Efficient classical computation of the neural tangent kernel of quantum neural networks
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AbstractWe propose an efficient classical algorithm to estimate the Neural Tangent Kernel (NTK) associated with a broad class of quantum neural networks. These networks consist of arbitrary unitary operators belonging to the Clifford group interleaved with parametric gates given by the time evolution generated by an arbitrary Hamiltonian belonging to the Pauli group. The proposed algorithm leverages a key insight: the average over the distribution of initialization parameters in the NTK definition can be exactly replaced by an average over just four discrete values, chosen such that the corresponding parametric gates are Clifford operations. This reduction enables an efficient classical simulation of the circuit. Combined with recent results establishing the equivalence between wide quantum neural networks and Gaussian processes [Girardi et al., Comm. Math. Phys. 406, 92 (2025); Melchor Hernandez et al., Ann. Henri Poincaré (2025)], our method enables efficient computation of the expected output of wide, trained quantum neural networks, and therefore shows that such networks cannot achieve quantum advantage.► BibTeX data@article{MelchorHernandez2026efficientclassical, doi = {10.22331/q-2026-05-29-2118}, url = {https://doi.org/10.22331/q-2026-05-29-2118}, title = {Efficient classical computation of the neural tangent kernel of quantum neural networks}, author = {Melchor Hernandez, Anderson and Pastorello, Davide and De Palma, Giacomo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2118}, month = may, year = {2026} }► References [1] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70 (5), November 2004. ISSN 1094-1622. 10.1103/physreva.70.052328. URL http://dx.doi.org/10.1103/PhysRevA.70.052328. https://doi.org/10.1103/physreva.70.052328 [2] Erfan Abedi, Salman Beigi, and Leila Taghavi.
Quantum Lazy Training. Quantum, 7: 989, April 2023. ISSN 2521-327X. 10.22331/q-2023-04-27-989. URL https://doi.org/10.22331/q-2023-04-27-989. https://doi.org/10.22331/q-2023-04-27-989 [3] Armando Angrisani, Alexander Schmidhuber, Manuel S. Rudolph, M. Cerezo, Zoë Holmes, and Hsin-Yuan Huang. Classically estimating observables of noiseless quantum circuits. arXiv preprint arXiv:2409.01706, 2024. https://doi.org/10.1103/lh6x-7rc3. https://doi.org/10.1103/lh6x-7rc3 arXiv:2409.01706 [4] Leonardo Banchi and Gavin E Crooks. Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule. Quantum, 5: 386, 2021. https://doi.org/10.22331/q-2021-01-25-386. https://doi.org/10.22331/q-2021-01-25-386 [5] E. W. Barankin. Locally best unbiased estimates. The Annals of Mathematical Statistics, 20 (4): 477–501, 1949. https://doi.org/10.1214/aoms/1177729943. https://doi.org/10.1214/aoms/1177729943 [6] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549 (7671): 195–202, 2017. https://doi.org/10.1038/nature23474. https://doi.org/10.1038/nature23474 [7] M. Cerezo, Martin Larocca, Diego García-Martín, N. L. Diaz, Paolo Braccia, Enrico Fontana, Manuel S. Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, Eric R. Anschuetz, and Zoë Holmes. Does provable absence of barren plateaus imply classical simulability? Nature Communications, 16 (1), August 2025. ISSN 2041-1723. 10.1038/s41467-025-63099-6. URL http://dx.doi.org/10.1038/s41467-025-63099-6. https://doi.org/10.1038/s41467-025-63099-6 [8] Lucas Pinheiro Cinelli, Matheus Araújo Marins, Eduardo Antonio Barros Da Silva, and Sérgio Lima Netto. Variational methods for machine learning with applications to deep networks, volume 15. Springer, 2021. https://doi.org/10.1007/978-3-030-70679-1. https://doi.org/10.1007/978-3-030-70679-1 [9] Franklin De Lima Marquezino, Renato Portugal, and Carlile Lavor. A primer on quantum computing. Springer, 2019. https://doi.org/10.1007/978-3-030-19066-8. https://doi.org/10.1007/978-3-030-19066-8 [10] Jeroen Dehaene and Bart De Moor. Clifford group, stabilizer states, and linear and quadratic operations over gf(2). Physical Review A, 68 (4), October 2003. ISSN 1094-1622. 