  We introduce and analyze an open quantum generalization of the q-state
Potts-Hopfield neural network, which is an associative memory model 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
investigate how classical fluctuations due to temperature and quantum
fluctuations effectuated by coherent spin rotations affect the ability of the
network to retrieve stored memory patterns. We construct the corresponding
phase diagram, which in the low temperature regime 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.
