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Learnable quantum spectral filters for hybrid graph neural networks

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 $N\times N$ 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.

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