Generalization despite overfitting in quantum machine learning models

Evan Peters; Maria Schuld

doi:10.22331/q-2023-12-20-1210

Generalization despite overfitting in quantum machine learning models

Evan Peters^1,2,3 and Maria Schuld⁴

¹Department of Physics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
²Institute for Quantum Computing, Waterloo, ON, N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
⁴Xanadu, Toronto, ON, M5G 2C8, Canada

Published:	2023-12-20, volume 7, page 1210
Eprint:	arXiv:2209.05523v2
Doi:	https://doi.org/10.22331/q-2023-12-20-1210
Citation:	Quantum 7, 1210 (2023).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The widespread success of deep neural networks has revealed a surprise in classical machine learning: very complex models often generalize well while simultaneously overfitting training data. This phenomenon of benign overfitting has been studied for a variety of classical models with the goal of better understanding the mechanisms behind deep learning. Characterizing the phenomenon in the context of quantum machine learning might similarly improve our understanding of the relationship between overfitting, overparameterization, and generalization. In this work, we provide a characterization of benign overfitting in quantum models. To do this, we derive the behavior of a classical interpolating Fourier features models for regression on noisy signals, and show how a class of quantum models exhibits analogous features, thereby linking the structure of quantum circuits (such as data-encoding and state preparation operations) to overparameterization and overfitting in quantum models. We intuitively explain these features according to the ability of the quantum model to interpolate noisy data with locally "spiky" behavior and provide a concrete demonstration example of benign overfitting.

► BibTeX data

@article{Peters2023generalization,
  doi = {10.22331/q-2023-12-20-1210},
  url = {https://doi.org/10.22331/q-2023-12-20-1210},
  title = {Generalization despite overfitting in quantum machine learning models},
  author = {Peters, Evan and Schuld, Maria},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1210},
  month = dec,
  year = {2023}
}

► References

[1] Michael A Nielsen. ``Neural networks and deep learning''. Determination Press. (2015). url: http://neuralnetworksanddeeplearning.com/.
http://neuralnetworksanddeeplearning.com/

[2] Stuart Geman, Elie Bienenstock, and René Doursat. ``Neural networks and the bias/variance dilemma''. Neural Comput. 4, 1–58 (1992).
https://doi.org/10.1162/neco.1992.4.1.1

[3] Trevor Hastie, Robert Tibshirani, Jerome H Friedman, and Jerome H Friedman. ``The elements of statistical learning: data mining, inference, and prediction''. Volume 2. Springer. (2009).
https://doi.org/10.1007/978-0-387-84858-7

[4] Peter L. Bartlett, Andrea Montanari, and Alexander Rakhlin. ``Deep learning: a statistical viewpoint''. Acta Numerica 30, 87–201 (2021).
https://doi.org/10.1017/S0962492921000027

[5] Mikhail Belkin. ``Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation''. Acta Numerica 30, 203–248 (2021).

[6] Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler. ``Benign overfitting in linear regression''. Proc. Natl. Acad. Sci. 117, 30063–30070 (2020).
https://doi.org/10.1073/pnas.1907378117

[7] Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. ``Reconciling modern machine-learning practice and the classical bias-variance trade-off''. Proc. Natl. Acad. Sci. 116, 15849–15854 (2019).
https://doi.org/10.1073/pnas.1903070116

[8] Mikhail Belkin, Alexander Rakhlin, and Alexandre B. Tsybakov. ``Does data interpolation contradict statistical optimality?''. In Proceedings of Machine Learning Research. Volume 89, pages 1611–1619. PMLR (2019). url: https://proceedings.mlr.press/v89/belkin19a.html.
https://proceedings.mlr.press/v89/belkin19a.html

[9] Vidya Muthukumar, Kailas Vodrahalli, Vignesh Subramanian, and Anant Sahai. ``Harmless interpolation of noisy data in regression''. IEEE Journal on Selected Areas in Information Theory 1, 67–83 (2020).
https://doi.org/10.1109/ISIT.2019.8849614

[10] Vidya Muthukumar, Adhyyan Narang, Vignesh Subramanian, Mikhail Belkin, Daniel Hsu, and Anant Sahai. ``Classification vs regression in overparameterized regimes: Does the loss function matter?''. J. Mach. Learn. Res. 22, 1–69 (2021). url: http://jmlr.org/papers/v22/20-603.html.
http://jmlr.org/papers/v22/20-603.html

