Quantum walks have been shown to be fruitful tools in analyzing the dynamic properties of quantum systems. This article proposes using quantum walks as an approach to quantum neural networks (QNNs). QNNs replace binary McCulloch-Pitts neurons with a qubit in order to use the advantages of quantum computing in neural networks. A quantum walk on the firing states of such a QNN is supposed to simulate the central properties of the dynamics of classical neural networks, such as associative memory. It is shown that a biased discrete Hadamard walk derived from the updating process of a biological neuron does not lead to a unitary walk. However, a stochastic quantum walk between the global firing states of a QNN can be constructed, and it is shown that it contains the feature of associative memory. The quantum contribution to the walk accounts for a modest speedup in some regimes.

• Received 5 February 2014

DOI:https://doi.org/10.1103/PhysRevA.89.032333