Inductive Graph Representation Learning with Quantum Graph Neural Networks

TL;DR

The paper addresses inductive learning on graphs using quantum computing by introducing a GraphSAGE–inspired Quantum Graph Neural Network (QGNN) that uses a modular quantum aggregator (QGCN) and a quantum feature map to process local node features. It demonstrates that a single minimal QGCN circuit can handle graphs with varying sizes, while extending to multiple per-hop quantum aggregators to improve trainability and performance. On QM9 molecular graphs, the QGNN shows stronger generalization as molecular complexity increases, and gradient-variance analyses indicate the absence of barren plateaus as qubit counts grow, supporting scalability. Overall, the work provides a flexible, scalable quantum framework for inductive graph representation learning with practical implications for complex graph-structured data.

Abstract

Quantum Graph Neural Networks (QGNNs) offer a promising approach to combining quantum computing with graph-structured data processing. While classical Graph Neural Networks (GNNs) are scalable and robust, existing QGNNs often lack flexibility due to graph-specific quantum circuit designs, limiting their applicability to diverse real-world problems. To address this, we propose a versatile QGNN framework inspired by GraphSAGE, using quantum models as aggregators. We integrate inductive representation learning techniques with parameterized quantum convolutional and pooling layers, bridging classical and quantum paradigms. The convolutional layer is flexible, allowing tailored designs for specific tasks. Benchmarked on a node regression task with the QM9 dataset, our framework, using a single minimal circuit for all aggregation steps, handles molecules with varying numbers of atoms without changing qubits or circuit architecture. While classical GNNs achieve higher training performance, our quantum approach remains competitive and often shows stronger generalization as molecular complexity increases. We also observe faster learning in early training epochs. To mitigate trainability limitations of a single-circuit setup, we extend the framework with multiple quantum aggregators on QM9. Assigning distinct circuits to each hop substantially improves training performance across all cases. Additionally, we numerically demonstrate the absence of barren plateaus as qubit numbers increase, suggesting that the proposed model can scale to larger, more complex graph-based problems.

Figures (8)

• Figure 1: Illustration of the GraphSAGE framework. (a) Aggregation steps: agg1 and agg2, generate the message which is then passed to the target node $v$ for $k=2$ hops. (b) Transformation of an input graph (left) into a computation graph (right), where nodes represent feature embeddings and edges indicate information flow. Gray boxes denote Multi-Layer Perceptrons (MLPs), which aggregate neighboring embeddings through modular transformations, enabling accurate node representations.
• Figure 2: QGNN framework.
• Figure 4: Schematic representation of the second convolutional unitary $W$, acting on two neighboring qubits. Each layer comprises of CZ gates applied to alternating qubit pairs, enclosed by parameterized general single-qubit rotations $R_G$. The single qubit gates: $R_Z, R_Y$, and, $R'_Y$ also are parametrized with different tunable angles.
• Figure 5: Schematic representation of the QGCN circuit. The QGCN takes as input an $n$-qubit quantum state $\rho_{\text{in}}$ and processes it through a sequence of $L$ alternating convolutional (C) and pooling (P) layers. In each convolutional layer, parameterized quantum gates perform unitary transformations $W$ on pairs of neighboring qubits. The pooling layers then reduce the number of qubits by tracing out half of them between layers. After the last pooling layer, the circuit produces the output state $\rho_{\mathrm{out}}^{\bm{\theta}_v}$.
• Figure 6: Comparison of training and testing performance for GNN (classical) and QGNN (quantum) models on molecules from the QM9 dataset, with all results obtained over 300 epochs. The markers are accompanied by dashed lines in their corresponding colors: circular and diamond markers distinguish the GNN's train and test results, respectively, while square and triangular markers represent the QGNN's train and test results. (a) and (b) present the R² score and Loss respectively for Case 1.