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  • Open Access

Neural quantum kernels: Training quantum kernels with quantum neural networks

Pablo Rodriguez-Grasa1,2,3,*, Yue Ban4,5,3, and Mikel Sanz1,2,6,7

  • *Contact author: pablojesus.rodriguez@ehu.eus
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Phys. Rev. Research 7, 023269 – Published 16 June, 2025

DOI: https://doi.org/10.1103/xphb-x2g4

Abstract

Quantum and classical machine learning have been naturally connected through kernel methods, which have also served as proof-of-concept for quantum advantage. Quantum embeddings encode classical data into quantum feature states, enabling the construction of embedding quantum kernels (EQKs) by measuring vector similarities and projected quantum kernels (PQKs) through projections of these states. However, in both approaches, the model is influenced by the choice of the embedding. In this work, we propose using the training of a quantum neural network (QNN) to construct neural quantum kernels, specifically neural EQKs and neural PQKs—problem-inspired kernel functions. Unlike previous approaches, our method requires the kernel matrix to be constructed only once, significantly reducing computational overhead. To achieve this, we introduce a scalable training method for an n-qubit data reuploading QNN. Furthermore, we demonstrate neural quantum kernels can alleviate exponential concentration and enhance generalization capabilities compared to problem-agnostic kernels, positioning them as a scalable and robust solution for quantum machine learning applications.

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References (72)

