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  • Open Access

Quantum feature-map learning with reduced resource overhead

Jonas Jäger1,2,*,†, Philipp Elsässer3,4,*,‡, and Elham Torabian4,2

  • *These authors contributed equally to this work.
  • †Contact author: jojaeger@cs.ubc.ca
  • ‡Contact author: philipp.elsaesser@physik.uni-freiburg.de
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Phys. Rev. Research 8, 023247 – Published 5 June, 2026

DOI: https://doi.org/10.1103/v29j-rh32

Abstract

Current quantum computers require algorithms that use limited resources economically. In quantum machine learning, success hinges on quantum feature maps, which embed classical data into the state space of qubits. We introduce Quantum Feature-Map Learning via Analytic Iterative Reconstructions (Q-FLAIR), an algorithm that reduces quantum resource overhead in iterative feature-map circuit construction. It shifts workloads to a classical computer via partial analytic reconstructions of the quantum model, using only a few evaluations. For each probed gate addition to the ansatz, the simultaneous selection and optimization of the data feature and weight parameter is then entirely classical. Integrated into quantum neural network and quantum kernel support vector classifiers, Q-FLAIR shows state-of-the-art benchmark performance. Since resource overhead decouples from feature dimension, we train a quantum model on a real IBM device in only 4 h, surpassing 90% accuracy on the full-resolution MNIST dataset (784 features, digits 3 versus 5). Such results were previously unattainable, as the feature dimension prohibitively drives hardware demands for fixed and search costs for adaptive ansätze. Furthermore, Q-FLAIR demonstrates dequantization robustness against direct classical modeling, satisfying a benchmark rare in the literature and a necessary condition for potential quantum advantage. By rethinking feature-map learning beyond black-box optimization, this work takes a concrete step toward enabling quantum machine learning for real-world problems and near-term quantum computers.

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References (126)

  1. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature (London) 549, 195 (2017).
  2. 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).
  3. 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).
  4. 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).
  5. M. Schuld and N. Killoran, Quantum machine learning in feature Hilbert spaces, Phys. Rev. Lett. 122, 040504 (2019).
  6. K. Huang, Z.-A. Wang, C. Song, K. Xu, H. Li, Z. Wang, Q. Guo, Z. Song, Z.-B. Liu, D. Zheng, et al., Quantum generative adversarial networks with multiple superconducting qubits, npj Quantum Inf. 7, 165 (2021).
  7. K. Sakka, K. Mitarai, and K. Fujii, Automating quantum feature map design via large language models, arXiv:2504.07396.
  8. D. Ristè, M. P. Da Silva, C. A. Ryan, A. W. Cross, A. D. Córcoles, J. A. Smolin, J. M. Gambetta, J. M. Chow, and B. R. Johnson, Demonstration of quantum advantage in machine learning, npj Quantum Inf. 3, 16 (2017).
  9. G. Cho and D. Kim, Machine learning on quantum experimental data toward solving quantum many-body problems, Nat. Commun. 15, 7552 (2024).
  10. Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nat. Phys. 17, 1013 (2021).
  11. J. Jäger and R. V. Krems, Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines, Nat. Commun. 14, 576 (2023).
  12. S. Y. Chang and M. Cerezo, A primer on quantum machine learning, arXiv:2511.15969.
  13. L. Pira and P. Rebentrost, Fundamentals of quantum machine learning and robustness, in Quantum Robustness in Artificial Intelligence Principles and Applications (Springer, 2026), pp. 1–19..
  14. 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).
  15. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  16. M. Watabe, K. Shiba, M. Sogabe, K. Sakamoto, and T. Sogabe, Quantum circuit parameters learning with gradient descent using backpropagation, arXiv:1910.14266.
  17. J. Qi, C.-H. Yang, and P.-Y. Chen, QTN-VQC: An end-to-end learning framework for quantum neural networks, Phys. Scr. 99, 015111 (2024).
  18. K. Terashi, M. Kaneda, T. Kishimoto, M. Saito, R. Sawada, and J. Tanaka, Event classification with quantum machine learning in high-energy physics, Comput. Software Big Sci. 5, 2 (2021).
  19. S. Y. C. Chen, C. M. Huang, C. W. Hsing, and Y. J. Kao, Hybrid quantum-classical classifier based on tensor network and variational quantum circuit, arXiv:2011.14651.