[11] Yehuda Dar, Vidya Muthukumar, and Richard G. Baraniuk. ``A farewell to the bias-variance tradeoff? an overview of the theory of overparameterized machine learning'' (2021). arXiv:2109.02355.
arXiv:2109.02355

[12] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. ``Parameterized quantum circuits as machine learning models''. Quantum Sci. Technol. 4, 043001 (2019).
https://doi.org/10.1088/2058-9565/ab4eb5

[13] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. ``Quantum circuit learning''. Phys. Rev. A 98, 032309 (2018).
https://doi.org/10.1103/physreva.98.032309

[14] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. ``Evaluating analytic gradients on quantum hardware''. Phys. Rev. A 99, 032331 (2019).
https://doi.org/10.1103/physreva.99.032331

[15] Maria Schuld and Nathan Killoran. ``Quantum machine learning in feature hilbert spaces''. Phys. Rev. Lett. 122, 040504 (2019).
https://doi.org/10.1103/physrevlett.122.040504

[16] Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta. ``Supervised learning with quantum-enhanced feature spaces''. Nature 567, 209–212 (2019).
https://doi.org/10.1038/s41586-019-0980-2

[17] Seth Lloyd and Christian Weedbrook. ``Quantum generative adversarial learning''. Phys. Rev. Lett. 121, 040502 (2018).
https://doi.org/10.1103/physrevlett.121.040502

[18] Pierre-Luc Dallaire-Demers and Nathan Killoran. ``Quantum generative adversarial networks''. Phys. Rev. A 98, 012324 (2018).
https://doi.org/10.1103/physreva.98.012324

[19] Amira Abbas, David Sutter, Christa Zoufal, Aurelien Lucchi, Alessio Figalli, and Stefan Woerner. ``The power of quantum neural networks''. Nat. Comput. Sci. 1, 403–409 (2021).
https://doi.org/10.1038/s43588-021-00084-1

[20] Logan G. Wright and Peter L. McMahon. ``The capacity of quantum neural networks''. In 2020 Conference on Lasers and Electro-Optics (CLEO). Pages 1–2. (2020). url: https://ieeexplore.ieee.org/document/9193529.
https://ieeexplore.ieee.org/document/9193529

[21] Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik. ``Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms''. Adv. Quantum Technol. 2, 1900070 (2019).
https://doi.org/10.1002/qute.201900070

[22] Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, and Koen Bertels. ``Evaluation of parameterized quantum circuits: on the relation between classification accuracy, expressibility and entangling capability''. Quantum Mach. Intell. 3, 1 (2021).
https://doi.org/10.1007/s42484-021-00038-w

[23] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nat. Commun. 9, 4812 (2018).
https://doi.org/10.1038/s41467-018-07090-4

[24] Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits''. Nat. Commun. 12, 1791 (2021).
https://doi.org/10.1038/s41467-021-21728-w

[25] Matthias C. Caro, Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, and Ryan Sweke. ``Encoding-dependent generalization bounds for parametrized quantum circuits''. Quantum 5, 582 (2021).
https://doi.org/10.22331/q-2021-11-17-582

[26] Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, and Jarrod R McClean. ``Power of data in quantum machine learning''. Nat. Commun. 12, 2631 (2021).
https://doi.org/10.1038/s41467-021-22539-9

[27] Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J. Coles. ``Generalization in quantum machine learning from few training data''. Nat. Commun. 13, 4919 (2022).
https://doi.org/10.1038/s41467-022-32550-3

[28] Leonardo Banchi, Jason Pereira, and Stefano Pirandola. ``Generalization in quantum machine learning: A quantum information standpoint''. PRX Quantum 2, 040321 (2021).
https://doi.org/10.1103/PRXQuantum.2.040321

[29] Francisco Javier Gil Vidal and Dirk Oliver Theis. ``Input redundancy for parameterized quantum circuits''. Front. Phys. 8, 297 (2020).
https://doi.org/10.3389/fphy.2020.00297

[30] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. ``Effect of data encoding on the expressive power of variational quantum-machine-learning models''. Phys. Rev. A 103, 032430 (2021).
https://doi.org/10.1103/physreva.103.032430

[31] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin. ``General parameter-shift rules for quantum gradients''. Quantum 6, 677 (2022).
https://doi.org/10.22331/q-2022-03-30-677

[32] Kendall E Atkinson. ``An introduction to numerical analysis''. John Wiley & Sons. (2008).

