🤖 AI Summary
Classical polynomial approximations for Laplacian spectral filtering in graph neural networks (GNNs) incur high computational overhead and limit scalability.
Method: This paper proposes a learnable quantum spectral filter embedded in a quantum-classical hybrid GNN architecture. Leveraging parameterized quantum Fourier circuits, the method encodes the geometric structure of the graph Laplacian into quantum circuit connectivity and directly learns the frequency-domain filtering function within an exponentially compressed parameter space—using only 1–100 trainable quantum parameters—bypassing classical approximation entirely. The model integrates quantum Fourier transforms, low-dimensional approximations of the Laplacian eigenspace, and a classical prediction head to enable efficient graph signal convolution and pooling.
Results: On the TUDataset benchmark, the approach achieves graph classification performance competitive with or superior to state-of-the-art GNNs, while significantly improving spectral filtering efficiency and scalability.
📝 Abstract
In this paper, we describe a parameterized quantum circuit that can be considered as convolutional and pooling layers for graph neural networks. The circuit incorporates the parameterized quantum Fourier circuit where the qubit connections for the controlled gates derived from the Laplacian operator. Specifically, we show that the eigenspace of the Laplacian operator of a graph can be approximated by using QFT based circuit whose connections are determined from the adjacency matrix. For an $N imes N$ Laplacian, this approach yields an approximate polynomial-depth circuit requiring only $n=log(N)$ qubits. These types of circuits can eliminate the expensive classical computations for approximating the learnable functions of the Laplacian through Chebyshev polynomial or Taylor expansions.
Using this circuit as a convolutional layer provides an $n-$ dimensional probability vector that can be considered as the filtered and compressed graph signal. Therefore, the circuit along with the measurement can be considered a very efficient convolution plus pooling layer that transforms an $N$-dimensional signal input into $n-$dimensional signal with an exponential compression. We then apply a classical neural network prediction head to the output of the circuit to construct a complete graph neural network. Since the circuit incorporates geometric structure through its graph connection-based approach, we present graph classification results for the benchmark datasets listed in TUDataset library. Using only [1-100] learnable parameters for the quantum circuit and minimal classical layers (1000-5000 parameters) in a generic setting, the obtained results are comparable to and in some cases better than many of the baseline results, particularly for the cases when geometric structure plays a significant role.