  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.
