A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks

TL;DR

The paper addresses jet tagging at the LHC by comparing four architectures—classical GNN, SE(2) EGNN, quantum GNN (QGNN), and equivariant quantum GNN (EQGNN)—under invariant and equivariant design principles. It encodes jets as graphs with physically motivated node and edge features, and evaluates on a binary quark-vs-gluon classification task using a Pythia8-based dataset, ensuring fair parameter budgets across models. Results show that EQGNN achieves the best test performance (~$75.17\%$ AUC) among the four, with QGNN offering competitive performance in some parameter regimes, while classical models lag behind in the quantum-augmented setting; however, quantum models require substantial training time on classical hardware due to simulation and API constraints. The study highlights the potential of quantum and equivariant methods for high-energy physics while outlining practical barriers and future directions toward broader node counts, attention mechanisms, and hardware-compatible implementations.

Abstract

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such as graph neural networks (GNNs), have been leveraged for various data analysis tasks in high-energy physics. One typical task is jet tagging, where jets are viewed as point clouds with distinct features and edge connections between their constituent particles. The increasing size and complexity of the LHC particle datasets, as well as the computational models used for their analysis, greatly motivate the development of alternative fast and efficient computational paradigms such as quantum computation. In addition, to enhance the validity and robustness of deep networks, one can leverage the fundamental symmetries present in the data through the use of invariant inputs and equivariant layers. In this paper, we perform a fair and comprehensive comparison between classical graph neural networks (GNNs) and equivariant graph neural networks (EGNNs) and their quantum counterparts: quantum graph neural networks (QGNNs) and equivariant quantum graph neural networks (EQGNN). The four architectures were benchmarked on a binary classification task to classify the parton-level particle initiating the jet. Based on their AUC scores, the quantum networks were shown to outperform the classical networks. However, seeing the computational advantage of the quantum networks in practice may have to wait for the further development of quantum technology and its associated APIs.

Figures (7)

• Figure S1: Distributions of the jet transverse momenta $p_T$, total momenta $p$, and energies $E$.
• Figure S2: A visualization of graphically structured data (left) and a sample jet shown in the $(\phi,y)$ plane (right) with each particle color-coded by its transverse momentum $p_{T,\alpha}^{(i)}$.
• Figure S3: Graph neural network (GNN, left) and equivariant graph neural network (EGNN, right) schematic diagrams.
• Figure S4: The $8 \times 8$ matrix representations of the coupling and transverse Hamiltonians defined in Equation (\ref{['eq:hamiltonian']}).
• Figure S5: Quantum graph neural network (QGNN, left) and equivariant quantum graph neural network (EQGNN, right) schematic diagrams.