  20. A. Blance and M. Spannowsky, Quantum machine learning for particle physics using a variational quantum classifier, J. High Energy Phys. 02 (2021) 212.
  21. Y. Kwak, W. J. Yun, S. Jung, J. K. Kim, and J. Kim, Introduction to quantum reinforcement learning: Theory and PennyLane-based implementation, in International Conference on ICT Convergence (IEEE, Jeju Island, 2021), pp. 416–420.
  22. D. Sierra-Sosa, J. Arcila-Moreno, B. Garcia-Zapirain, C. Castillo-Olea, and A. Elmaghraby, Dementia prediction applying variational quantum classifier, arXiv:2007.08653.
  23. S. Y. C. Chen, C. H. H. Yang, J. Qi, P. Y. Chen, X. Ma, and H. S. Goan, Variational quantum circuits for deep reinforcement learning, IEEE Access 8, 141007 (2020).
  24. S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran, Quantum embeddings for machine learning, arXiv:2001.03622.
  25. 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).
  26. L. P. Henry, S. Thabet, C. Dalyac, and L. Henriet, Quantum evolution kernel: Machine learning on graphs with programmable arrays of qubits, Phys. Rev. A 104, 032416 (2021).
  27. 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).
  28. Y. Du, T. Huang, S. You, M. Hsieh, and D. Tao, Quantum circuit architecture search for variational quantum algorithms, npj Quantum Inf. 8, 62 (2022).
  29. E. Torabian and R. V. Krems, Compositional optimization of quantum circuits for quantum kernels of support vector machines, Phys. Rev. Res. 5, 013211 (2023).
  30. S. Zahid and M. A. Tahir, Unlocking quantum SVM potential: Optimal feature map generation and feature selection, Phys. Scr. 100, 015120 (2025).
  31. T. Fösel, M. Y. Niu, F. Marquardt, and L. Li, Quantum circuit optimization with deep reinforcement learning, arXiv:2103.07585.
  32. S. Chakraborty, S. H. Shaikh, A. Chakrabarti, and R. Ghosh, A hybrid quantum feature selection algorithm using a quantum inspired graph theoretic approach, Appl. Intell. 50, 1775 (2020).
  33. Z. Lu, P. X. Shen, and D. L. Deng, Markovian quantum neuroevolution for machine learning, Phys. Rev. Appl. 16, 044039 (2021).
  34. H. Wang, Y. Ding, J. Gu, Y. Lin, D. Z. Pan, F. T. Chong, and S. Han, QuantumNAS: Noise-adaptive search for robust quantum circuits, in Proceedings of the IEEE International Symposium on High-Performance Computer Architecture (IEEE, Seoul, 2022), pp. 692–708.
  35. S. Altares-López, A. Ribeiro, and J. J. García-Ripoll, Automatic design of quantum feature maps, Quantum Sci. Technol. 6, 045015 (2021).
  36. R. Pellow-Jarman, A. Pillay, I. Sinayskiy, and F. Petruccione, Hybrid genetic optimization for quantum feature map design, Quantum Mach. Intell. 6, 45 (2024).
  37. Y. Liu, F. Meng, L. Wang, Y. Hu, S. Li, Z. Zhang, and X. Yu, Hardware-aware quantum kernel design based on graph neural networks, arXiv:2506.21161.
  38. M. Ostaszewski, E. Grant, and M. Benedetti, Structure optimization for parameterized quantum circuits, Quantum 5, 391 (2021).
  39. K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for quantum-classical hybrid algorithms, Phys. Rev. Res. 2, 043158 (2020).
  40. R. M. Parrish, J. T. Iosue, A. Ozaeta, and P. L. McMahon, A Jacobi diagonalization and Anderson acceleration algorithm for variational quantum algorithm parameter optimization, arXiv:1904.03206.
  41. J. G. Vidal and D. O. Theis, Calculus on parameterized quantum circuits, arXiv:1812.06323.
  42. C. Feniou, M. Hassan, B. Claudon, A. Courtat, O. Adjoua, Y. Maday, and J.-P. Piquemal, Greedy gradient-free adaptive variational quantum algorithms on a noisy intermediate scale quantum computer, Sci. Rep. 15, 18689 (2025).
  43. J. Jäger, T. N. Kaldenbach, M. Haas, and E. Schultheis, Fast gradient-free optimization of excitations in variational quantum eigensolvers, Commun. Phys. 8, 418 (2025).
