Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression

TL;DR

This paper presents a theoretical analysis of the error performance of a novel quantum neural network combining tensor-train networks and variational quantum circuits, with experimental validation on digit classification.

Contribution

It introduces the TTN-VQC model and provides the first theoretical error analysis of its representation, generalization, and optimization properties.

Findings

Theoretical bounds on the error performance of TTN-VQC.

Validation of theoretical analysis through experiments on digit classification.

Insights into the optimization landscape of TTN-VQC.

Abstract

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.