[33] Ali Rahimi and Benjamin Recht. ``Random features for large-scale kernel machines''. In Advances in Neural Information Processing Systems. Volume 20. (2007). url: https://papers.nips.cc/paper_files/paper/2007/hash/013a006f03dbc5392effeb8f18fda755-Abstract.html.
https://papers.nips.cc/paper_files/paper/2007/hash/013a006f03dbc5392effeb8f18fda755-Abstract.html

[34] Walter Rudin. ``The basic theorems of fourier analysis''. John Wiley & Sons, Ltd. (1990).
https://doi.org/10.1002/9781118165621.ch1

[35] Song Mei and Andrea Montanari. ``The generalization error of random features regression: Precise asymptotics and the double descent curve''. Commun. Pure Appl. Math. 75, 667–766 (2022).
https://doi.org/10.1002/cpa.22008

[36] Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J. Tibshirani. ``Surprises in high-dimensional ridgeless least squares interpolation''. Ann. Stat. 50, 949 – 986 (2022).
https://doi.org/10.1214/21-AOS2133

[37] Tengyuan Liang, Alexander Rakhlin, and Xiyu Zhai. ``On the multiple descent of minimum-norm interpolants and restricted lower isometry of kernels''. In Proceedings of Machine Learning Research. Volume 125, pages 1–29. PMLR (2020). url: http://proceedings.mlr.press/v125/liang20a.html.
http://proceedings.mlr.press/v125/liang20a.html

[38] Edward Farhi and Hartmut Neven. ``Classification with quantum neural networks on near term processors'' (2018). arXiv:1802.06002.
arXiv:1802.06002

[39] Maria Schuld, Alex Bocharov, Krysta M. Svore, and Nathan Wiebe. ``Circuit-centric quantum classifiers''. Phys. Rev. A 101, 032308 (2020).
https://doi.org/10.1103/physreva.101.032308

[40] Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I. Latorre. ``Data re-uploading for a universal quantum classifier''. Quantum 4, 226 (2020).
https://doi.org/10.22331/q-2020-02-06-226

[41] Sofiene Jerbi, Lukas J Fiderer, Hendrik Poulsen Nautrup, Jonas M Kübler, Hans J Briegel, and Vedran Dunjko. ``Quantum machine learning beyond kernel methods''. Nat. Commun. 14, 517 (2023).
https://doi.org/10.1038/s41467-023-36159-y

[42] Casper Gyurik, Dyon Vreumingen, van, and Vedran Dunjko. ``Structural risk minimization for quantum linear classifiers''. Quantum 7, 893 (2023).
https://doi.org/10.22331/q-2023-01-13-893

[43] Maria Schuld. ``Supervised quantum machine learning models are kernel methods'' (2021). arXiv:2101.11020.
arXiv:2101.11020

[44] S. Shin, Y. S. Teo, and H. Jeong. ``Exponential data encoding for quantum supervised learning''. Phys. Rev. A 107, 012422 (2023).
https://doi.org/10.1103/PhysRevA.107.012422

[45] Sophie Piccard. ``Sur les ensembles de distances des ensembles de points d'un espace euclidien.''. Memoires de l'Universite de Neuchatel. Secretariat de l'Universite. (1939).

[46] Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak, and Matthias Troyer. ``Solving strongly correlated electron models on a quantum computer''. Phys. Rev. A 92, 062318 (2015).
https://doi.org/10.1103/PhysRevA.92.062318

[47] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. ``Quantum simulation of electronic structure with linear depth and connectivity''. Phys. Rev. Lett. 120, 110501 (2018).
https://doi.org/10.1103/PhysRevLett.120.110501

[48] Martín Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo. ``Group-invariant quantum machine learning''. PRX Quantum 3, 030341 (2022).
https://doi.org/10.1103/PRXQuantum.3.030341

[49] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert. ``Exploiting symmetry in variational quantum machine learning''. PRX Quantum 4, 010328 (2023).
https://doi.org/10.1103/PRXQuantum.4.010328

[50] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J Coles, and Marco Cerezo. ``Theory of overparametrization in quantum neural networks''. Nat. Comput. Sci. 3, 542–551 (2023).
https://doi.org/10.1038/s43588-023-00467-6

[51] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. ``Expressive power of parametrized quantum circuits''. Phys. Rev. Res. 2, 033125 (2020).
https://doi.org/10.1103/physrevresearch.2.033125

[52] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus''. PRX Quantum 3, 010313 (2022).
https://doi.org/10.1103/PRXQuantum.3.010313

[53] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nat. Commun. 12, 6961 (2021).
https://doi.org/10.1038/s41467-021-27045-6

