Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias C. Caro1,2, Elies Gil-Fuster3,4, Johannes Jakob Meyer3, Jens Eisert3,4,5, and Ryan Sweke3

1Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
2Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
4Fraunhofer Heinrich Hertz Institute, 10587 Berlin, Germany
5Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany

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Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:
35 pages, 3 figures; corrected a mistake in Eq. (38) of the previous version, results remain unchanged except for a restriction to frequency vectors with integer entries; added a conjecture to recover results in full


A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

In a typical supervised machine learning problem, one is given a fixed-size dataset consisting of labelled objects. For example, the dataset might contain images labelled as dogs or cats. The goal is then to use the dataset to learn some classification algorithm, which when given appropriate new objects – ones not already present in the dataset but of the same type – can label them correctly. Ideally, once one has obtained a classification algorithm, one would like to be able to have some sort of guarantee on how well this classification algorithm will perform – i.e., on how accurate it will be on previously unseen objects. Such guarantees on the “out-of-sample" performance of classification algorithms – i.e., on their accuracy on previously unseen data – are known as generalization bounds. Over time though it has been realized that various notions of complexity are critical for determining how well a class of classification algorithms will generalize. Intuitively, the more complex a classification algorithm can be, the more likely it is to "overfit" the available data, and therefore generalize badly. Indeed, we now know that one can derive a variety of generalization bounds for classification algorithms, which allow us to place guarantees on the "out-of-sample" performance of classification algorithms, in terms of their performance on the available data, and the complexity of the class of algorithms that they belong to.

Given the emerging availability of "noisy intermediate scale quantum" devices, a variety of approaches have been proposed for using such devices in supervised learning. One particularly important way in which these approaches differ, is in the way the classical data is entered into the quantum classification algorithm. In light of this, and given the importance of generalization bounds, it is natural to ask whether one can determine how the appropriate notions of complexity for generalization depend on the different strategies for encoding classical data into the quantum classifier.

In this work, we answer this question by quantifying rigorously how various notions of complexity depend on the strategy for encoding data into a quantum classifier. These results immediately allow us to state generalization bounds for such quantum classification algorithms. With these, we can give concrete recommendations for choosing classical-to-quantum encoding strategies.

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► References

[1] Vedran Dunjkoand Hans J Briegel ``Machine learning & artificial intelligence in the quantum domain: A review of recent progress'' Rep. Prog. Phys. 81, 074001 (2018).

[2] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, ``Quantum machine learning'' Nature 549, 195–202 (2017).

[3] S. Arunachalamand R. de Wolf ``Guest Column: A Survey of Quantum Learning Theory'' SIGACT News 48 (2017).

[4] N. Wiebe, A. Kapoor, and K. Svore, ``Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning'' Quant. Inf. Comp. 15, 0318 (2015).

[5] S. Lloyd, M. Mohseni, and P. Rebentrost, ``Quantum algorithms for supervised and unsupervised machine learning'' arXiv:1307.0411 (2013).

[6] R. Sweke, J.-P. Seifert, D. Hangleiter, and J. Eisert, ``On the quantum versus classical learnability of discrete distributions'' Quantum 5, 417 (2021).

[7] Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme, ``A rigorous and robust quantum speed-up in supervised machine learning'' Nature Physics 17, 1013–1017 (2021) Bandiera_abtest: a Cg_type: Nature Research Journals Number: 9 Primary_atype: Research Publisher: Nature Publishing Group Subject_term: Computational science;Information theory and computation;Quantum information Subject_term_id: computational-science;information-theory-and-computation;quantum-information.

[8] F. Arute ``Quantum supremacy using a programmable superconducting processor'' Nature 574, 505–510 (2019).

[9] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, ``Noisy intermediate-scale quantum (NISQ) algorithms'' arXiv:2101.08448 (2021).

[10] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, ``The theory of variational hybrid quantum-classical algorithms'' New J. Phys. 18, 023023 (2016).

