  Graph Neural Networks (GNNs) are powerful machine learning models that excel
at analyzing structured data represented as graphs, demonstrating remarkable
performance in applications like social network analysis and recommendation
systems. However, classical GNNs face scalability challenges when dealing with
large-scale graphs. This paper proposes frameworks for implementing GNNs on
quantum computers to potentially address the challenges. We devise quantum
algorithms corresponding to the three fundamental types of classical GNNs:
Graph Convolutional Networks, Graph Attention Networks, and Message-Passing
GNNs. A complexity analysis of our quantum implementation of the Simplified
Graph Convolutional (SGC) Network shows potential quantum advantages over its
classical counterpart, with significant improvements in time and space
complexities. Our complexities can have trade-offs between the two: when
optimizing for minimal circuit depth, our quantum SGC achieves logarithmic time
complexity in the input sizes (albeit at the cost of linear space complexity).
When optimizing for minimal qubit usage, the quantum SGC exhibits space
complexity logarithmic in the input sizes, offering an exponential reduction
compared to classical SGCs, while still maintaining better time complexity.
These results suggest our Quantum GNN frameworks could efficiently process
large-scale graphs. This work paves the way for implementing more advanced
Graph Neural Network models on quantum computers, opening new possibilities in
quantum machine learning for analyzing graph-structured data.