10.1103/physreva.68.042318. URL http://dx.doi.org/10.1103/PhysRevA.68.042318. https://doi.org/10.1103/physreva.68.042318 [11] Filippo Girardi and Giacomo De Palma. Trained quantum neural networks are gaussian processes. Communications in Mathematical Physics, 406 (4), April 2025. ISSN 1432-0916. 10.1007/s00220-025-05238-0. URL http://dx.doi.org/10.1007/s00220-025-05238-0. https://doi.org/10.1007/s00220-025-05238-0 [12] Paul R Halmos. The theory of unbiased estimation. The Annals of Mathematical Statistics, 17 (1): 34–43, 1946. https://doi.org/10.1214/aoms/1177731020. https://doi.org/10.1214/aoms/1177731020 [13] 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 (7747): 209–212, 2019. https://doi.org/10.1038/s41586-019-0980-2. https://doi.org/10.1038/s41586-019-0980-2 [14] Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. https://doi.org/10.48550/arXiv.1806.07572. URL https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf. https://doi.org/10.48550/arXiv.1806.07572 https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf [15] Junyu Liu, Francesco Tacchino, Jennifer R. Glick, Liang Jiang, and Antonio Mezzacapo. Representation learning via quantum neural tangent kernels. PRX Quantum, 3: 030323, Aug 2022. 10.1103/PRXQuantum.3.030323. URL https://doi.org/10.1103/PRXQuantum.3.030323. https://doi.org/10.1103/PRXQuantum.3.030323 [16] Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme. A rigorous and robust quantum speed-up in supervised machine learning. Nature Physics, 17 (9): 1013–1017, 2021. https://doi.org/10.1038/s41567-021-01287-z. https://doi.org/10.1038/s41567-021-01287-z [17] Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran. Quantum embeddings for machine learning. arXiv preprint arXiv:2001.03622, 2020. https://doi.org/10.48550/arXiv.2001.03622. https://doi.org/10.48550/arXiv.2001.03622 arXiv:2001.03622 [18] Victor Martinez, Armando Angrisani, Ekaterina Pankovets, Omar Fawzi, and Daniel Stilck França. Efficient simulation of parametrized quantum circuits under nonunital noise through pauli backpropagation. Phys. Rev. Lett., 134: 250602, Jun 2025. 10.1103/j1gg-s6zb. URL https://doi.org/10.1103/j1gg-s6zb. https://doi.org/10.1103/j1gg-s6zb [19] Kieran Mastel. The clifford theory of the $ n $-qubit clifford group. Journal of Mathematical Physics, 2026. https://doi.org/10.1063/5.0311547. https://doi.org/10.1063/5.0311547 [20] Anderson Melchor Hernandez, Filippo Girardi, Davide Pastorello, and Giacomo De Palma. Quantitative convergence of trained quantum neural networks to a gaussian process: A. melchor hernandez et al.
Annales Henri Poincaré, pages 1–57, 2025. https://doi.org/10.1007/s00023-025-01631-6. https://doi.org/10.1007/s00023-025-01631-6 [21] Davide Pastorello. Concise guide to quantum machine learning. Springer, 2023. https://doi.org/10.1007/978-981-19-6897-6. https://doi.org/10.1007/978-981-19-6897-6 [22] Oliver Reardon-Smith, Michał Oszmaniec, and Kamil Korzekwa. Improved simulation of quantum circuits dominated by free fermionic operations. Quantum, 8: 1549, December 2024. ISSN 2521-327X. 10.22331/q-2024-12-04-1549. URL https://doi.org/10.22331/q-2024-12-04-1549. https://doi.org/10.22331/q-2024-12-04-1549 [23] Francesco Scala, Christa Zoufal, Dario Gerace, and Francesco Tacchino. Towards practical quantum neural network diagnostics with neural tangent kernels. arXiv preprint arXiv:2503.01966, 2025. https://doi.org/10.48550/arXiv.2503.01966. https://doi.org/10.48550/arXiv.2503.01966 arXiv:2503.01966 [24] Maria Schuld and Francesco Petruccione. Supervised learning with quantum computers, volume 17. Springer, 2018. https://doi.org/10.1007/978-3-319-96424-9. https://doi.org/10.1007/978-3-319-96424-9 [25] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. An introduction to quantum machine learning. Contemporary Physics, 56 (2): 172–185, 2015. https://doi.org/10.1080/00107514.2014.964942. https://doi.org/10.1080/00107514.2014.964942 [26] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models. Physical Review A, 103 (3): 032430, 2021. https://doi.org/10.1103/PhysRevA.103.032430. https://doi.org/10.1103/PhysRevA.103.032430 [27] Norihito Shirai, Kenji Kubo, Kosuke Mitarai, and Keisuke Fujii. Quantum tangent kernel. Phys. Rev. Res., 6 (3): 033179, 2024. 