[54] Abdulkadir Canatar, Evan Peters, Cengiz Pehlevan, Stefan M. Wild, and Ruslan Shaydulin. ``Bandwidth enables generalization in quantum kernel models''. Transactions on Machine Learning Research (2023). url: https://openreview.net/forum?id=A1N2qp4yAq.
https://openreview.net/forum?id=A1N2qp4yAq

[55] Hsin-Yuan Huang, Michael Broughton, Jordan Cotler, Sitan Chen, Jerry Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John Preskill, and Jarrod R. McClean. ``Quantum advantage in learning from experiments''. Science 376, 1182–1186 (2022).
https://doi.org/10.1126/science.abn7293

[56] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. ``Exponential separations between learning with and without quantum memory''. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). Pages 574–585. (2022).
https://doi.org/10.1109/FOCS52979.2021.00063

[57] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Information-theoretic bounds on quantum advantage in machine learning''. Phys. Rev. Lett. 126, 190505 (2021).
https://doi.org/10.1103/PhysRevLett.126.190505

[58] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, M. Sohaib Alam, Shahnawaz Ahmed, Juan Miguel Arrazola, Carsten Blank, Alain Delgado, Soran Jahangiri, Keri McKiernan, Johannes Jakob Meyer, Zeyue Niu, Antal Száva, and Nathan Killoran. ``Pennylane: Automatic differentiation of hybrid quantum-classical computations'' (2018). arXiv:1811.04968.
arXiv:1811.04968

[59] Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler. ``Benign overfitting in linear regression''. Proc. Natl. Acad. Sci. 117, 30063–30070 (2020).
https://doi.org/10.1073/pnas.1907378117

[60] Vladimir Koltchinskii and Karim Lounici. ``Concentration inequalities and moment bounds for sample covariance operators''. Bernoulli 23, 110 – 133 (2017).
https://doi.org/10.3150/15-BEJ730

[61] Zbigniew Puchała and Jarosław Adam Miszczak. ``Symbolic integration with respect to the haar measure on the unitary group''. Bull. Pol. Acad. Sci. 65, 21–27 (2017).
https://doi.org/10.1515/bpasts-2017-0003

[62] Daniel A. Roberts and Beni Yoshida. ``Chaos and complexity by design''. J. High Energy Phys. 2017, 121 (2017).
https://doi.org/10.1007/jhep04(2017)121

[63] Wallace C. Babcock. ``Intermodulation interference in radio systems frequency of occurrence and control by channel selection''. Bell Syst. tech. j. 32, 63–73 (1953).
https://doi.org/10.1002/j.1538-7305.1953.tb01422.x

[64] M. Atkinson, N. Santoro, and J. Urrutia. ``Integer sets with distinct sums and differences and carrier frequency assignments for nonlinear repeaters''. IEEE Trans. Commun. 34, 614–617 (1986).
https://doi.org/10.1109/TCOM.1986.1096587

[65] J. Robinson and A. Bernstein. ``A class of binary recurrent codes with limited error propagation''. IEEE Trans. Inf. 13, 106–113 (1967).
https://doi.org/10.1109/TIT.1967.1053951

[66] R. J. F. Fang and W. A. Sandrin. ``Carrier frequency assignment for nonlinear repeaters''. COMSAT Technical Review 7, 227–245 (1977).

Cited by

[1] Diego H Useche, Sergio Quiroga-Sandoval, Sebastian L Molina, Vladimir Vargas-Calderón, Juan E Ardila-García, and Fabio A González, "Quantum generative classification with mixed states", Quantum Science and Technology 10 4, 045024 (2025).

[2] Daeyeun Kim, Seungcheol Oh, and Joongheon Kim, 2026 40th International Conference on Information Networking (ICOIN) 1019 (2026) ISBN:979-8-3315-7896-1.

[3] Zhihao Zhang, Peng Lei, Jinhua Hu, Jiayu Wu, Qi Wu, Changhui Huang, and Xiqi Zhu, "Diagnostic performance and generalizability of preoperative prediction models for lymph node metastasis in intrahepatic cholangiocarcinoma: a multimodal evidence synthesis", BMC Cancer 26 1, 673 (2026).

[4] Slimane Thabet, Léo Monbroussou, Eliott Z. Mamon, and Jonas Landman, "When quantum and classical models disagree: learning beyond minimum norm least square", npj Quantum Information 12 1, 81 (2026).

