Theoretical error performance analysis for variational quantum circuit based functional regression

The noisy intermediate-scale quantum 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 QNN, 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 condition. Moreover, we conduct the experiments of functional regression on a handwritten digit classification dataset to justify our theoretical analysis.

Automated summary

In this work, an end-to-end quantum neural network TTN-VQC is proposed, which combines a quantum tensor network based on tensor-train network (TTN) for dimensionality reduction and a VQC for functional regression. The error performance analysis is conducted to evaluate the representation and generalization powers of TTN-VQC, and the optimization properties are characterized using the Polyak-Lojasiewicz condition. Additionally, experiments on a handwritten digit classification dataset are conducted to validate the theoretical analysis.