10.1103/PhysRevResearch.6.033179. https://doi.org/10.1103/PhysRevResearch.6.033179 [28] Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13 (4): 354–356, 1969. https://doi.org/10.1007/BF02165411. https://doi.org/10.1007/BF02165411 [29] Michel Talagrand. The missing factor in hoeffding's inequalities. Annales de l'IHP Probabilités et statistiques, 31 (4): 689–702, 1995. URL https://www.numdam.org/item/AIHPB_1995__31_4_689_0/. https://www.numdam.org/item/AIHPB_1995__31_4_689_0/ [30] Joel A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 12 (4): 389–434, August 2011. ISSN 1615-3383. 10.1007/s10208-011-9099-z. URL http://dx.doi.org/10.1007/s10208-011-9099-z. https://doi.org/10.1007/s10208-011-9099-z [31] Joel A. Tropp. An introduction to matrix concentration inequalities. 2015. https://doi.org/10.48550/arXiv.1501.01571. URL https://arxiv.org/abs/1501.01571. https://doi.org/10.48550/arXiv.1501.01571 arXiv:1501.01571 [32] 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, oct 2024. 10.1088/1361-6633/ad82cf. URL https://dx.doi.org/10.1088/1361-6633/ad82cf. https://doi.org/10.1088/1361-6633/ad82cf [33] Yifan Zhang and Yuxuan Zhang. Classical simulability of quantum circuits with shallow magic depth. PRX Quantum, 6: 010337, Feb 2025. 10.1103/PRXQuantum.6.010337. URL https://doi.org/10.1103/PRXQuantum.6.010337. https://doi.org/10.1103/PRXQuantum.6.010337Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-29 07:32:56: Could not fetch cited-by data for 10.22331/q-2026-05-29-2118 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-29 07:32:56: Cannot retrieve data from ADS due to rate limitations.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. AbstractWe propose an efficient classical algorithm to estimate the Neural Tangent Kernel (NTK) associated with a broad class of quantum neural networks. These networks consist of arbitrary unitary operators belonging to the Clifford group interleaved with parametric gates given by the time evolution generated by an arbitrary Hamiltonian belonging to the Pauli group. The proposed algorithm leverages a key insight: the average over the distribution of initialization parameters in the NTK definition can be exactly replaced by an average over just four discrete values, chosen such that the corresponding parametric gates are Clifford operations. This reduction enables an efficient classical simulation of the circuit. Combined with recent results establishing the equivalence between wide quantum neural networks and Gaussian processes [Girardi et al., Comm. Math. Phys. 406, 92 (2025); Melchor Hernandez et al., Ann. Henri Poincaré (2025)], our method enables efficient computation of the expected output of wide, trained quantum neural networks, and therefore shows that such networks cannot achieve quantum advantage.► BibTeX data@article{MelchorHernandez2026efficientclassical, doi = {10.22331/q-2026-05-29-2118}, url = {https://doi.org/10.22331/q-2026-05-29-2118}, title = {Efficient classical computation of the neural tangent kernel of quantum neural networks}, author = {Melchor Hernandez, Anderson and Pastorello, Davide and De Palma, Giacomo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2118}, month = may, year = {2026} }► References [1] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70 (5), November 2004. ISSN 1094-1622. 10.1103/physreva.70.052328. URL http://dx.doi.org/10.1103/PhysRevA.70.052328. https://doi.org/10.1103/physreva.70.052328 [2] Erfan Abedi, Salman Beigi, and Leila Taghavi.
