  Graph convolutional neural networks (GCNs) operate by aggregating messages
over local neighborhoods given the prediction task under interest. Many GCNs
can be understood as a form of generalized diffusion of input features on the
graph, and significant work has been dedicated to improving predictive accuracy
by altering the ways of message passing. In this work, we propose a new
convolution kernel that effectively rewires the graph according to the
occupation correlations of the vertices by trading on the generalized diffusion
paradigm for the propagation of a quantum particle over the graph. We term this
new convolution kernel the Quantum Diffusion Convolution (QDC) operator. In
addition, we introduce a multiscale variant that combines messages from the QDC
operator and the traditional combinatorial Laplacian. To understand our method,
we explore the spectral dependence of homophily and the importance of quantum
dynamics in the construction of a bandpass filter. Through these studies, as
well as experiments on a range of datasets, we observe that QDC improves
predictive performance on the widely used benchmark datasets when compared to
similar methods.
