Recent neural networks designed to operate on graph-structured data have proven effective in many domains. These graph neural networks often diffuse information using the spatial structure of the graph. We propose a quantum walk neural network that learns a diffusion operation that is not only dependent on the geometry of the graph but also on the features of the nodes and the learning task. A quantum walk neural network is based on learning the coin operators that determine the behavior of quantum random walks, the quantum parallel to classical random walks. We demonstrate the effectiveness of our method on multiple classification and regression tasks at both node and graph levels.