  This paper proposes a new Quantum Spatial Graph Convolutional Neural Network
(QSGCNN) model that can directly learn a classification function for graphs of
arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network
(GCNN) models, the proposed QSGCNN model incorporates the process of
identifying transitive aligned vertices between graphs, and transforms
arbitrary sized graphs into fixed-sized aligned vertex grid structures. In
order to learn representative graph characteristics, a new quantum spatial
graph convolution is proposed and employed to extract multi-scale vertex
features, in terms of quantum information propagation between grid vertices of
each graph. Since the quantum spatial convolution preserves the grid structures
of the input vertices (i.e., the convolution layer does not change the original
spatial sequence of vertices), the proposed QSGCNN model allows to directly
employ the traditional convolutional neural network architecture to further
learn from the global graph topology, providing an end-to-end deep learning
architecture that integrates the graph representation and learning in the
quantum spatial graph convolution layer and the traditional convolutional layer
for graph classifications. We demonstrate the effectiveness of the proposed
QSGCNN model in relation to existing state-of-the-art methods. The proposed
QSGCNN model addresses the shortcomings of information loss and imprecise
information representation arising in existing GCN models associated with the
use of SortPooling or SumPooling layers. Experiments on benchmark graph
classification datasets demonstrate the effectiveness of the proposed QSGCNN
model.
