Jet Discrimination with Quantum Complete Graph Neural Network

TL;DR

The paper introduces the Quantum Complete Graph Neural Network (QCGNN), a variational quantum algorithm tailored for learning on complete graphs, and applies it to jet discrimination where jets are modeled as complete graphs. By encoding node features into a two-register quantum circuit and employing data re-uploading, QCGNN achieves permutation-invariant graph-level outputs with potential polynomial speedups over classical graph neural networks. Empirical results show QCGNN achieving competitive performance to classical baselines with similar parameter budgets, and greater training stability across seeds, though real-device quantum noise currently limits practical demonstrations. Noise-aware simulations indicate that achieving a quantum advantage will require substantially reduced hardware noise and potentially deeper quantum circuits, while encoding and runtime characteristics suggest linear scaling in the encoded graph size when the parametrized layers are sufficiently expressive.

Abstract

Machine learning, particularly deep neural networks, has been widely used in high-energy physics, demonstrating remarkable results in various applications. Furthermore, the extension of machine learning to quantum computers has given rise to the emerging field of quantum machine learning. In this paper, we propose the Quantum Complete Graph Neural Network (QCGNN), which is a variational quantum algorithm based model designed for learning on complete graphs. QCGNN with deep parametrized operators offers a polynomial speedup over its classical and quantum counterparts, leveraging the property of quantum parallelism. We investigate the application of QCGNN with the challenging task of jet discrimination, where the jets are represented as complete graphs. Additionally, we conduct a comparative analysis with classical models to establish a performance benchmark.

Figures (7)

• Figure 1: Example of a QCGNN ansatz for learning a 3-node complete graph, with $n_I=2$ and $n_Q=4$. The quantum state is initially prepared in a uniform quantum state as described in Eq. \ref{['psi_0']} via a uniform state oracle (blue block). The dashed box contains the encoding (red blocks) and parametrized operators (green blocks), which may be re-uploaded $R$ times reupload. The encoding operators consist of multi-controlled operators that correspond to Eq. \ref{['mc_gates']}, transforming the quantum state to Eq. \ref{['psi_1']}, or, in this example, Eq. \ref{['psi_1_ex3']}. The empty ($\circ$) and filled ($\bullet$) circles in the IR represent the controlled conditions for the controlled qubits: the encoding operators are applied if the corresponding qubit in IR is in state $\ket{0}$ and $\ket{1}$, respectively. Since the encoding operators are operated with different controlled conditions, the red controlled encoding blocks are mutually commutable, i.e., the quantum state stays the same if one first encode $\mathbf{x}_1$ then $\mathbf{x}_0$ with their corresponding controlled condition. The quantum state then evolves to Eq. \ref{['psi_2']} through the parametrized operators. If the dashed box is re-uploaded $R$ times, the quantum state evolves to Eq. \ref{['psi_final']}. Finally, the qubits in the IR are measured in the $X$ basis as described in Eq. \ref{['Jdecompose']}, while the NR qubits are measured using Pauli-$Z$ observables to calculate the result in Eq. \ref{['eq_JP']}. The details of the encoding and parametrized operators used are provided in Sec. \ref{['sec_cq_model_setup']}.
• Figure 2: The ansatz of encoding operator used in QCGNN with $n_Q=4$. The rotation gates $R_x$ and $R_y$ are defined in Eq. \ref{['sin_rotgate']}. The rotation angle $\mathbf{x}^{(0)}_{i,j}$ corresponds to the $j$-th feature of the $i$-th particle, with the three features being the transformed particle flow information described in Eq. \ref{['QCGNN_feature']}. Since the data encoding method used in each NR qubit is identical, this ansatz can be generalized to any number of qubits.
• Figure 3: The ansatz of strongly entangling layers strong_ent used in QCGNN is illustrated with $n_Q=4$ as an example. The three-angle rotation gate $R$ is defined in Eq. \ref{['tri_rotgate']}. The parameters $\boldsymbol{\theta}$ are tunable, with distinct parameters for each repetition $l$. Specifically, $\boldsymbol{\theta}_{l,1}$, $\boldsymbol{\theta}_{l,2}$, $\boldsymbol{\theta}_{l,3}$, and $\boldsymbol{\theta}_{l,4}$ are all three-dimensional parameters corresponding to the arguments of $R(\alpha, \beta, \gamma)$ in Eq. \ref{['tri_rotgate']}. Note that this ansatz can be naturally generalized for $n_Q \ge 3$. For $n_Q < 3$, alternative ansatz should be considered.
• Figure 4: The detailed structure of the quantum model based on QCGNN and the classical model based on MPGNN used for benchmarking are described. The particle flow features are defined in Sec. \ref{['sec_data_setup']}, and the hyperparameters of the models are discussed in Sec. \ref{['sec_train']}. The classical model on the left is based on MPGNN, with the aggregation function chosen to be SUM. The number of hidden neurons in MPGNN is denoted as $n_M$, set equivalently to $n_Q$ (3 or 6) for comparison with QCGNN. The quantum model on the right is based on QCGNN, with feature preprocessed as described in Eq. \ref{['QCGNN_feature']}. The ansatz for the parametrized operators follows the pattern depicted in Fig. \ref{['fig:exvqc']}. Note that the data re-upload technique introduced in reupload is used, which involves encoding the data twice and applying the parametrized operators with different parameters each time. The dimension of the final output is denoted as $n_C$, with $n_C=1$ for the Top dataset and $n_C=5$ for the JetNet dataset. The final output is passed through a Sigmoid function for binary classification ($n_C=1$) or a Softmax function for multi-class classification ($n_C=5$).
• Figure 5: Histograms of the number of particles per jet for the Top and the JetNet datasets. A detailed description of both datasets is provided in Sec. \ref{['sec_data_setup']}. The original JetNet dataset exhibits a sharp distribution at 30 particles, as the original data retain only the first 30 particles with the highest transverse momentum. The histograms for particles with a relative transverse momentum $z_i \ge 0.025$ are also given. This threshold is selected to ensure that the majority of the distribution of the number of particles per jet falls between 4 and 16.