  44. H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, An adaptive variational algorithm for exact molecular simulations on a quantum computer, Nat. Commun. 10, 3007 (2019).
  45. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, NY, 2010).
  46. M. Grossi, N. Ibrahim, V. Radescu, R. Loredo, K. Voigt, C. Von Altrock, and A. Rudnik, Mixed quantum—classical method for fraud detection with quantum feature selection, IEEE Trans. Quantum Eng. 3, 1 (2022).
  47. A. S. Albino, O. M. Pires, M. Q. Nooblath, and E. G. S. Nascimento, Evolutionary quantum feature selection, arXiv:2303.07131.
  48. L. Wang, Z. Y. Chen, F. Y. Le, Z. Q. Yu, C. Xue, X. N. Zhuang, Q. Yan, Y. Yang, Y. C. Wu, and G. P. Guo, A quantum feature selection framework via ground state preparation, Phys. Scr. 98, 115121 (2023).
  49. S. Mücke, R. Heese, S. Müller, M. Wolter, and N. Piatkowski, Feature selection on quantum computers, Quantum Mach. Intell. 5, 11 (2023).
  50. R. Battiti, Using mutual information for selecting features in supervised neural net learning, IEEE Trans. Neural Networks 5, 537 (1994).
  51. M. A. F. A. Fida, T. Ahmad, and M. Ntahobari, Variance threshold as early screening to Boruta feature selection for intrusion detection system, in Proceedings of the 13th International Conference on Information and Communication Technology and Systems (IEEE, Surabaya, 2021), pp. 46–50.
  52. I. Guyon, J. Weston, S. Barnhill, and V. Vapnik, Gene selection for cancer classification using support vector machines, Mach. Learn. 46, 389 (2002).
  53. R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. B 58, 267 (1996).
  54. J. Bowles, S. Ahmed, and M. Schuld, Better than classical? The subtle art of benchmarking quantum machine learning models, arXiv:2403.07059.
  55. F. J. Schreiber, J. Eisert, and J. J. Meyer, Classical surrogates for quantum learning models, Phys. Rev. Lett. 131, 100803 (2023).
  56. J. Stein, M. Poppel, P. Adamczyk, R. Fabry, Z. Wu, M. Kölle, J. Nüßlein, D. Schuman, P. Altmann, T. Ehmer, V. Narasimhan, and C. Linnhoff-Popien, Benchmarking quantum surrogate models on scarce and noisy data, in Proceedings of the 16th International Conference on Agents and Artificial Intelligence: ICAART, INSTICC (SciTePress, Rome, Italy, 2024), Vol. 3, pp. 352–359.
  57. E. Farhi and H. Neven, Classification with quantum neural networks on near term processors, arXiv:1802.06002.
  58. M. Schuld, A. Bocharov, K. M. Svore, and N. Wiebe, Circuit-centric quantum classifiers, Phys. Rev. A 101, 032308 (2020).
  59. This loss function draws motivation from statistics and information theory, also known as log loss, logistic loss, or cross-entropy loss, and has been established as the standard choice in the classical machine learning literature—including extensions to multiclass—classification [72, 109, 126].
  60. 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).
  61. B. E. Boser, I. M. Guyon, and V. N. Vapnik, A training algorithm for optimal margin classifiers, in Proceedings of the 5th Annual Workshop on Computational Learning Theory, COLT ’92 (Association for Computing Machinery, New York, NY, 1992), pp. 144–152.
  62. Note that given the evaluated quantum kernel values, commonly aggregated in the kernel matrix (also referred to as Gram matrix), the training of the QSVM does not differ from that of classical SVMs [72, 109, 126].
  63. N. Cristianini, J. Shawe-Taylor, A. Elisseeff, and J. Kandola, On kernel-target alignment, Adv. Neural Inf. Proc. Syst. 14 (2001).
  64. A. E. Paine, V. E. Elfving, and O. Kyriienko, Quantum kernel methods for solving regression problems and differential equations, Phys. Rev. A 107, 032428 (2023).
  65. 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 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Montreal, 2024), Vol. 1, pp. 1663–1671.
  66. The required form to apply Q-FLAIR is still maintained when appending a fixed gate along with a (weight-) data-dependent gate because the fixed gate is absorbed by the state and observable in Eq. (28).