[11] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, ``Variational quantum algorithms'' Nature Reviews Physics 3, 625–644 (2021) Bandiera_abtest: a Cg_type: Nature Research Journals Number: 9 Primary_atype: Reviews Publisher: Nature Publishing Group Subject_term: Computer science;Quantum information;Quantum simulation Subject_term_id: computer-science;quantum-information;quantum-simulation.

[12] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, ``Parameterized quantum circuits as machine learning models'' Quant. Sc. Tech. 4, 043001 (2019).

[13] S. Sim, P. D. Johnson, and A. Aspuru-Guzik, ``Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms'' Adv. Quant. Tech. 2, 1900070 (2019).

[14] Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, and Koen Bertels, ``Evaluation of parameterized quantum circuits: on the relation between classification accuracy, expressibility, and entangling capability'' Quant. Mach. Int. 3, 1–19 (2021).

[15] F. J. Gil Vidaland D. O. Theis ``Input Redundancy for Parameterized Quantum Circuits'' Front. Phys. 8 (2020) Publisher: Frontier.

[16] M. Schuld ``Quantum machine learning models are kernel methods'' arXiv:2101.11020 (2021).

[17] A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, ``Data re-uploading for a universal quantum classifier'' Quantum 4, 226 (2020).

[18] C. Bishop ``Pattern recognition and machine learning'' Springer (2006).

[19] B. Schölkopfand A. J. Smola ``Learning with kernels: support vector machines, regularization, optimization, and beyond'' MIT Press (2002).

[20] M. Mohri, A. Rostamizadeh, and A. Talwalkar, ``Foundations of machine learning'' MIT Press (2018).

[21] Olivier Bousquetand André Elisseeff ``Stability and generalization'' J. Mach. Learn. Res. 2, 499–526 (2002).

[22] Nick Littlestoneand Manfred Warmuth ``Relating data compression and learnability'' Technical report, University of California Santa Cruz (1986).

[23] David A McAllester ``Some pac-bayesian theorems'' Machine Learning 37, 355–363 (1999).

[24] M. C. Caroand I. Datta ``Pseudo-dimension of quantum circuits'' Quant. Mach. Int. 2, 172 (2020).

[25] Amira Abbas, David Sutter, Christa Zoufal, Aurélien Lucchi, Alessio Figalli, and Stefan Woerner, ``The power of quantum neural networks'' Nature Computational Science 1, 403–409 (2021).

[26] K. Bu, D. E. Koh, L. Li, Q. Luo, and Y. Zhang, ``On the statistical complexity of quantum circuits'' arXiv:2101.06154 (2021).

[27] K. Bu, D. E. Koh, L. Li, Q. Luo, and Y. Zhang, ``Effects of quantum resources on the statistical complexity of quantum circuits'' arXiv:2102.03282 (2021).

[28] K. Bu, D. E. Koh, L. L., Q. Luo, and Y. Zhang, ``Rademacher complexity of noisy quantum circuits'' arXiv:2103.03139 (2021).

[29] Y. Du, Z. Tu, X. Yuan, and D. Tao, ``An efficient measure for the expressivity of variational quantum algorithms'' arXiv:2104.09961 (2021).

[30] H.-Y. Huang, M. Broughton, M. Mohseni, R. Babbush, S. Boixo, H. Neven, and J. R. McClean, ``Power of data in quantum machine learning'' Nature Comm. 12, 1–9 (2021).

[31] L. Banchi, J. Pereira, and S. Pirandola, ``Generalization in quantum machine learning: A quantum information perspective'' arXiv:2102.08991 (2021).

[32] C. Gyurik, D. van Vreumingen, and V. Dunjko, ``Structural risk minimization for quantum linear classifiers'' arXiv:2105.05566 (2021).

[33] M. Schuld, R. Sweke, and J. J. Meyer, ``Effect of data encoding on the expressive power of variational quantum-machine-learning models'' Phys. Rev. A 103, 032430 (2021).

