- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Learning complexity gradually in quantum machine learning models

Erik Recio-Armengol1,2, Franz J. Schreiber3, Jens Eisert3,4,5, and Carlos Bravo-Prieto3,*

  • *Contact author: c.bravo.prieto@fu-berlin.de
PDF

Phys. Rev. Research 8, 033006 – Published 6 July, 2026

DOI: https://doi.org/10.1103/wnlr-q9xd

Abstract

Quantum machine learning is an emergent field that continues to draw significant interest for its potential to offer improvements over classical algorithms in certain areas. However, training quantum models remains a challenging task, largely because of the difficulty in establishing an effective inductive bias when solving high-dimensional problems. In this work, we propose a training framework that prioritizes informative data points over the entire training set. This approach draws inspiration from classical techniques such as curriculum learning and hard example mining to introduce an additional inductive bias through the training data itself. By selectively focusing on informative samples, we aim to steer the optimization process toward more favorable regions of the parameter space. This data-centric approach complements existing strategies such as warm-start initialization methods, providing an additional pathway to address performance challenges in quantum machine learning. We provide theoretical insights into the benefits of prioritizing informative data for quantum models, and we validate our methodology with numerical experiments on selected recognition tasks of quantum phases of matter. Our findings indicate that this strategy could be a valuable approach for improving the performance of quantum machine learning models.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. R. Sweke, J.-P. Seifert, D. Hangleiter, and J. Eisert, On the quantum versus classical learnability of discrete distributions, Quantum 5, 417 (2021).
  2. Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nat. Phys. 17, 1013 (2021).
  3. H.-Y. Huang, R. Kueng, and J. Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021).
  4. N. Pirnay, R. Sweke, J. Eisert, and J.-P. Seifert, Superpolynomial quantum-classical separation for density modeling, Phys. Rev. A 107, 042416 (2023).
  5. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15, 7172 (2024).
  6. 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).
  7. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2024).
  8. E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nat. Commun. 13, 7760 (2022).
  9. A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. S. França, and Y. Quek, Noise-induced shallow circuits and absence of barren plateaus, Nat. Phys. 22, 751 (2026).
  10. N. A. Nemkov, E. O. Kiktenko, and A. K. Fedorov, Barren plateaus are swamped with traps, Phys. Rev. A 111, 012441 (2025).
  11. 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).
  12. J. M. Kübler, S. Buchholz, and B. Schölkopf, The inductive bias of quantum kernels, in Proceedings of the 35th International Conference on Neural Information Processing Systems, NIPS'21 (Curran Associates Inc., Red Hook, NY, 2021), pp. 12661–12673.
  13. M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Nat. Comput. Sci. 2, 567 (2022).
  14. J. Bowles, V. J. Wright, M. Farkas, N. Killoran, and M. Schuld, Contextuality and inductive bias in quantum machine learning, arXiv:2302.01365.
  15. 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).
  16. 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).
  17. J. Bowles, D. Wierichs, and C.-Y. Park, Backpropagation scaling in parameterised quantum circuits, Quantum 9, 1873 (2023).
  18. C.-Y. Park and N. Killoran, Hamiltonian variational ansatz without barren plateaus, Quantum 8, 1239 (2024).
  19. S. Jerbi, C. Gyurik, S. C. Marshall, R. Molteni, and V. Dunjko, Shadows of quantum machine learning, Nat. Commun. 15, 5676 (2024).
  20. L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation-equivariant quantum neural networks, npj Quantum Inf. 10, 12 (2024).