  67. J. Bowles, S. Ahmed, and M. Schuld, PennyLane datasets for better than classical? The subtle art of benchmarking quantum machine learning models, PennyLane (2024), https://pennylane.ai/datasets/linearly-separable, https://pennylane.ai/datasets/two-curves, https://pennylane.ai/datasets/bars-and-stripes, https://pennylane.ai/datasets/downscaled-mnist.
  68. Y. LeCun, The MNIST database of handwritten digits (1998), http://yan.lecun.com/exdb/mnist/.
  69. S. Buchanan, D. Gilboa, and J. Wright, Deep networks and the multiple manifold problem, arXiv:2008.11245.
  70. A. C. Lorena, A. I. Maciel, P. B. de Miranda, I. G. Costa, and R. B. Prudêncio, Data complexity meta-features for regression problems, Mach. Learn. 107, 209 (2018).
  71. X. Pan, X. Cao, W. Wang, Z. Hua, W. Cai, X. Li, H. Wang, J. Hu, Y. Song, D.-L. Deng, et al., Experimental quantum end-to-end learning on a superconducting processor, npj Quantum Inf. 9, 18 (2023).
  72. K. P. Murphy, Machine Learning: A Probabilistic Perspective, 2nd ed. (MIT Press, Cambridge, MA, 2012).
  73. R. Coelho, G. Kruse, and A. Rosskopf, Quantum-efficient kernel target alignment, arXiv:2502.08225.
  74. W. Fedus, B. Zoph, and N. Shazeer, Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity, J. Mach. Learn. Res. 23, 1 (2022).
  75. D. Reinsel, J. Gantz, and J. Rydning, The Digitization of the World from Edge to Core (International Data Corporation, Framingham, 2018), Vol. 16, p. 1.
  76. 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).
  77. 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).
  78. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  79. S. Mittal, M. Kumar, Y. Chand, R. Krishnakumar, and N. K. Kundu, Experimental validation of dequantization of hybrid quantum machine learning models using classical surrogates, in 2025 5th International Conference on AI-ML-Systems (AIMLSystems) (IEEE, Bangalore, 2025), pp. 412–417.
  80. P. A. Hernicht, A. Sakhnenko, C. O’Meara, G. Cortiana, and J. M. Lorenz, Enhancing the scalability of classical surrogates for real-world quantum machine learning applications, arXiv:2508.06131.
  81. Due to qubit demands, we do not directly compare Q-FLAIR feature maps with fixed-ansatz maps matching the qubit count to the feature count, e.g., angle embedding, Z feature map, and ZZ feature map [4].
  82. I. Kerenidis and A. Luongo, Classification of the MNIST data set with quantum slow feature analysis, Phys. Rev. A 101, 062327 (2020).
  83. M. Slysz, K. Kurowski, G. Waligóra, and J. Węglarz, Exploring the capabilities of quantum support vector machines for image classification on the MNIST benchmark, in International Conference on Computational Science, edited by J. Mikyka, C. de Mulatier, M. Paszynski, V. V. Krzhizhanovskaya, J. J. Dongarra, and P. M. Sloot (Springer Nature Switzerland, Cham, 2023), pp. 193–200.
  84. A. Senokosov, A. Sedykh, A. Sagingalieva, B. Kyriacou, and A. Melnikov, Quantum machine learning for image classification, Mach. Learn.: Sci. Technol. 5, 015040 (2024).
  85. R. Zhou, S. Sarkar, S. Bose, and A. Bayat, Enhanced image classification via hybridizing quantum dynamics with classical neural networks, arXiv:2507.13587.
  86. Z. Erkan, J. Rahebi, and A. Yelghi, Quantum image dataset transform (QIDT) for image processing, Quantum Inf. Proc. 24, 156 (2025).
  87. K. Chen, J. Liu, and F. Yan, Exploring quantum neural networks for binary classification on MNIST dataset: A swap test approach, Neural Networks 188, 107442 (2025).
  88. P. A. X. Tognini, L. Banchi, and G. De Palma, Solving MNIST with a globally trained mixture of quantum experts, arXiv:2505.14789.
  89. S. Manko and D. Frolovtsev, Classification and reconstruction for single-pixel imaging with classical and quantum neural networks, Signal Image Video Process. 19, 277 (2025).
  90. P. Wang, C. R. Myers, L. C. Hollenberg, and U. Parampalli, Quantum Hamiltonian embedding of images for data reuploading classifiers, Quantum Mach. Intell. 7, 35 (2025).