[34] S. Shalev-Shwartzand S. Ben-David ``Understanding machine learning: From theory to algorithms'' Cambridge University Press (2014).

[35] M. M Wolf ``Mathematical foundations of machine learning'' (2020).

[36] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals, ``Understanding deep learning requires rethinking generalization'' arXiv:1611.03530 (2016).

[37] Y. Jiang, B. Neyshabur, H. Mobahi, D. Krishnan, and S. Bengio, ``Fantastic generalization measures and where to find them'' arXiv:1912.02178 (2019).

[38] V. N. Vapnikand A. Ya. Chervonenkis ``On the uniform convergence of relative frequencies of events to their probabilities'' Th. Prob. App. 16, 264–280 (1971).

[39] P. L. Bartlettand S. Mendelson ``Rademacher and Gaussian complexities: Risk bounds and structural results'' J. Mach. Learn. Res. 3, 463–482 (2002).

[40] D. Pollard ``Convergence of stochastic processes'' Springer (1984).

[41] H. Abraham ``Qiskit: An Open-source Framework for Quantum Computing'' (2019) 10.5281/​zenodo.2562110.

[42] Cirq Developers ``Cirq'' (2021) https:/​/​zenodo.org/​record/​4586899#.YJqDAi5fiUk.

[43] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, M. S. Alam, S. Ahmed, J. M. Arrazola, C. Blank, A. Delgado, S. Jahangiri, K. McKiernan, J. J. Meyer, Z. Niu, A. Száva, and N. Killoran, ``PennyLane: Automatic differentiation of hybrid quantum-classical computations'' arXiv:1811.04968 (2020).

[44] Claudiu Marius Popescu ``Learning bounds for quantum circuits in the agnostic setting'' Quantum Information Processing 20, 1–24 (2021).

[45] Chih-Chieh Chen, Masaya Watabe, Kodai Shiba, Masaru Sogabe, Katsuyoshi Sakamoto, and Tomah Sogabe, ``On the Expressibility and Overfitting of Quantum Circuit Learning'' ACM Transactions on Quantum Computing 2, 1–24 (2021).

[46] M. Ledouxand M. Talagrand ``Probability in Banach spaces: Isoperimetry and processes'' Springer-Verlag (1991).

[47] R. Vershynin ``High-dimensional probability: An introduction with applications in data science'' Cambridge University Press (2018).

[48] P. Massart ``Some applications of concentration inequalities to statistics'' Annales de la Faculté des sciences de Toulouse : Mathématiques Ser. 6, 9, 245–303 (2000).

[49] R. M. Dudley ``Uniform central limit theorems'' Cambridge University Press (1999).

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[1] Marco Fanizza, Yihui Quek, and Matteo Rosati, "Learning Quantum Processes Without Input Control", PRX Quantum 5 2, 020367 (2024).

[2] Charles Moussa, Jan N. van Rijn, Thomas Bäck, and Vedran Dunjko, Lecture Notes in Computer Science 13601, 32 (2022) ISBN:978-3-031-18839-8.

[3] Masahiro Kobayashi, Kouhei Nakaji, and Naoki Yamamoto, "Overfitting in quantum machine learning and entangling dropout", Quantum Machine Intelligence 4 2, 30 (2022).

[4] Charles Moussa, Yash J. Patel, Vedran Dunjko, Thomas Bäck, and Jan N. van Rijn, "Hyperparameter importance and optimization of quantum neural networks across small datasets", Machine Learning 113 4, 1941 (2024).

[5] Sofiene Jerbi, Casper Gyurik, Simon C. Marshall, Riccardo Molteni, and Vedran Dunjko, "Shadows of quantum machine learning", Nature Communications 15 1, 5676 (2024).

