Node embeddings act as the information interface for graph neural networks, yet their empirical impact is often reported under mismatched backbones, splits, and training budgets. This paper provides a controlled benchmark of embedding choices for graph classification, comparing classical baselines with quantum-oriented node representations under a unified pipeline. We evaluate two classical baselines alongside quantum-oriented alternatives, including a circuit-defined variational embedding and quantum-inspired embeddings computed via graph operators and linear-algebraic constructions. All variants are trained and tested with the same backbone, stratified splits, identical optimization and early stopping, and consistent metrics. Experiments on five different TU datasets and on QM9 converted to classification via target binning show clear dataset dependence: quantum-oriented embeddings yield the most consistent gains on structure-driven benchmarks, while social graphs with limited node attributes remain well served by classical baselines. The study highlights practical trade-offs between inductive bias, trainability, and stability under a fixed training budget, and offers a reproducible reference point for selecting quantum-oriented embeddings in graph learning.
Journal reference
2026 International Joint Conference on Neural Networks (IJCNN), Maastricht, Netherlands, 2026, pp. 1-6
Node embeddings act as the information interface for graph neural networks, yet their empirical impact is often reported under mismatched backbones, splits, and training budgets. This paper provides a controlled benchmark of embedding choices for graph classification, comparing classical baselines with quantum-oriented node representations under a unified pipeline. We evaluate two classical baselines alongside quantum-oriented alternatives, including a circuit-defined variational embedding and quantum-inspired embeddings computed via graph operators and linear-algebraic constructions. All variants are trained and tested with the same backbone, stratified splits, identical optimization and early stopping, and consistent metrics. Experiments on five different TU datasets and on QM9 converted to classification via target binning show clear dataset dependence: quantum-oriented embeddings yield the most consistent gains on structure-driven benchmarks, while social graphs with limited node attributes remain well served by classical baselines. The study highlights practical trade-offs between inductive bias, trainability, and stability under a fixed training budget, and offers a reproducible reference point for selecting quantum-oriented embeddings in graph learning.
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 to generate quantum-native node embeddings. 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 enhanced 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 remains trainable to larger, more complex graph-based problems.
Graph Neural Networks (GNNs) achieve state-of-the-art performance on graph-structured data, but training and inference on large graphs are often bottlenecked by memory constraints and sparse linear-algebra workloads. Quantum computing offers an alternative set of primitives that may improve scalability for graph learning. Building on the quantum graph neural network (QGNN) framework of Liao \textit{et al.}, this work implements two representative architectures --- the Simplified Graph Convolution (SGC) and Linear Graph Convolution (LGC) models --- and evaluates them on open benchmark graph datasets and semi-supervised learning tasks using quantum simulation. We compare predictive performance and optimization behavior against classical baselines, showing that the quantum models achieve competitive performance with fewer parameters. Finally, we present a cost gradient analysis that identifies the tasks for which the models showcased are trainable. This is followed by a classical simulability study to find regimes in which the proposed circuits remain robust during training.
Background As graph datasets—including social networks, supply chains, and bioinformatics data—grow in size and complexity, researchers are driven to search for solutions enhancing model efficiency and speed. One avenue that may provide a solution is Quantum Graph Learning (QGL), a subfield of Quantum Machine Learning (QML) that applies machine learning inspired or powered by quantum computing to graph learning tasks. Methods We reevaluate Quantum Feature Embeddings (QFE), a QGL methodology published by Xu et al. earlier this year. QFE uses Variational Quantum Circuits to preprocess node features and then sends them to a classical Graph Neural Network (GNN), with the goal of increasing performance and/or decreasing total model size. Xu et al. evaluated this methodology by comparing its performance with the performance of variously-sized classical models on the benchmark datasets PROTEINS and ENZYMES, and they report success. Our core methodology and learning task remain unchanged. However, we have made several changes to the experimental design that enhance the rigor of the study: 1) we include the testing of models with no embedder; 2) we conduct a thorough hyperparameter search using a state-of-the-art optimization algorithm; and 3) we conduct stratified five-fold cross-validation, which mitigates the bias produced by our small datasets and provides multiple test statistics from which we can calculate a confidence interval. Results We produce classical models that perform comparably to QFE and significantly outperform the small classical models used in Xu et al.’s comparison. Notably, many of our classical models achieve this using fewer parameters than the QFE models we trained. Xu et al. do not report their total model sizes. Conclusion Our study sheds doubt on the efficacy of QFE by demonstrating that small, well-tuned classical models can perform just as well as QFE, highlighting the importance of hyperparameter tuning and rigorous experimental design.
Quantum computing offers a promising avenue to reduce growing machine learning model complexity as required in large language models and simulation models for weather forecasts, financial forecasts, or engineering.Graph neural networks are a particular class of machine learning models that have garnered much attention for their ability to deal well with structured data.We investigate how to enhance existing GNNs and find through the inductive bias that quantum circuits are used best to encode node features.The proposed Quantum Feature Embeddings (QFEs) turn raw input features into quantum states, enabling non-linear and entangled representations.In particular, QFEs provide normalized, non-redundant weight matrices in an exponentially larger feature space and require much fewer qubits than fully quantum graph neural networks.On standard graph benchmark datasets, we showcase that for the same parameter count QFEs perform better than their classical counterpart, and are able to match the performance of an exponentially larger model.Finally, we study the potential benefit of using a hybrid quantum graph neural network over a classic alternative on a concrete use case, laser cutting.We find that the proposed model has the performance and thus the near-term potential to uplift these business applications.
Quantum computing offers a promising avenue to reduce growing machine learning model complexity as required in large language models and simulation models for weather forecasts, financial forecasts, or engineering. Graph neural networks are a particular class of machine learning models that have garnered much attention for their ability to deal well with structured data. We investigate how to enhance existing GNNs and find through the inductive bias that quantum circuits are used best to encode node features. The proposed Quantum Feature Embeddings (QFEs) turn raw input features into quantum states, enabling non-linear and entangled representations. In particular, QFEs provide normalized, non-redundant weight matrices in an exponentially larger feature space and require much fewer qubits than fully quantum graph neural networks. On standard graph benchmark datasets, we showcase that for the same parameter count QFEs perform better than their classical counterpart, and are able to match the performance of an exponentially larger model. Finally, we study the potential benefit of using a hybrid quantum graph neural network over a classic alternative on a concrete use case, laser cutting. We find that the proposed model has the performance and thus the near-term potential to uplift these business applications.
Deep Equilibrium Models (DEQs) replace a stack of explicit layers with a single operator whose fixed point defines the output, giving the expressive power of an arbitrarily deep network at the memory cost of a single layer. Quantum Deep Equilibrium Models (QDEQs) bring this idea to quantum machine learning, offering an alternative to Parameterized Quantum Circuits (PQCs), whose depth is limited by hardware coherence and trainability. Here, we introduce, formulate, and compare three ways of coupling a quantum signal to graph DEQs, differing in where the signal enters the fixed-point operator. \textit{Independent} injection computes the quantum signal once per graph and forward fixed-point solve, and holds it fixed throughout the solve. \textit{State-dependent} injection instead recomputes the signal at every solver step and applies it to the current iterate. \textit{Backbone-dependent} injection likewise recomputes at every iteration but applies the signal to the classical backbone's output evaluated at the current iterate. We establish contraction guarantees for each variant under explicit assumptions on the Lipschitz constants of the classical backbone and the quantum signal. On the TU Dortmund graph-classification benchmarks NCI1, PROTEINS, and MUTAG, independent injection achieves the best test accuracy while using fewer forward-solver iterations than both the classical equilibrium baseline and the two dependent variants.