  1. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature (London) 549, 195 (2017).
  2. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  3. V. Dunjko and H. J. Briegel, Machine learning & artificial intelligence in the quantum domain: A review of recent progress, Rep. Prog. Phys. 81, 074001 (2018).
  4. A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.-F. Chen, A. Gilyén, C. T. Hann, M. J. Kastoryano, E. T. Khabiboulline, A. Kubica, G. Salton, S. Wang, and F. G. S. L. Brandão, Quantum Algorithms: A Survey of Applications and End-To-End Complexities (Cambridge University Press, Cambridge, 2025)
  5. M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019).
  6. A. Skolik, S. Jerbi, and V. Dunjko, Quantum agents in the gym: A variational quantum algorithm for deep Q-learning, Quantum 6, 720 (2022).
  7. M. Schuld, A. Bocharov, K. M. Svore, and N. Wiebe, Circuit-centric quantum classifiers, Phys. Rev. A 101, 032308 (2020).
  8. E. Farhi and H. Neven, Classification with quantum neural networks on near term processors, arXiv:1802.06002.
  9. R. Sweke, J.-P. Seifert, D. Hangleiter, and J. Eisert, On the quantum versus classical learnability of discrete distributions, Quantum 5, 417 (2021).
  10. S. Jerbi, L. M. Trenkwalder, H. Poulsen Nautrup, H. J. Briegel, and V. Dunjko, Quantum enhancements for deep reinforcement learning in large spaces, PRX Quantum 2, 010328 (2021).
  11. N. Pirnay, R. Sweke, J. Eisert, and J.-P. Seifert, Superpolynomial quantum-classical separation for density modeling, Phys. Rev. A 107, 042416 (2023).
  12. C. Gyurik and V. Dunjko, On establishing learning separations between classical and quantum machine learning with classical data, arXiv:2208.06339.
  13. V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature (London) 567, 209 (2019).
  14. M. Schuld and N. Killoran, Quantum machine learning in feature hilbert spaces, Phys. Rev. Lett. 122, 040504 (2019).
  15. A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-reuploading for a universal quantum classifier, Quantum 4, 226 (2020).
  16. 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).
  17. M. C. Caro, E. Gil-Fuster, J. J. Meyer, J. Eisert, and R. Sweke, Encoding-dependent generalization bounds for parametrized quantum circuits, Quantum 5, 582 (2021).
  18. T. Ono, W. Roga, K. Wakui, M. Fujiwara, S. Miki, H. Terai, and M. Takeoka, Demonstration of a bosonic quantum classifier with data reuploading, Phys. Rev. Lett. 131, 013601 (2023).
  19. H.-Y. Huang, M. Broughton, M. Mohseni, R. Babbush, S. Boixo, H. Neven, and J. R. McClean, Power of data in quantum machine learning, Nat. Commun. 12, 2631 (2021).
  20. E. Peters, J. Caldeira, A. Ho, S. Leichenauer, M. Mohseni, H. Neven, P. Spentzouris, D. Strain, and G. N. Perdue, Machine learning of high dimensional data on a noisy quantum processor, npj Quantum Inf. 7, 161 (2021).
  21. K. Bartkiewicz, C. Gneiting, A. Černoch, K. Jiráková, K. Lemr, and F. Nori, Experimental kernel-based quantum machine learning in finite feature space, Sci. Rep. 10, 12356 (2020).
  22. T. Kusumoto, K. Mitarai, K. Fujii, M. Kitagawa, and M. Negoro, Experimental quantum kernel trick with nuclear spins in a solid, npj Quantum Inf. 7, 94 (2021).
  23. Y. Wu, B. Wu, J. Wang, and X. Yuan, Quantum phase recognition via quantum kernel methods, Quantum 7, 981 (2023).
  24. O. Kyriienko and E. B. Magnusson, Unsupervised quantum machine learning for fraud detection, arXiv:2208.01203.
  25. J. Tomasi, S. Anthoine, and H. Kadri, Benign overfitting with quantum kernels, arXiv:2503.17020.
  26. M. Schuld, Supervised quantum machine learning models are kernel methods, arXiv:2101.11020.
  27. M. Schuld and F. Petruccione, Quantum models as kernel methods, in Machine Learning with Quantum Computers (Springer International Publishing, Cham, 2021), pp. 217–245.
  28. Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nat. Phys. 17, 1013 (2021).
  29. B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, Adaptive Computation and Machine Learning (MIT Press, Cambridge, MA, 2002), p. 644.
  30. S. Jerbi, L. J. Fiderer, H. P. Nautrup, J. M. Kübler, H. J. Briegel, and V. Dunjko, Quantum machine learning beyond kernel methods, Nat. Commun. 14, 517 (2023).
  31. C. Blank, D. K. Park, J.-K. K. Rhee, and F. Petruccione, Quantum classifier with tailored quantum kernel, npj Quantum Inf. 6, 41 (2020).
  32. I. Salmenperä, I. Kuhtarskis, A. M. van de Griend, and J. K. Nurminen, The impact of feature embedding placement in the ansatz of a quantum kernel in QSVMs, in Proceedings of IEEE International Conference Quantum Computer Engineering (QCE) (2024).
  33. N. Shirai, K. Kubo, K. Mitarai, and K. Fujii, Quantum tangent kernel, Phys. Rev. Res. 6, 033179 (2024).
  34. J. M. Kübler, S. Buchholz, and B. Schölkopf, The inductive bias of quantum kernels, Adv. Neural Inf. Process. Syst. 34, 12661 (2021).
  35. Y. Wang and L. Cao, Quantum phase transition detection via quantum support vector machine, Quantum Sci. Technol. 10, 015043 (2025).
  36. V. Belis, K. A. Woźniak, E. Puljak, P. Barkoutsos, G. Dissertori, M. Grossi, M. Pierini, F. Reiter, I. Tavernelli, and S. Vallecorsa, Quantum anomaly detection in the latent space of proton collision events at the LHC, Commun. Phys. 7, 334 (2024).