Quantum Lazy Training. Quantum, 7: 989, April 2023. ISSN 2521-327X. 10.22331/q-2023-04-27-989. URL https://doi.org/10.22331/q-2023-04-27-989. https://doi.org/10.22331/q-2023-04-27-989 [3] Armando Angrisani, Alexander Schmidhuber, Manuel S. Rudolph, M. Cerezo, Zoë Holmes, and Hsin-Yuan Huang. Classically estimating observables of noiseless quantum circuits. arXiv preprint arXiv:2409.01706, 2024. https://doi.org/10.1103/lh6x-7rc3. https://doi.org/10.1103/lh6x-7rc3 arXiv:2409.01706 [4] Leonardo Banchi and Gavin E Crooks. Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule. Quantum, 5: 386, 2021. https://doi.org/10.22331/q-2021-01-25-386. https://doi.org/10.22331/q-2021-01-25-386 [5] E. W. Barankin. Locally best unbiased estimates. The Annals of Mathematical Statistics, 20 (4): 477–501, 1949. https://doi.org/10.1214/aoms/1177729943. https://doi.org/10.1214/aoms/1177729943 [6] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549 (7671): 195–202, 2017. https://doi.org/10.1038/nature23474. https://doi.org/10.1038/nature23474 [7] M. Cerezo, Martin Larocca, Diego García-Martín, N. L. Diaz, Paolo Braccia, Enrico Fontana, Manuel S. Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, Eric R. Anschuetz, and Zoë Holmes. Does provable absence of barren plateaus imply classical simulability? Nature Communications, 16 (1), August 2025. ISSN 2041-1723. 10.1038/s41467-025-63099-6. URL http://dx.doi.org/10.1038/s41467-025-63099-6. https://doi.org/10.1038/s41467-025-63099-6 [8] Lucas Pinheiro Cinelli, Matheus Araújo Marins, Eduardo Antonio Barros Da Silva, and Sérgio Lima Netto. Variational methods for machine learning with applications to deep networks, volume 15. Springer, 2021. https://doi.org/10.1007/978-3-030-70679-1. https://doi.org/10.1007/978-3-030-70679-1 [9] Franklin De Lima Marquezino, Renato Portugal, and Carlile Lavor. A primer on quantum computing. Springer, 2019. https://doi.org/10.1007/978-3-030-19066-8. https://doi.org/10.1007/978-3-030-19066-8 [10] Jeroen Dehaene and Bart De Moor. Clifford group, stabilizer states, and linear and quadratic operations over gf(2). Physical Review A, 68 (4), October 2003. ISSN 1094-1622. 10.1103/physreva.68.042318. URL http://dx.doi.org/10.1103/PhysRevA.68.042318. https://doi.org/10.1103/physreva.68.042318 [11] Filippo Girardi and Giacomo De Palma. Trained quantum neural networks are gaussian processes. Communications in Mathematical Physics, 406 (4), April 2025. ISSN 1432-0916. 10.1007/s00220-025-05238-0. URL http://dx.doi.org/10.1007/s00220-025-05238-0. https://doi.org/10.1007/s00220-025-05238-0 [12] Paul R Halmos. The theory of unbiased estimation. The Annals of Mathematical Statistics, 17 (1): 34–43, 1946. https://doi.org/10.1214/aoms/1177731020. https://doi.org/10.1214/aoms/1177731020 [13] 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 (7747): 209–212, 2019. https://doi.org/10.1038/s41586-019-0980-2. https://doi.org/10.1038/s41586-019-0980-2 [14] Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. https://doi.org/10.48550/arXiv.1806.07572. URL https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf. https://doi.org/10.48550/arXiv.1806.07572 https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf [15] Junyu Liu, Francesco Tacchino, Jennifer R. Glick, Liang Jiang, and Antonio Mezzacapo. Representation learning via quantum neural tangent kernels. PRX Quantum, 3: 030323, Aug 2022. 10.1103/PRXQuantum.3.030323. URL https://doi.org/10.1103/PRXQuantum.3.030323. https://doi.org/10.1103/PRXQuantum.3.030323 [16] Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme. A rigorous and robust quantum speed-up in supervised machine learning. Nature Physics, 17 (9): 1013–1017, 2021. https://doi.org/10.1038/s41567-021-01287-z. https://doi.org/10.1038/s41567-021-01287-z [17] Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran. Quantum embeddings for machine learning. arXiv preprint arXiv:2001.03622, 2020. https://doi.org/10.48550/arXiv.2001.03622. https://doi.org/10.48550/arXiv.2001.03622 arXiv:2001.03622 [18] Victor Martinez, Armando Angrisani, Ekaterina Pankovets, Omar Fawzi, and Daniel Stilck França. Efficient simulation of parametrized quantum circuits under nonunital noise through pauli backpropagation. Phys. Rev. Lett., 134: 250602, Jun 2025. 10.1103/j1gg-s6zb. URL https://doi.org/10.1103/j1gg-s6zb. https://doi.org/10.1103/j1gg-s6zb [19] Kieran Mastel. The clifford theory of the $ n $-qubit clifford group. Journal of Mathematical Physics, 2026. https://doi.org/10.1063/5.0311547. https://doi.org/10.1063/5.0311547 [20] Anderson Melchor Hernandez, Filippo Girardi, Davide Pastorello, and Giacomo De Palma. Quantitative convergence of trained quantum neural networks to a gaussian process: A. melchor hernandez et al.