[5] Ben Jaderberg, Antonio A. Gentile, Atiyo Ghosh, Vincent E. Elfving, Caitlin Jones, Davide Vodola, John Manobianco, and Horst Weiss, "Potential of quantum scientific machine learning applied to weather modeling", Physical Review A 110 5, 052423 (2024).

[6] Diego Alvarez-Estevez, "Benchmarking Quantum Machine Learning Kernel Training for Classification Tasks", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[7] Domenico Pomarico, Alfonso Monaco, Nicola Amoroso, Loredana Bellantuono, Antonio Lacalamita, Marianna La Rocca, Tommaso Maggipinto, Ester Pantaleo, Sabina Tangaro, Sebastiano Stramaglia, and Roberto Bellotti, "Emerging generalization advantage of quantum-inspired machine learning in the diagnosis of hepatocellular carcinoma", Discover Applied Sciences 7 3, 205 (2025).

[8] Muhammad AbuGhanem, "Toward scalable fault-tolerant photonic quantum computers", The Journal of Supercomputing 82 2, 51 (2026).

[9] Vincenzo Lipardi, Domenica Dibenedetto, Georgios Stamoulis, and Mark H.M. Winands, "Quantum Circuit Design using a Progressive Widening Enhanced Monte Carlo Tree Search", Advanced Quantum Technologies 8 10, e2500093 (2025).

[10] Dirk Heimann, Hans Hohenfeld, Gunnar Schönhoff, Elie Mounzer, and Frank Kirchner, "Learning Fourier series with parametrized quantum circuits", Physical Review Research 7 2, 023151 (2025).

[11] Abdul Basit, Ali Hassan Jamal, Kashif Hameed, Saif Ullah, and Rab Nawaz, 2023 20th International Bhurban Conference on Applied Sciences and Technology (IBCAST) 606 (2023) ISBN:979-8-3503-0825-9.

[12] Giuliana Siddi Moreau, Lorenzo Pisani, Manuela Profir, Carlo Podda, Lidia Leoni, and Giacomo Cao, "Quantum Artificial Intelligence Scalability in the NISQ Era: Pathways to Quantum Utility", Advanced Quantum Technologies 8 10, 2400716 (2025).

[13] Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, and Jonas Landman, "Constrained and Vanishing Expressivity of Quantum Fourier Models", Quantum 9, 1847 (2025).

[14] Seungcheol Oh, Emily Jimin Roh, Athanasios V. Vasilakos, Soohyun Park, and Joongheon Kim, "Fourier Analysis Perspective on Quantum Neural Networks", Communications Physics 9 1, 176 (2026).

[15] Kurudi V Vedavyasa and Ashok Kumar, "Classification Analysis of Transition Metal Compounds Using Quantum Machine Learning", Advanced Quantum Technologies 8 5, 2400081 (2025).

[16] Brian Coyle, Snehal Raj, Natansh Mathur, El Amine Cherrat, Nishant Jain, Skander Kazdaghli, and Iordanis Kerenidis, "Training-efficient density quantum machine learning", npj Quantum Information 11 1, 172 (2025).

[17] Philip Anton Hernicht, Alona Sakhnenko, Corey O’Meara, Giorgio Cortiana, and Jeanette Miriam Lorenz, "Enhancing the scalability of classical surrogates for real-world quantum machine learning applications", Quantum Machine Intelligence 8 2, 83 (2026).

[18] R Moretti, A Giachero, V Radescu, and M Grossi, "Enhanced feature encoding and classification on distributed quantum hardware", Machine Learning: Science and Technology 6 1, 015056 (2025).

[19] Ben Jaderberg, Antonio A. Gentile, Youssef Achari Berrada, Elvira Shishenina, and Vincent E. Elfving, "Let quantum neural networks choose their own frequencies", Physical Review A 109 4, 042421 (2024).

[20] Patrick Holzer and Ivica Turkalj, "Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks", Quantum Machine Intelligence 8 1, 33 (2026).

[21] Wei Gao, Feilong Li, Yanpeng Cai, and Xikang Hou, "Enhancing streamflow prediction in a dam-regulated river by integrating mechanism and machine learning models", Journal of Hydrology: Regional Studies 62, 102799 (2025).

[22] M. A. Obisa, E. C. Too, and B. O. Osero, Signals and Communication Technology 1 (2025) ISBN:978-3-031-91801-8.

[23] Rihab Hoceini and Ahmed Bouida, Lecture Notes in Computer Science 16830, 244 (2027) ISBN:978-3-032-32334-7.