  37. J. R. Glick, T. P. Gujarati, A. D. Córcoles, Y. Kim, A. Kandala, J. M. Gambetta, and K. Temme, Covariant quantum kernels for data with group structure, Nat. Phys. 20, 479 (2024).
  38. J. J. Meyer, M. Mularski, E. Gil-Fuster, A. A. Mele, F. Arzani, A. Wilms, and J. Eisert, Exploiting symmetry in variational quantum machine learning, PRX Quantum 4, 010328 (2023).
  39. M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-invariant quantum machine learning, PRX Quantum 3, 030341 (2022).
  40. S. Thanasilp, S. Wang, M. Cerezo, and Z. Holmes, Exponential concentration in quantum kernel methods, Nat. Commun. 15, 5200 (2024).
  41. S. S. Vedaie, M. Noori, J. S. Oberoi, B. C. Sanders, and E. Zahedinejad, Quantum multiple kernel learning, arXiv:2011.09694.
  42. A. Ghukasyan, J. S. Baker, O. Goktas, J. Carrasquilla, and S. K. Radha, Quantum-classical multiple kernel learning, arXiv:2305.17707.
  43. T. Hubregtsen, D. Wierichs, E. Gil-Fuster, P.-J. H. S. Derks, P. K. Faehrmann, and J. J. Meyer, Training quantum embedding kernels on near-term quantum computers, Phys. Rev. A 106, 042431 (2022).
  44. P. Rodriguez-Grasa, R. Farzan-Rodriguez, G. Novelli, Y. Ban, and M. Sanz, Satellite image classification with neural quantum kernels, Mach. Learn.: Sci. Technol. 6, 015043 (2025).
  45. J. Bowles, S. Ahmed, and M. Schuld, Better than classical? The subtle art of benchmarking quantum machine learning models, arXiv:2403.07059.
  46. P. Rebentrost, M. Mohseni, and S. Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett. 113, 130503 (2014).
  47. S. Park, D. K. Park, and J.-K. K. Rhee, Variational quantum approximate support vector machine with inference transfer, Sci. Rep. 13, 3288 (2023).
  48. E. Gil-Fuster, J. Eisert, and V. Dunjko, On the expressivity of embedding quantum kernels, Mach. Learn.: Sci. Technol. 5, 025003 (2024).
  49. H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, Quantum fingerprinting, Phys. Rev. Lett. 87, 167902 (2001).
  50. M. Fanizza, M. Rosati, M. Skotiniotis, J. Calsamiglia, and V. Giovannetti, Beyond the swap test: Optimal estimation of quantum state overlap, Phys. Rev. Lett. 124, 060503 (2020).
  51. L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, Learning the quantum algorithm for state overlap, New J. Phys. 20, 113022 (2018).
  52. R. Shaydulin and S. M. Wild, Importance of kernel bandwidth in quantum machine learning, Phys. Rev. A 106, 042407 (2022).
  53. A. Canatar, E. Peters, C. Pehlevan, S. M. Wild, and R. Shaydulin, Bandwidth enables generalization in quantum kernel models, Trans. Mach. Learn. Res. (2023).
  54. Y. Suzuki, H. Kawaguchi, and N. Yamamoto, Quantum Fisher kernel for mitigating the vanishing similarity issue, Quantum Sci. Technol. 9, 035050 (2024).
  55. F. Sauvage, M. Larocca, P. J. Coles, and M. Cerezo, Building spatial symmetries into parameterized quantum circuits for faster training, Quantum Sci. Technol. 9, 015029 (2024).
  56. S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran, Quantum embeddings for machine learning, arXiv:2001.03622.
  57. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  58. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15, 7172 (2024).
  59. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  60. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1791 (2021).
  61. S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nat. Commun. 12, 6961 (2021).
  62. S. Thanasilp, S. Wang, N. A. Nghiem, P. Coles, and M. Cerezo, Subtleties in the trainability of quantum machine learning models, Quantum Mach. Intell. 5, 21 (2023).
  63. G. Li, R. Ye, X. Zhao, and X. Wang, Concentration of data encoding in parameterized quantum circuits, Adv. Neural Inf. Process. Syst. 35, 19456 (2022).
  64. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  65. E. R. Anschuetz, A unified theory of quantum neural network loss landscapes, arXiv:2408.11901.
  66. B. Casas and A. Cervera-Lierta, Multidimensional Fourier series with quantum circuits, Phys. Rev. A 107, 062612 (2023).
  67. A. Barthe and A. Pérez-Salinas, Gradients and frequency profiles of quantum re-uploading models, Quantum 8, 1523 (2024).
  68. A. Pérez-Salinas, D. López-Núñez, A. García-Sáez, P. Forn-Díaz, and J. I. Latorre, One qubit as a universal approximant, Phys. Rev. A 104, 012405 (2021).
  69. I. Panadero, Y. Ban, H. Espinós, R. Puebla, J. Casanova, and E. Torrontegui, Regressions on quantum neural networks at maximal expressivity, arXiv:2311.06090.
  70. H. Xiao, K. Rasul, and R. Vollgraf, Fashion-MNIST: A novel image data set for benchmarking machine learning algorithms, arXiv:1708.07747.
  71. E. Gil-Fuster, J. Eisert, and C. Bravo-Prieto, Understanding quantum machine learning also requires rethinking generalization, Nat. Commun. 15, 2277 (2024).
  72. D. Holmes and L. Jain, Innovations in Machine Learning: Theory and Applications (Springer, New York, 2006).

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Phys. Rev. Research 7, 023269– Published 16 June, 2025
  • Received 15 January 2024
  • Accepted 20 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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