Annales Henri Poincaré, pages 1–57, 2025. https://doi.org/10.1007/s00023-025-01631-6. https://doi.org/10.1007/s00023-025-01631-6 [21] Davide Pastorello. Concise guide to quantum machine learning. Springer, 2023. https://doi.org/10.1007/978-981-19-6897-6. https://doi.org/10.1007/978-981-19-6897-6 [22] Oliver Reardon-Smith, Michał Oszmaniec, and Kamil Korzekwa. Improved simulation of quantum circuits dominated by free fermionic operations. Quantum, 8: 1549, December 2024. ISSN 2521-327X. 10.22331/q-2024-12-04-1549. URL https://doi.org/10.22331/q-2024-12-04-1549. https://doi.org/10.22331/q-2024-12-04-1549 [23] Francesco Scala, Christa Zoufal, Dario Gerace, and Francesco Tacchino. Towards practical quantum neural network diagnostics with neural tangent kernels. arXiv preprint arXiv:2503.01966, 2025. https://doi.org/10.48550/arXiv.2503.01966. https://doi.org/10.48550/arXiv.2503.01966 arXiv:2503.01966 [24] Maria Schuld and Francesco Petruccione. Supervised learning with quantum computers, volume 17. Springer, 2018. https://doi.org/10.1007/978-3-319-96424-9. https://doi.org/10.1007/978-3-319-96424-9 [25] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. An introduction to quantum machine learning. Contemporary Physics, 56 (2): 172–185, 2015. https://doi.org/10.1080/00107514.2014.964942. https://doi.org/10.1080/00107514.2014.964942 [26] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models. Physical Review A, 103 (3): 032430, 2021. https://doi.org/10.1103/PhysRevA.103.032430. https://doi.org/10.1103/PhysRevA.103.032430 [27] Norihito Shirai, Kenji Kubo, Kosuke Mitarai, and Keisuke Fujii. Quantum tangent kernel. Phys. Rev. Res., 6 (3): 033179, 2024. 10.1103/PhysRevResearch.6.033179. https://doi.org/10.1103/PhysRevResearch.6.033179 [28] Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13 (4): 354–356, 1969. https://doi.org/10.1007/BF02165411. https://doi.org/10.1007/BF02165411 [29] Michel Talagrand. The missing factor in hoeffding's inequalities. Annales de l'IHP Probabilités et statistiques, 31 (4): 689–702, 1995. URL https://www.numdam.org/item/AIHPB_1995__31_4_689_0/. https://www.numdam.org/item/AIHPB_1995__31_4_689_0/ [30] Joel A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 12 (4): 389–434, August 2011. ISSN 1615-3383. 10.1007/s10208-011-9099-z. URL http://dx.doi.org/10.1007/s10208-011-9099-z. https://doi.org/10.1007/s10208-011-9099-z [31] Joel A. Tropp. An introduction to matrix concentration inequalities. 2015. https://doi.org/10.48550/arXiv.1501.01571. URL https://arxiv.org/abs/1501.01571. https://doi.org/10.48550/arXiv.1501.01571 arXiv:1501.01571 [32] 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, oct 2024. 10.1088/1361-6633/ad82cf. URL https://dx.doi.org/10.1088/1361-6633/ad82cf. https://doi.org/10.1088/1361-6633/ad82cf [33] Yifan Zhang and Yuxuan Zhang. Classical simulability of quantum circuits with shallow magic depth. PRX Quantum, 6: 010337, Feb 2025. 10.1103/PRXQuantum.6.010337. URL https://doi.org/10.1103/PRXQuantum.6.010337. https://doi.org/10.1103/PRXQuantum.6.010337Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-29 07:32:56: Could not fetch cited-by data for 10.22331/q-2026-05-29-2118 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-29 07:32:56: Cannot retrieve data from ADS due to rate limitations.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.