[24] Pablo Rodriguez-Grasa, Robert Farzan-Rodriguez, Gabriele Novelli, Yue Ban, and Mikel Sanz, "Satellite image classification with neural quantum kernels", Machine Learning: Science and Technology 6 1, 015043 (2025).

[25] Aakash Ravindra Shinde, Charu Jain, and Amir Kalev, "Post-training approach for mitigating overfitting in quantum convolutional neural networks", Physical Review A 110 4, 042409 (2024).

[26] Dahiru Adamu Aliyu, Emelia Akashah Patah Akhir, Yahaya Saidu, Shamsuddeen Adamu, Kabir Ismail Umar, Abubakar Sadiq Bunu, and Hussaini Mamman, "Optimization Techniques for Asthma Exacerbation Prediction Models: A Systematic Literature Review", IEEE Access 12, 110862 (2024).

[27] Alice Barthe and Adrián Pérez-Salinas, "Gradients and frequency profiles of quantum re-uploading models", Quantum 8, 1523 (2024).

[28] Martyna Czuba, Patrick Holzer, and Hein Zay Yar Oo, "Benchmarking Encoding Families in Quantum Neural Networks Under Fixed Circuit Area for Frequency Spectrum and Trainability", (2026).

[29] Tobias Haug and M. S. Kim, "Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric", Physical Review Letters 133 5, 050603 (2024).

[30] Hiroshi Ohno, "Adaptive pruning algorithm using a quantum Fisher information matrix for parameterized quantum circuits", Quantum Machine Intelligence 6 2, 77 (2024).

[31] Massimiliano Incudini, Michele Grossi, Antonio Mandarino, Sofia Vallecorsa, Alessandra Di Pierro, and David Windridge, "The Quantum Path Kernel: A Generalized Neural Tangent Kernel for Deep Quantum Machine Learning", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[32] Minh Duc Nguyen, Tuan Hai Vu, Bin Ho Le, and Lan Nguyen Tran, "Flexible genetic algorithm for quantum support vector machines", Machine Learning: Science and Technology 7 4, 045030 (2026).

[33] Mo Kordzanganeh, Daria Kosichkina, and Alexey Melnikov, "Parallel Hybrid Networks: An Interplay between Quantum and Classical Neural Networks", Intelligent Computing 2, 0028 (2023).

[34] Lorenzo Pastori, Veronika Eyring, and Mierk Schwabe, "Fisher information, training and bias in Fourier regression models", Machine Learning: Science and Technology 7 4, 045048 (2026).

[35] Olga Tsurkan, Aleksandra Konstantinova, Arsenii Senokosov, Asel Sagingalieva, and Alexey Melnikov, "Hybrid Quantum Recurrent Neural Network for Remaining Useful Life Prediction of Turbofan Engines", Algorithms 19 8, 663 (2026).

[36] Fanyi Meng, Sihan Xia, Qi Fan, Zijin Xiao, and Kangkang Mu, Proceedings of the 2024 5th International Symposium on Artificial Intelligence for Medicine Science 364 (2024) ISBN:9798400717826.

[37] Marco Mordacci, Davide Ferrari, and Michele Amoretti, Proceedings of the 2024 Workshop on Quantum Search and Information Retrieval 9 (2024) ISBN:9798400706462.

[38] Asel Sagingalieva, Stefan Komornyik, Arsenii Senokosov, Ayush Joshi, Christopher Mansell, Olga Tsurkan, Karan Pinto, Markus Pflitsch, and Alexey Melnikov, "Photovoltaic power forecasting using quantum machine learning", Solar Energy 302, 114016 (2025).

[39] Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto, "Understanding quantum machine learning also requires rethinking generalization", Nature Communications 15 1, 2277 (2024).

[40] Mattia Bacigalupo and Pier Giuseppe Giribone, "Support Vector Machines for Credit Scoring: a performance comparison between classical Machine Learning and Quantum-enhanced approach", Risk Management Magazine 21 2, 23 (2026).

[41] Yannick Couzinié, Shunsuke Daimon, Hirofumi Nishi, Natsuki Ito, Yusuke Harazono, and Yu-ichiro Matsushita, "Towards improved quantum machine learning for molecular force fields", Physical Review A 112 6, 062442 (2025).

[42] Seongwook Shin, Yong Siah Teo, and Hyunseok Jeong, "Dequantizing quantum machine learning models using tensor networks", Physical Review Research 6 2, 023218 (2024).

