mlsanityData to 7 Oct 2026

Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits

arXiv 2310.07528 ↗First posted 11 Oct 2023quant-ph cs.LGcited by 9

Not on the map. Why?

Abstract (arXiv)

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 34 of its 76 references that are in the map. First posted on arXiv 11 Oct 2023.

Builds on

34 of 76 references.

References · 34

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2018BERT: Pre-training of Deep Bidirectional Transformers for Language Understandingcited by 40,204
  3. 2015Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networkscited by 9,617
  4. 2019XLNet: Generalized Autoregressive Pretraining for Language Understandingcited by 2,801
  5. 2018Supervised learning with quantum enhanced feature spacescited by 654
  6. 2016Error bounds for approximations with deep ReLU networkscited by 567
  7. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 431
  8. 2020The power of quantum neural networkscited by 377
  9. 2020Power of data in quantum machine learningcited by 336
  10. 2019Parameterized quantum circuits as machine learning modelscited by 330
  11. 2020The effect of data encoding on the expressive power of variational quantum machine learning modelscited by 328
  12. 2020A rigorous and robust quantum speed-up in supervised machine learningcited by 270
22 more
  1. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  2. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 165
  3. 2020Deep Network Approximation for Smooth Functionscited by 163
  4. 2019Deep Network Approximation Characterized by Number of Neuronscited by 120
  5. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  6. 2021Information-theoretic bounds on quantum advantage in machine learningcited by 95
  7. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 95
  8. 2021Quantum machine learning beyond kernel methodscited by 91
  9. 2021Parametrized quantum policies for reinforcement learningcited by 87
  10. 2019The phase diagram of approximation rates for deep neural networkscited by 86
  11. 2020Experimental Quantum Generative Adversarial Networks for Image Generationcited by 83
  12. 2020Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spacescited by 44
  13. 2022Experimental quantum adversarial learning with programmable superconducting qubitscited by 38
  14. 2019Input Redundancy for Parameterized Quantum Circuitscited by 33
  15. 2023Deep Network Approximation: Beyond ReLU to Diverse Activation Functionscited by 33
  16. 2020Pseudo-dimension of quantum circuitscited by 30
  17. 2022Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machinescited by 28
  18. 2022Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regressioncited by 26
  19. 2021On the statistical complexity of quantum circuitscited by 25
  20. 2022Power and limitations of single-qubit native quantum neural networkscited by 19
  21. 2023Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirscited by 15
  22. 2021Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Classcited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 6; 2026: 3.

9 papers.

  1. 2025Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networkscited by 10
  2. 2025Feedback-driven recurrent quantum neural network universalitycited by 5
  3. 2025Mitigating Exponential Mixed Frequency Growth through Frequency Selectioncited by 1
  4. 2025Hybrid Quantum-Classical Recurrent Neural Networkscited by 1
  5. 2026Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEscited by 1
  6. 2025Learning AC Power Flow Solutions using a Data-Dependent Variational Quantum Circuitcited by 0
  7. 2025Approximation rates of quantum neural networks for periodic functions via Jackson's inequalitycited by 0
  8. 2026SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximatorcited by 0
  9. 2026SQUARE: Structured Quantum Representation Adapters as Compact Quadratic Feature Maps for Frozen Language Modelscited by 0