  Quantum neural networks form one pillar of the emergent field of quantum
machine learning. Here, quantum generalisations of classical networks realizing
associative memories - capable of retrieving patterns, or memories, from
corrupted initial states - have been proposed. It is a challenging open problem
to analyze quantum associative memories with an extensive number of patterns,
and to determine the maximal number of patterns the quantum networks can
reliably store, i.e. their storage capacity. In this work, we propose and
explore a general method for evaluating the maximal storage capacity of quantum
neural network models. By generalizing what is known as Gardner's approach in
the classical realm, we exploit the theory of classical spin glasses for
deriving the optimal storage capacity of quantum networks with quenched pattern
variables. As an example, we apply our method to an open-system quantum
associative memory formed of interacting spin-1/2 particles realizing coupled
artificial neurons. The system undergoes a Markovian time evolution resulting
from a dissipative retrieval dynamics that competes with a coherent quantum
dynamics. We map out the non-equilibrium phase diagram and study the effect of
temperature and Hamiltonian dynamics on the storage capacity. Our method opens
an avenue for a systematic characterization of the storage capacity of quantum
associative memories.
