# Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression

arXiv: https://arxiv.org/abs/2206.04804 · HTML: https://mlsanity.com/paper/2206.04804/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 55 references that are in the map.
- First posted on arXiv 8 Jun 2022.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Variational Quantum Circuits for Deep Reinforcement Learning](https://mlsanity.com/paper/1907.00397/) (2019)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [Differentially Private Empirical Risk Minimization Revisited: Faster and More General](https://mlsanity.com/paper/1802.05251/) (2018)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [Reinforcement learning for optimization of variational quantum circuit architectures](https://mlsanity.com/paper/2103.16089/) (2021)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [Tensor-Train Recurrent Neural Networks for Video Classification](https://mlsanity.com/paper/1707.01786/) (2017)
- [Decentralizing Feature Extraction with Quantum Convolutional Neural Network for Automatic Speech Recognition](https://mlsanity.com/paper/2010.13309/) (2020)
- [An end-to-end trainable hybrid classical-quantum classifier](https://mlsanity.com/paper/2102.02416/) (2021)
- [When BERT Meets Quantum Temporal Convolution Learning for Text Classification in Heterogeneous Computing](https://mlsanity.com/paper/2203.03550/) (2022)
- [On Mean Absolute Error for Deep Neural Network Based Vector-to-Vector Regression](https://mlsanity.com/paper/2008.07281/) (2020)
- [QTN-VQC: An End-to-End Learning framework for Quantum Neural Networks](https://mlsanity.com/paper/2110.03861/) (2021)
- [Classical-to-Quantum Transfer Learning for Spoken Command Recognition Based on Quantum Neural Networks](https://mlsanity.com/paper/2110.08689/) (2021)
- [A Grover-search Based Quantum Learning Scheme for Classification](https://mlsanity.com/paper/1809.06056/) (2018)
- [Analyzing Upper Bounds on Mean Absolute Errors for Deep Neural Network Based Vector-to-Vector Regression](https://mlsanity.com/paper/2008.05459/) (2020)
- [Exploiting Low-Rank Tensor-Train Deep Neural Networks Based on Riemannian Gradient Descent With Illustrations of Speech Processing](https://mlsanity.com/paper/2203.06031/) (2022)
- [Tensor-to-Vector Regression for Multi-channel Speech Enhancement based on Tensor-Train Network](https://mlsanity.com/paper/2002.00544/) (2020)

## Built on by (most cited first, 26 of 26)

- [Certified Robustness of Quantum Classifiers against Adversarial Examples through Quantum Noise](https://mlsanity.com/paper/2211.00887/) (2022)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Federated Quantum Machine Learning with Differential Privacy](https://mlsanity.com/paper/2310.06973/) (2023)
- [Federated Quantum Natural Gradient Descent for Quantum Federated Learning](https://mlsanity.com/paper/2209.00564/) (2022)
- [Parallel Hybrid Networks: an interplay between quantum and classical neural networks](https://mlsanity.com/paper/2303.03227/) (2023)
- [Optimizing Quantum Federated Learning Based on Federated Quantum Natural Gradient Descent](https://mlsanity.com/paper/2303.08116/) (2023)
- [Efficient quantum recurrent reinforcement learning via quantum reservoir computing](https://mlsanity.com/paper/2309.07339/) (2023)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [Quantum Federated Learning With Quantum Networks](https://mlsanity.com/paper/2310.15084/) (2023)
- [Generalization Error Bound for Quantum Machine Learning in NISQ Era -- A Survey](https://mlsanity.com/paper/2409.07626/) (2024)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [TensorHyper-VQC: A Tensor-Train-Guided Hypernetwork for Robust and Scalable Variational Quantum Computing](https://mlsanity.com/paper/2508.01116/) (2025)
- [A hybrid quantum-classical fusion neural network to improve protein-ligand binding affinity predictions for drug discovery](https://mlsanity.com/paper/2309.03919/) (2023)
- [Lean classical-quantum hybrid neural network model for image classification](https://mlsanity.com/paper/2412.02059/) (2024)
- [Quantum Machine Learning: An Interplay Between Quantum Computing and Machine Learning](https://mlsanity.com/paper/2411.09403/) (2024)
- [Pre-training Tensor-Train Networks Facilitates Machine Learning with Variational Quantum Circuits](https://mlsanity.com/paper/2306.03741/) (2023)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [Leveraging Pre-Trained Neural Networks to Enhance Machine Learning with Variational Quantum Circuits](https://mlsanity.com/paper/2411.08552/) (2024)
- [Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noise](https://mlsanity.com/paper/2412.11451/) (2024)
- [VQC-MLPNet: An Unconventional Hybrid Quantum-Classical Architecture for Scalable and Robust Quantum Machine Learning](https://mlsanity.com/paper/2506.10275/) (2025)
- [Joint Tensor-Train Parameterization for Efficient and Expressive Low-Rank Adaptation](https://mlsanity.com/paper/2506.16456/) (2025)
- [Quantum Circuit-Based Learning Models: Bridging Quantum Computing and Machine Learning](https://mlsanity.com/paper/2602.00048/) (2026)
- [Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits](https://mlsanity.com/paper/2601.01877/) (2026)
- [Continual Quantum Architecture Search with Tensor-Train Encoding: Theory and Applications to Signal Processing](https://mlsanity.com/paper/2601.06392/) (2026)
- [Quantum LEGO Learning: A Modular Design Principle for Hybrid Artificial Intelligence](https://mlsanity.com/paper/2601.21780/) (2026)
- [Multitask Learning for Earth Observation Data Classification with Hybrid Quantum Network](https://mlsanity.com/paper/2601.22195/) (2026)

## Abstract (arXiv)

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.