  We propose a new framework to understand how quantum effects may impact on
the dynamics of neural networks. We implement the dynamics of neural networks
in terms of Markovian open quantum systems, which allows us to treat thermal
and quantum coherent effects on the same footing. In particular, we propose an
open quantum generalisation of the celebrated Hopfield neural network, the
simplest toy model of associative memory. We determine its phase diagram and
show that quantum fluctuations give rise to a qualitatively new non-equilibrium
phase. This novel phase is characterised by limit cycles corresponding to
high-dimensional stationary manifolds that may be regarded as a generalisation
of storage patterns to the quantum domain.
