Scalable Message-Passing Quantum Graph Neural Networks in the Weisfeiler-Leman Hierarchy

2026-06-25

AI Summary

This work proposes a scalable quantum graph neural network that establishes, for the first time, a rigorous correspondence with the Weisfeiler–Leman (WL) hierarchy. By employing variational quantum circuits to implement permutation-equivariant message passing, the model explicitly aligns with a specified WL test level, thereby offering theoretical guarantees on expressivity. To address key limitations of existing approaches—namely, weak connections to classical message-passing schemes and insufficient theoretical foundations regarding trainability and scalability—the architecture integrates a low-complexity readout mechanism and a pretraining strategy generalizable across graph sizes. Empirical evaluations on simulations up to 56 qubits demonstrate the model’s ability to distinguish graph structures indistinguishable by classical methods, while achieving superior performance in molecular property prediction and the traveling salesman problem.

Abstract

Graphs provide a natural language for relational data in chemistry, biology and optimisation. Graph neural networks (GNNs) have driven much of the recent progress in learning from such data through message passing, a single primitive that generalises convolution and attention. Quantum counterparts have been proposed, but with limited connection to message passing and few guarantees on performance or scalability. More broadly, the trainability of variational quantum circuits is a recognised bottleneck for their wide applicability, and pre-training has emerged as one way to address it. Yet for a quantum model to be useful, it must offer expressivity guarantees along with demonstrable scalability. Here we show how a quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy, the standard measure of how finely a model can tell graphs apart. We show that, as for classical GNNs, the training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows. We validate the framework in large-scale simulations of up to 56 qubits across three datasets, on synthetic graphs that ordinary message passing cannot separate, on molecular property prediction, and on the travelling salesperson problem. Our framework opens a path for near-term quantum algorithms with theoretical guarantees and practical scalability, bringing the principles of graph learning into quantum circuit design.

Problem

• quantum graph neural networks
• message passing
• Weisfeiler-Leman hierarchy
• scalability
• trainability

Innovation

• Quantum Graph Neural Networks
• Message Passing
• Weisfeiler-Leman Hierarchy
• Pre-training
• Scalability