynamics can be generalized into the quantum domain via the formalism of open quantum systems. 

Firstly, while focussing on the widely used Hebb’s learning rule, we introduce and analyze an open quantum generalization of the q-state Potts-Hopfield neural network, which is based on multi-level classical spins. The dynamics of this many-body system is formulated in terms of a Markovian master equation of Lindblad type, which allows to incorporate both probabilistic classical and coherent quantum processes on an equal footing. By employing a mean field description we analyze the corresponding phase diagram. In the low temperature regime it displays pattern retrieval, in analogy to the classical Potts-Hopfield neural network; when increasing quantum fluctuations, however, a limit cycle phase emerges, which has no classical counterpart. This shows that quantum effects can qualitatively alter the structure of the stationary state manifold with respect to the classical model, and potentially allow one to encode and retrieve novel types of patterns.

Further, we address the more sophisticated question concerning the issue of identifying the storage capacity αc, i.e., how many patterns can be stored in a network with N constituents. For this purpose, we generalize a classical approach that has been first introduced by Gardner for the zero-temperature case, and eventually adapted for dealing with thermal noise. In this case, while a set of memories is required to be a stationary distribution, the weights encoding the learning rule are left as free parameters. We show how to evaluate the storage capacity for the open quantum formulation of Hopfield-type NNs. Our results highlight that such NNs reproduce the classical storage capacity behavior in absence of the coherent contribution. In addition, the NNs shows robustness to the presence of the latter: even though the storage capacity diminishes as the Hamiltonian strength is present, we could identify a range of temperature and coherent control parameter where the storage capacity stays finite.