  With the overwhelming success in the field of quantum information in the last
decades, the "quest" for a Quantum Neural Network (QNN) model began in order to
combine quantum computing with the striking properties of neural computing.
This article presents a systematic approach to QNN research, which so far
consists of a conglomeration of ideas and proposals. It outlines the challenge
of combining the nonlinear, dissipative dynamics of neural computing and the
linear, unitary dynamics of quantum computing. It establishes requirements for
a meaningful QNN and reviews existing literature against these requirements. It
is found that none of the proposals for a potential QNN model fully exploits
both the advantages of quantum physics and computing in neural networks. An
outlook on possible ways forward is given, emphasizing the idea of Open Quantum
Neural Networks based on dissipative quantum computing.