[43] Marie Kempkes, Aroosa Ijaz, Elies Gil-Fuster, Carlos Bravo-Prieto, Jakob Spiegelberg, Evert van Nieuwenburg, and Vedran Dunjko, "Double Descent in Quantum Kernel Methods", PRX Quantum 7 1, 010312 (2026).

[44] Julian Berberich, Daniel Fink, Daniel Pranjić, Christian Tutschku, and Christian Holm, "Training robust and generalizable quantum models", Physical Review Research 6 4, 043326 (2024).

[45] 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).

[46] Domenico Pomarico, Roberto Cilli, Alfonso Monaco, Loredana Bellantuono, Marianna La Rocca, Tommaso Maggipinto, Giuseppe Magnifico, Marlis Ontivero Ortega, Ester Pantaleo, Sabina Tangaro, Sebastiano Stramaglia, Roberto Bellotti, and Nicola Amoroso, "Transfer Entropy and O-Information to Detect Grokking in Tensor Network Multi-Class Classification Problems", Technologies 13 10, 438 (2025).

[47] Marco Wiedmann, Maniraman Periyasamy, and Daniel D. Scherer, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1785 (2025) ISBN:979-8-3315-5736-2.

[48] Mekena Metcalf, Pablo Andrés-Martínez, and Nathan Fitzpatrick, SC24: International Conference for High Performance Computing, Networking, Storage and Analysis 1 (2024) ISBN:979-8-3503-5291-7.

[49] Linna Sun, Shengchang Liu, Liming Zhang, Keqiang He, and Xiuzheng Yan, "Prediction of the displacement in a foundation pit based on neural network model fusion error and variational modal decomposition methods", Measurement 240, 115534 (2025).

[50] Shreeya Sanjeev Gokhale, Rishit Ravi Dhote, and Radhakrishnan Delhibabu, "A review of quantum machine learning algorithms, applications, and emerging advantages", Discover Computing 29 1, 226 (2026).

[51] Yan Gu, Shuhong Dai, Cong Liu, Qing-Le Wang, and Long Cheng, Quantum Computational AI 117 (2026) ISBN:9780443302596.

[52] Julian Berberich, Tobias Fellner, and Christian Holm, Quantum Science and Technology 209 (2026) ISBN:978-3-032-11152-4.

[53] Lajos Hanzo, Zunaira Babar, Zhenyu Cai, Daryus Chandra, Ivan B. Djordjevic, Balint Koczor, Soon Xin Ng, Mohsen Razavi, and Osvaldo Simeone, "Quantum Information Processing, Sensing, and Communications: Their Myths, Realities, and Futures", IEEE Proceedings 113 9, 1024 (2025).

[54] Yuxuan Du, Yibo Yang, Dacheng Tao, and Min-Hsiu Hsieh, "Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification", Physical Review Letters 131 14, 140601 (2023).

[55] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee, "Quantum machine learning: from physics to software engineering", Advances in Physics X 8 1, 2165452 (2023).

[56] Jonas Landman, Slimane Thabet, Constantin Dalyac, Hela Mhiri, and Elham Kashefi, "Classically Approximating Variational Quantum Machine Learning with Random Fourier Features", arXiv:2210.13200, (2022).

[57] Elies Gil-Fuster, Jens Eisert, and Vedran Dunjko, "On the expressivity of embedding quantum kernels", Machine Learning: Science and Technology 5 2, 025003 (2024).

[58] M. AbuGhanem, "Photonic Quantum Computers", arXiv:2409.08229, (2024).

[59] Berta Casas and Alba Cervera-Lierta, "Multidimensional Fourier series with quantum circuits", Physical Review A 107 6, 062612 (2023).

[60] Elies Gil-Fuster, Casper Gyurik, Adrián Pérez-Salinas, and Vedran Dunjko, "On the relation between trainability and dequantization of variational quantum learning models", arXiv:2406.07072, (2024).

[61] Mo Kordzanganeh, Pavel Sekatski, Leonid Fedichkin, and Alexey Melnikov, "An exponentially-growing family of universal quantum circuits", Machine Learning: Science and Technology 4 3, 035036 (2023).

[62] S. Shin, Y. S. Teo, and H. Jeong, "Exponential data encoding for quantum supervised learning", Physical Review A 107 1, 012422 (2023).

[63] Jason Iaconis and Sonika Johri, "Tensor Network Based Efficient Quantum Data Loading of Images", arXiv:2310.05897, (2023).

[64] Shun Okumura and Masayuki Ohzeki, "Fourier Analysis of Parameterized Quantum Circuits and the Barren Plateau Problem", arXiv:2309.06740, (2023).

[65] Melvin Strobl, M. Emre Sahin, Lucas van der Horst, Eileen Kuehn, Achim Streit, and Ben Jaderberg, "Fourier Fingerprints of Ansatzes in Quantum Machine Learning", arXiv:2508.20868, (2025).

[66] Mo Kordzanganeh, Daria Kosichkina, and Alexey Melnikov, "Parallel Hybrid Networks: an interplay between quantum and classical neural networks", arXiv:2303.03227, (2023).

[67] Francesco Scala, Andrea Ceschini, Massimo Panella, and Dario Gerace, "A General Approach to Dropout in Quantum Neural Networks", arXiv:2310.04120, (2023).

[68] Lucas Slattery, Ruslan Shaydulin, Shouvanik Chakrabarti, Marco Pistoia, Sami Khairy, and Stefan M. Wild, "Numerical evidence against advantage with quantum fidelity kernels on classical data", Physical Review A 107 6, 062417 (2023).

[69] Aikaterini, Gratsea, and Patrick Huembeli, "The effect of the processing and measurement operators on the expressive power of quantum models", arXiv:2211.03101, (2022).

[70] Mazen Ali and Matthias Kabel, "Piecewise Polynomial Tensor Network Quantum Feature Encoding", arXiv:2402.07671, (2024).

[71] Stefano Mangini, "Variational quantum algorithms for machine learning: theory and applications", arXiv:2306.09984, (2023).

[72] Domenico Pomarico, Alfonso Monaco, Giuseppe Magnifico, Antonio Lacalamita, Ester Pantaleo, Loredana Bellantuono, Sabina Tangaro, Tommaso Maggipinto, Marianna La Rocca, Ernesto Picardi, Nicola Amoroso, Graziano Pesole, Sebastiano Stramaglia, and Roberto Bellotti, "Grokking as an entanglement transition in tensor network machine learning", arXiv:2503.10483, (2025).

[73] Michael Poppel, David Bucher, Maximilian Zorn, Nico Kraus, Claudia Linnhoff-Popien, Philipp Altmann, and Jonas Stein, "Mitigating Exponential Mixed Frequency Growth through Frequency Selection", arXiv:2508.10533, (2025).

[74] Patrick Holzer and Ivica Turkalj, "Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks", arXiv:2402.14515, (2024).

[75] Evan Peters, "On the coherent extension of some Fano-type learning bounds", arXiv:2404.07277, (2024).

[76] Sashwat Anagolum, Narges Alavisamani, Poulami Das, Moinuddin Qureshi, Eric Kessler, and Yunong Shi, "Élivágar: Efficient Quantum Circuit Search for Classification", arXiv:2401.09393, (2024).

[77] Shahram Dehdashti, Prayag Tiwari, Kareem H. El Safty, Peter Bruza, and Janis Notzel, "Enhancing Quantum Machine Learning: The Power of Non-Linear Optical Reproducing Kernels", arXiv:2407.13809, (2024).

[78] Hongni Jin, Gurinder Singh, and Kenneth M. Merz, "QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representations", arXiv:2504.19053, (2025).

[79] Jorja J. Kirk, Matthew D. Jackson, Daniel J. M. King, Philip Intallura, and Mekena Metcalf, "Emergent Order in Classical Data Representations on Ising Spin Models", arXiv:2303.01461, (2023).

[80] Richard A. Wolf, "Why we care (about quantum machine learning)", arXiv:2401.07547, (2024).

[81] Michael Poppel, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, and Jonas Stein, "Long Range Frequency Tuning for QML", arXiv:2602.23409, (2026).

[82] Mekena Metcalf, Pablo Andrés-Martínez, and Nathan Fitzpatrick, "Realizing Quantum Kernel Models at Scale with Matrix Product State Simulation", arXiv:2411.09336, (2024).

[83] Maja Franz, Melvin Strobl, Jonathan Hunz, Lukas Scheller, Lucas van der Horst, Eileen Kuehn, Achim Streit, and Wolfgang Mauerer, "Software Between Quantum and Machine Learning -- And Down to Pulses", arXiv:2605.21286, (2026).

[84] Melvin Strobl, Maja Franz, Lukas Scheller, Eileen Kuehn, Wolfgang Mauerer, and Achim Streit, "Beyond Gates: Pulse Level Quantum Fourier Models", arXiv:2605.04945, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 16:13:55) and SAO/NASA ADS (last updated successfully 2026-08-13 16:13:56). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.