Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression
This work addresses the need for better theoretical understanding of quantum neural networks in machine learning, particularly for high-dimensional inputs, but it is incremental as it builds on existing VQC and tensor network methods.
The authors tackled the problem of analyzing representation and generalization powers in variational quantum circuits for functional regression by proposing TTN-VQC, a quantum neural network combining tensor-train networks for dimensionality reduction and VQC, and provided theoretical error analysis and experimental validation on a handwritten digit dataset.
The noisy intermediate-scale quantum (NISQ) devices enable the implementation of the variational quantum circuit (VQC) for quantum neural networks (QNN). Although the VQC-based QNN has succeeded in many machine learning tasks, the representation and generalization powers of VQC still require further investigation, particularly when the dimensionality of classical inputs is concerned. In this work, we first put forth an end-to-end quantum neural network, TTN-VQC, which consists of a quantum tensor network based on a tensor-train network (TTN) for dimensionality reduction and a VQC for functional regression. Then, we aim at the error performance analysis for the TTN-VQC in terms of representation and generalization powers. We also characterize the optimization properties of TTN-VQC by leveraging the Polyak-Lojasiewicz (PL) condition. Moreover, we conduct the experiments of functional regression on a handwritten digit classification dataset to justify our theoretical analysis.