  21. P. Rodriguez-Grasa, Y. Ban, and M. Sanz, Training embedding quantum kernels with data re-uploading quantum neural networks, Phys. Rev. Res. 7, 023269 (2025).
  22. K. Zhang, L. Liu, M.-H. Hsieh, and D. Tao, Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits, Adv. Neural Inf. Process. Syst. 35, 18612 (2022).
  23. H. R. Grimsley, G. S. Barron, E. Barnes, S. E. Economou, and N. J. Mayhall, Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus, npj Quantum Inf. 9, 19 (2023).
  24. J. Dborin, F. Barratt, V. Wimalaweera, L. Wright, and A. G. Green, Matrix product state pre-training for quantum machine learning, Quantum Sci. Technol. 7, 035014 (2022).
  25. M. S. Rudolph, J. Miller, D. Motlagh, J. Chen, A. Acharya, and A. Perdomo-Ortiz, Synergistic pretraining of parametrized quantum circuits via tensor networks, Nat. Commun. 14, 8367 (2023).
  26. A. A. Mele, G. B. Mbeng, G. E. Santoro, M. Collura, and P. Torta, Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz, Phys. Rev. A 106, L060401 (2022).
  27. R. Puig, M. Drudis, S. Thanasilp, and Z. Holmes, Variational quantum simulation: A case study for understanding warm starts, PRX Quantum 6, 010317 (2025).
  28. Y. Wang, B. Qi, C. Ferrie, and D. Dong, Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization, Phys. Rev. Appl. 22, 054005 (2024).
  29. C.-Y. Park, M. Kang, and J. Huh, Hardware-efficient ansatz without barren plateaus in any depth, arXiv:2403.04844.
  30. X. Shi and Y. Shang, Avoiding barren plateaus via Gaussian mixture model, New J. Phys. 27, 104501 (2025).
  31. Y. Bengio, J. Louradour, R. Collobert, and J. Weston, Curriculum learning, in Proceedings of the 26th Annual International Conference on Machine Learning (Association for Computing Machinery, Montreal, Quebec, Canada, 2009), pp. 41–48.
  32. Q. H. Tran, Y. Endo, and H. Oshima, Quantum curriculum learning, Phys. Rev. A 112, 032431 (2025).
  33. A. Shrivastava, A. Gupta, and R. Girshick, Training region-based object detectors with online hard example mining, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, Piscataway, NJ, 2016), pp. 761–769.
  34. X. Wang, Y. Chen, and W. Zhu, A survey on curriculum learning, IEEE Trans. Pattern Anal. Mach. Intell. 44, 4555 (2021).
  35. P. Soviany, R. T. Ionescu, P. Rota, and N. Sebe, Curriculum learning: A survey, Int. J. Comp. Vis. 130, 1526 (2022).
  36. R. Tudor Ionescu, B. Alexe, M. Leordeanu, M. Popescu, D. P. Papadopoulos, and V. Ferrari, How hard can it be? Estimating the difficulty of visual search in an image, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, Piscataway, NJ, 2016), pp. 2157–2166.
  37. S. Guo, W. Huang, H. Zhang, C. Zhuang, D. Dong, M. R. Scott, and D. Huang, CurriculumNet: Weakly supervised learning from large-scale web images, in Proceedings of the European Conference on Computer Vision (ECCV), edited by V. Ferrari, M. Hebert, C. Sminchisescu, and Y. Weiss, Lecture Notes in Computer Science, Vol. 11214 (Springer, Cham, 2018), pp. 139–154.
  38. L. Jiang, D. Meng, T. Mitamura, and A. G. Hauptmann, Easy samples first: Self-paced reranking for zero-example multimedia search, in Proceedings of the 22nd ACM international conference on Multimedia (Association for Computing Machinery, Orlando, FL, 2014), pp. 547–556.
  39. E. A. Platanios, O. Stretcu, G. Neubig, B. Poczos, and T. Mitchell, Competence-based curriculum learning for neural machine translation, in Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Association for Computational Linguistics, 2019), pp. 1162–1172.
  40. T. Matiisen, A. Oliver, T. Cohen, and J. Schulman, Teacher–student curriculum learning, IEEE Trans. Neural Netw. Learning Syst. 31, 3732 (2019).
  41. X. Zhang, G. Kumar, H. Khayrallah, K. Murray, J. Gwinnup, M. J. Martindale, P. McNamee, K. Duh, and M. Carpuat, An empirical exploration of curriculum learning for neural machine translation, arXiv:1811.00739.
  42. 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).
  43. 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).