[6] Chih-Chieh Chen, Masaru Sogabe, Kodai Shiba, Katsuyoshi Sakamoto, and Tomah Sogabe, "General Vapnik–Chervonenkis dimension bounds for quantum circuit learning", Journal of Physics: Complexity 3 4, 045007 (2022).

[7] Matthias C. Caro, "Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix", ACM Transactions on Quantum Computing 5 2, 1 (2024).

[8] Francesco Scala, Andrea Ceschini, Massimo Panella, and Dario Gerace, "A General Approach to Dropout in Quantum Neural Networks", Advanced Quantum Technologies 2300220 (2023).

[9] Franz J. Schreiber, Jens Eisert, and Johannes Jakob Meyer, "Classical Surrogates for Quantum Learning Models", Physical Review Letters 131 10, 100803 (2023).

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

[11] André J. Ferreira-Martins, Leandro Silva, Alberto Palhares, Rodrigo Pereira, Diogo O. Soares-Pinto, Rafael Chaves, and Askery Canabarro, "Detecting quantum phase transitions in a frustrated spin chain via transfer learning of a quantum classifier algorithm", Physical Review A 109 5, 052623 (2024).

[12] Sharu Theresa Jose and Osvaldo Simeone, 2023 IEEE Information Theory Workshop (ITW) 532 (2023) ISBN:979-8-3503-0149-6.

[13] Fan Chen, Lei Jiang, Hausi Müller, Philip Richerme, Cheng Chu, Zhenxiao Fu, and Min Yang, "NISQ Quantum Computing: A Security-Centric Tutorial and Survey [Feature]", IEEE Circuits and Systems Magazine 24 1, 14 (2024).

[14] Leonardo Banchi, Jason Luke Pereira, Sharu Theresa Jose, and Osvaldo Simeone, "Statistical Complexity of Quantum Learning", Advanced Quantum Technologies 2300311 (2024).

[15] Gennaro De Luca and Yinong Chen, 2022 6th Asian Conference on Artificial Intelligence Technology (ACAIT) 1 (2022) ISBN:978-1-6654-5311-0.

[16] Simon C. Marshall, Casper Gyurik, and Vedran Dunjko, "High Dimensional Quantum Machine Learning With Small Quantum Computers", Quantum 7, 1078 (2023).

[17] 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).

[18] 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).

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

[20] Yang Qian, Xinbiao Wang, Yuxuan Du, Xingyao Wu, and Dacheng Tao, "The Dilemma of Quantum Neural Networks", IEEE Transactions on Neural Networks and Learning Systems 35 4, 5603 (2024).

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

[22] 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", Nature Communications 13 1, 4919 (2022).

[23] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", IEEE Transactions on Quantum Engineering 4, 1 (2023).

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

[25] Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, "Fock state-enhanced expressivity of quantum machine learning models", EPJ Quantum Technology 9 1, 16 (2022).

[26] Wenhui Ren, Weikang Li, Shibo Xu, Ke Wang, Wenjie Jiang, Feitong Jin, Xuhao Zhu, Jiachen Chen, Zixuan Song, Pengfei Zhang, Hang Dong, Xu Zhang, Jinfeng Deng, Yu Gao, Chuanyu Zhang, Yaozu Wu, Bing Zhang, Qiujiang Guo, Hekang Li, Zhen Wang, Jacob Biamonte, Chao Song, Dong-Ling Deng, and H. Wang, "Experimental quantum adversarial learning with programmable superconducting qubits", Nature Computational Science 2 11, 711 (2022).

[27] Weikang Li, Zhi-de Lu, and Dong-Ling Deng, "Quantum Neural Network Classifiers: A Tutorial", SciPost Physics Lecture Notes 61 (2022).

[28] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time‐Series", Advanced Quantum Technologies 6 10, 2300065 (2023).

[29] Maxwell T. West, Shu-Lok Tsang, Jia S. Low, Charles D. Hill, Christopher Leckie, Lloyd C. L. Hollenberg, Sarah M. Erfani, and Muhammad Usman, "Towards quantum enhanced adversarial robustness in machine learning", Nature Machine Intelligence 5 6, 581 (2023).

