Approximation and Generalization Capacities of Parametrized Quantum Circuits for Functions in Sobolev Spaces
1Department of Mathematics and CITIC, Universidade da Coruña, Campus de Elviña s/n, A Coruña, Spain
2$\langle aQa^L\rangle$ Applied Quantum Algorithms Leiden, The Netherlands
3Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
4LIACS, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, Netherlands
| Published: | 2025-03-10, volume 9, page 1658 |
| Editor: | Patrick Rebentrost |
| Eprint: | arXiv:2307.14792v3 |
| Doi: | https://doi.org/10.22331/q-2025-03-10-1658 |
| Citation: | Quantum 9, 1658 (2025). |
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Abstract
Parametrized quantum circuits (PQC) are quantum circuits which consist of both fixed and parametrized gates. In recent approaches to quantum machine learning (QML), PQCs are essentially ubiquitous and play the role analogous to classical neural networks. They are used to learn various types of data, with an underlying expectation that if the PQC is made sufficiently deep, and the data plentiful, the generalization error will vanish, and the model will capture the essential features of the distribution. While there exist results proving the approximability of square-integrable functions by PQCs under the $L^2$ distance, the approximation for other function spaces and under other distances has been less explored. In this work we show that PQCs can approximate the space of continuous functions, $p$-integrable functions and the $H^k$ Sobolev spaces under specific distances. Moreover, we develop generalization bounds that connect different function spaces and distances. These results provide a theoretical basis for different applications of PQCs, for example for solving differential equations. Furthermore, they provide us with new insight on the role of the data normalization in PQCs and of loss functions which better suit the specific needs of the users.
Featured image: This figure illustrates how the input re-scaling strategy affects the ability of the PQC to approximate a linear function. Even in this simple case, the approximation deteriorates under the re-scalings applied in the middle and right panels. This is particularly visible near the boundaries of the domain. In contrast, with the re-scaling applied in the left panel, the PQC accurately approximates both the function and its derivatives, as predicted by our theoretical analysis.
Popular summary
A key challenge is the capacity of PQCs to approximate both functions and their derivatives sufficiently well. We prove that naive strategies relying on simple fitting must in general fail for fundamental reasons that are similar to the Gibbs phenomenon from harmonic analysis. They lead to not only poor approximations of the function derivatives but also cause large errors in the function approximation at the boundary of the domain. We propose a solution involving a simple but specific input re-scaling strategy, which enables significant improvements in the simultaneous approximation of function values and derivatives.
Beyond function approximation, ensuring good generalization is another key challenge in quantum machine learning. In particular, training a PQC with the commonly used $L^2$-loss function, which computes the average of the squared differences between approximated and true function values over all training points, does not allow for arbitrary precision at every function value across the entire domain. However, we prove that including both function values and their derivatives in the training allows PQCs to achieve this in principle, which is especially valuable in practical settings where derivative data is often available at little or no extra cost, such as in physical measurements or financial modelling.
Our results broaden the potential application of PQCs in fields where understanding both behavior and trends is crucial, from solving differential equations in physics to analyzing financial risks.
► BibTeX data
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Cited by
[1] Adrián Pérez-Salinas, Mahtab Yaghubi Rad, Alice Barthe, and Vedran Dunjko, "Universal approximation of continuous functions with minimal quantum circuits", Physical Review Research 7 4, 043282 (2025).
[2] Said Lantigua, Gilson Giraldi, and Renato Portugal, "Classical-quantum hybrid architecture for physics-informed neural networks", Physical Review A 113 4, 042446 (2026).
[3] Shreyan Basu Ray and Soujanya Ray, Information Systems Engineering and Management 65, 205 (2025) ISBN:978-3-031-99785-3.
[4] Haimeng Zhao, Laura Lewis, Ishaan Kannan, Yihui Quek, Hsin-Yuan Huang, and Matthias C. Caro, "Learning Quantum States and Unitaries of Bounded Gate Complexity", PRX Quantum 5 4, 040306 (2024).
[5] Li-Wei Yu, Weikang Li, Qi Ye, Zhide Lu, Zizhao Han, and Dong-Ling Deng, "Expressibility-induced concentration of quantum neural tangent kernels", Reports on Progress in Physics 87 11, 110501 (2024).
[6] Zhan Yu, Qiuhao Chen, Yuling Jiao, Yinan Li, Xiliang Lu, Xin Wang, and Jerry Zhijian Yang, "Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits", arXiv:2310.07528, (2023).
[7] Qiuhao Chen, Yuling Jiao, Yinan Li, Xiliang Lu, and Jerry Zhijian Yang, "Near-optimal Prediction Error Estimation for Quantum Machine Learning Models", arXiv:2510.18208, (2025).
[8] Laszlo Gyongyosi and Sandor Imre, "Networked Quantum Services", Quantum Information & Computation 25 2, 97 (2025).
[9] Xinliang Zhai, Tailong Xiao, Jingzheng Huang, Jianping Fan, and Guihua Zeng, "Quantum neural compressive sensing for ghost imaging", Physical Review Applied 23 1, 014018 (2025).
[10] Junaid Aftab and Haizhao Yang, "Approximating Korobov Functions via Quantum Circuits", arXiv:2404.14570, (2024).
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