  44. G. Penha and C. Hauff, Curriculum learning strategies for IR: An empirical study on conversation response ranking, in Advances in Information Retrieval (Springer-Verlag, Lisbon, Portugal, 2020), pp. 699–713.
  45. M. C. Caro, H.-Y. Huang, M. Cerezo, K. Sharma, A. Sornborger, L. Cincio, and P. J. Coles, Generalization in quantum machine learning from few training data, Nat. Commun. 13, 4919 (2022).
  46. Y. Wu, B. Wu, J. Wang, and X. Yuan, Quantum phase recognition via quantum kernel methods, Quantum 7, 981 (2023).
  47. Y.-J. Liu, A. Smith, M. Knap, and F. Pollmann, Model-independent learning of quantum phases of matter with quantum convolutional neural networks, Phys. Rev. Lett. 130, 220603 (2023).
  48. E. Gil-Fuster, J. Eisert, and C. Bravo-Prieto, Understanding quantum machine learning also requires rethinking generalization, Nat. Commun. 15, 2277 (2024).
  49. C. Umeano, A. E. Paine, V. E. Elfving, and O. Kyriienko, What can we learn from quantum convolutional neural networks? Adv. Quantum Tech. 8, 2400325 (2025).
  50. P. Zapletal, N. A. McMahon, and M. J. Hartmann, Error-tolerant quantum convolutional neural networks for symmetry-protected topological phases, Phys. Rev. Res. 6, 033111 (2024).
  51. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  52. S. Sachdev, Quantum Phases of Matter (Cambridge University Press, Cambridge, MA, 2023).
  53. R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017).
  54. N. Schuch, D. Pérez-García, and I. Cirac, Classifying quantum phases using matrix product states and projected entangled pair states, Phys. Rev. B 84, 165139 (2011).
  55. A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch, Many-body topological invariants from randomized measurements in synthetic quantum matter, Sci. Adv. 6, eaaz3666 (2020).
  56. I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nat. Phys. 15, 1273 (2019).
  57. E. Recio-Armengol and C. Bravo-Prieto, Learning complexity gradually in QML models repository, 2025, https://github.com/erik-armengol/learning-complexity-gradually
  58. V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, et al., Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv:1811.04968.
  59. D. P. Kingma, Adam: A method for stochastic optimization, arXiv:1412.6980.
  60. G. Aguilar, S. Cichy, J. Eisert, and L. Bittel, Full classification of Pauli Lie algebras, arXiv:2408.00081.
  61. D. Gross, K. M. R. Audenaert, and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48, 052104 (2007).
  62. M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for variational quantum computing, Phys. Rev. Res. 7, 033266 (2025).
  63. M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Diagnosing barren plateaus with tools from quantum optimal control, Quantum 6, 824 (2022).
  64. L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, Learning the quantum algorithm for state overlap, New J. Phys. 20, 113022 (2018).
  65. K. Beer, D. Bondarenko, T. Farrelly, T. J. Osborne, R. Salzmann, D. Scheiermann, and R. Wolf, Training deep quantum neural networks, Nat. Commun. 11, 808 (2020).
  66. L. Cincio, K. Rudinger, M. Sarovar, and P. J. Coles, Machine learning of noise-resilient quantum circuits, PRX Quantum 2, 010324 (2021).
  67. Z. Yu, X. Zhao, B. Zhao, and X. Wang, Optimal quantum dataset for learning a unitary transformation, Phys. Rev. Appl. 19, 034017 (2023).
  68. D. F. Locher, L. Cardarelli, and M. Müller, Quantum error correction with quantum autoencoders, Quantum 7, 942 (2023).
  69. S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms (Cambridge University Press, Cambridge, 2014).
  70. S.-X. Zhang, J. Allcock, Z.-Q. Wan, S. Liu, J. Sun, H. Yu, X.-H. Yang, J. Qiu, Z. Ye, Y.-Q. Chen, et al., Tensorcircuit: A quantum software framework for the NISQ era, Quantum 7, 912 (2023).
  71. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  72. D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, General parameter-shift rules for quantum gradients, Quantum 6, 677 (2022).

Outline

Information

Phys. Rev. Research 8, 033006– Published 6 July, 2026
  • Received 9 January 2026
  • Accepted 19 May 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation