• REQGNN with rotational and permutation equivariance are considered. • An optimized method for constructing equivariant quantum gatesets with reduced computational complexity is proposed. • The trainable quantum graph compression encoder effectively captures low-dimensional structural representations while preserving graph information. • The entanglement-enhanced aggregation strategy effectively captures edge information while preserving graph rotation symmetry. • An auxiliary entanglement layer mitigates the over-smoothing issue induced by multiple entanglement layers. • Experiments on standard graph classification and regression datasets validate the model’s effectiveness. The integration of symmetry, such as permutation equivariance, into Quantum Graph Neural Networks (QGNNs), referred to as Equivariant Quantum Graph Neural Networks (EQGNNs), markedly improves the model’s generalization performance on graph-structured data. Despite this advancement, current research has not yet extended rotational equivariance to QGNN frameworks. Furthermore, processing large-scale graph data increases computational complexity due to numerous inter-node connections, significantly raising the required number of qubits. To address these challenges, a novel Rotationally Equivariant Quantum Graph Neural Network (REQGNN) with trainable compression encoder and entanglement-enhanced aggregation mechanism is proposed. By adopting quantum fidelity as the evaluation metric, we design a quantum autoencoder to effectively compress feature dimensionality, substantially lowering the qubit requirements of the model while preserving essential global structural details. To achieve rotational equivariance in the model, we propose an entanglement-enhanced layer that incorporates distance and angle information between nodes. This layer performs entanglement by extracting diverse edge information, thereby further refining edge feature extraction. Additionally, an auxiliary entanglement layer is introduced to mitigate the over-smoothing issue. Experimental results demonstrate REQGNN is significantly better for graph classification tasks than GIN, Gra+QSVM, and Gra+QCNN on four datasets in all metrics and achieves better results than egoGQNN in accuracy on PTC dataset, and it also has advantage for graph regression tasks over the classical models, including EGNN and EquiformerV2, and reduces the MAE of C v task unit by 20% on average compared with a previous quantum model QGCNN. Our approach offers an effective solution for achieving rotational equivariance while providing a novel perspective for exploring symmetry in graph neural networks (GNNs).

Equivariant quantum graph neural networks (EQGNNs) offer a potentially powerful method to process graph data. However, existing EQGNN models only consider the permutation symmetry of graphs, and failing to fully exploit the geometric and non-geometric information in graphs, resulting in suboptimal performance when processing 3D graph data. To address these limitations, we derive constraints of rotation and permutation equivariance, and then propose a novel rotation- and permutation-equivariant quantum graph neural network (RP-EQGNN). An equivariant module is designed to extract the geometric information. Then, a convolution and entanglement module is constructed to extract non-geometric information. To improve performance of our model, an edge entanglement strategy is designed to perform distinguishable entanglement operations based on edge heterogeneity. The experiment results demonstrate that RP-EQGNN is significantly better for graph regression on the QM9 dataset and the OC20 dataset than Q3DGL and EQC in MAE and achieves results comparable to those of EquiformerV2, Geoformer, SO3KRATES and HEGNN. It also has advantage for point cloud classification on the ModelNet40 dataset over quantum models, including sQCNN-3D and PI-QSVM. RP-EQGNN introduces an innovative approach for processing 3D graph data, establishing a basis for future investigations into symmetries within graph neural networks.