[30] Maniraman Periyasamy, Nico Meyer, Christian Ufrecht, Daniel D. Scherer, Axel Plinge, and Christopher Mutschler, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 31 (2022) ISBN:978-1-6654-9113-6.

[31] Chih-Hung Wu and Ching-Che Yen, "The expressivity of classical and quantum neural networks on entanglement entropy", The European Physical Journal C 84 2, 192 (2024).

[32] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[33] Joe Gibbs, Zoë Holmes, Matthias C. Caro, Nicholas Ezzell, Hsin-Yuan Huang, Lukasz Cincio, Andrew T. Sornborger, and Patrick J. Coles, "Dynamical simulation via quantum machine learning with provable generalization", Physical Review Research 6 1, 013241 (2024).

[34] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration in quantum kernel methods", Nature Communications 15 1, 5200 (2024).

[35] Matthias C. Caro, Hsin-Yuan Huang, Nicholas Ezzell, Joe Gibbs, Andrew T. Sornborger, Lukasz Cincio, Patrick J. Coles, and Zoë Holmes, "Out-of-distribution generalization for learning quantum dynamics", Nature Communications 14 1, 3751 (2023).

[36] Evan Peters and Maria Schuld, "Generalization despite overfitting in quantum machine learning models", Quantum 7, 1210 (2023).

[37] Jingwei Wen, Zhiguo Huang, Dunbo Cai, and Ling Qian, "Enhancing the expressivity of quantum neural networks with residual connections", Communications Physics 7 1, 220 (2024).

[38] Yuxuan Du, Yang Qian, Xingyao Wu, and Dacheng Tao, "A Distributed Learning Scheme for Variational Quantum Algorithms", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[39] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond, "Quantum machine learning on near-term quantum devices: Current state of supervised and unsupervised techniques for real-world applications", Physical Review Applied 21 6, 067001 (2024).

[40] Haoyuan Cai, Qi Ye, and Dong-Ling Deng, "Sample complexity of learning parametric quantum circuits", Quantum Science and Technology 7 2, 025014 (2022).

[41] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[42] Anurag Anshu and Srinivasan Arunachalam, "A survey on the complexity of learning quantum states", Nature Reviews Physics 6 1, 59 (2023).

[43] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[44] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics, and Astronomy 65 2, 220301 (2022).

[45] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, and Dacheng Tao, "Learnability of Quantum Neural Networks", PRX Quantum 2 4, 040337 (2021).

[46] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[47] Weikang Li, Zhide Lu, and Dong-Ling Deng, "Quantum Neural Network Classifiers: A Tutorial", arXiv:2206.02806, (2022).

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

[49] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljačić, "QuACK: Accelerating Gradient-Based Quantum Optimization with Koopman Operator Learning", arXiv:2211.01365, (2022).

[50] Jingwei Wen, Zhiguo Huang, Dunbo Cai, and Ling Qian, "Enhancing the expressivity of quantum neural networks with residual connections", arXiv:2401.15871, (2024).

[51] Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, and Jonas Landman, "Constrained and Vanishing Expressivity of Quantum Fourier Models", arXiv:2403.09417, (2024).

[52] Yuxuan Du, Yang Qian, and Dacheng Tao, "Accelerating variational quantum algorithms with multiple quantum processors", arXiv:2106.12819, (2021).

[53] Alberto Manzano, David Dechant, Jordi Tura, and Vedran Dunjko, "Parametrized Quantum Circuits and their approximation capacities in the context of quantum machine learning", arXiv:2307.14792, (2023).

[54] Julian Berberich, Daniel Fink, Daniel Pranjić, Christian Tutschku, and Christian Holm, "Training robust and generalizable quantum models", arXiv:2311.11871, (2023).

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