  Recently, the implementation of quantum neural networks is based on noisy
intermediate-scale quantum (NISQ) devices. Parameterized quantum circuit (PQC)
is such the method, and its current design just can handle linear data
classification. However, data in the real world often shows a topological
structure. In the machine learning field, the classical graph convolutional
layer (GCL)-based graph convolutional network (GCN) can well handle the
topological data. Inspired by the architecture of a classical GCN, in this
paper, to expand the function of the PQC, we design a novel PQC architecture to
realize a quantum GCN (QGCN). More specifically, we first implement an adjacent
matrix based on linear combination unitary and a weight matrix in a quantum
GCL, and then by stacking multiple GCLs, we obtain the QGCN. In addition, we
first achieve gradients decent on quantum circuit following the parameter-shift
rule for a GCL and then for the QGCN. We evaluate the performance of the QGCN
by conducting a node classification task on Cora dataset with topological data.
The numerical simulation result shows that QGCN has the same performance as its
classical counterpart, the GCN, in contrast, requires less tunable parameters.
Compared to a traditional PQC, we also verify that deploying an extra adjacent
matrix can significantly improve the classification performance for quantum
topological data.