  91. K. Shen, B. Jobst, E. Shishenina, and F. Pollmann, Classification of the Fashion-MNIST dataset on a quantum computer, arXiv:2403.02405.
  92. F. J. Kiwit, B. Jobst, A. Luckow, F. Pollmann, and C. A. Riofrío, Typical machine learning datasets as low-depth quantum circuits, Quantum Sci. Technol. 10, 045035 (2025).
  93. P. Röseler, O. Schaudt, H. Berg, C. Bauckhage, and M. Koch, Efficient quantum convolutional neural networks for image classification: Overcoming hardware constraints, arXiv:2505.05957.
  94. Y. Zeng, H. Wang, J. He, Q. Huang, and S. Chang, A multi-classification hybrid quantum neural network using an all-qubit multi-observable measurement strategy, Entropy 24, 394 (2022).
  95. B. Dhara, M. Agrawal, and S. D. Roy, Multi-class classification using quantum transfer learning, Quantum Inf. Proc. 23, 34 (2024).
  96. F. Riaz, S. Abdulla, H. Suzuki, S. Ganguly, R. C. Deo, and S. Hopkins, Application of quantum pre-processing filter for binary image classification with small samples, J. Data Sci. Intell. Syst. 3, 109 (2023).
  97. 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).
  98. A. Skolik, J. R. McClean, M. Mohseni, P. van der Smagt, and M. Leib, Layerwise learning for quantum neural networks, Quantum Mach. Intell. 3, 5 (2021).
  99. S. R. Nair and C. Ferrie, Local surrogates for quantum machine learning, arXiv:2506.09425.
  100. 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).
  101. D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, General parameter-shift rules for quantum gradients, Quantum 6, 677 (2022).
  102. M. Schuld and F. Petruccione, Machine Learning with Quantum Computers, Quantum Science and Technology (Springer International Publishing, Cham, 2021).
  103. The discrepancy introduced by classical nonlinear preprocessing is another reason why we refrained from directly comparing the feature maps composed by Q-FLAIR with such fixed-ansatz quantum feature maps as the ZZ feature map.
  104. J. Jäger, P. Elsässer, and E. Torabian, Q-FLAIR: Quantum feature-map learning with reduced resource overhead, Zenodo (2026), https://zenodo.org/records/20101655.
  105. J. Li, X. Yang, X. Peng, and C.-P. Sun, Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118, 150503 (2017).
  106. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  107. One could further consider purely weight-dependent gates as a special case of the weight-data-dependent gates if a constant feature π is included (considering a parameter range of [−1,1]) in the input data points, which is a common approach for convenience in machine learning practice.
  108. P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta, Scalable mitigation of measurement errors on quantum computers, PRX Quantum 2, 040326 (2021).
  109. C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics (Springer, New York, 2006).
  110. S. Thanasilp, S. Wang, M. Cerezo, and Z. Holmes, Exponential concentration in quantum kernel methods, Nat. Commun. 15, 5200 (2024).
  111. J. Barzen and F. Leymann, On the differential topology of expressivity of parameterized quantum circuits, AppliedMath 5, 121 (2025).
  112. J. Yao, Learning to maximize quantum neural network expressivity via effective rank, arXiv:2506.15375.
  113. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with Qiskit, arXiv:2405.08810.
  114. Qiskit contributors, Qiskit 0.44.0, Zenodo (2023), https://zenodo.org/records/8190968.
  115. L. Monbroussou, E. Z. Mamon, J. Landman, A. B. Grilo, R. Kukla, and E. Kashefi, Trainability and expressivity of Hamming-weight preserving quantum circuits for machine learning, Quantum 9, 1745 (2025).
  116. T. Haug and M. S. Kim, Generalization of quantum machine learning models using quantum Fisher information metric, Phys. Rev. Lett. 133, 050603 (2024).
  117. J. J. Meyer, Fisher information in noisy intermediate-scale quantum applications, Quantum 5, 539 (2021).
  118. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  119. M. Schuld, Supervised quantum machine learning models are kernel methods, arXiv:2101.11020.
  120. A. Abbas, D. Sutter, A. Figalli, and S. Woerner, Effective dimension of machine learning models, arXiv:2112.04807.
  121. A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, The power of quantum neural networks, Nat. Comput. Sci. 1, 403 (2021).
  122. O. Berezniuk, A. Figalli, R. Ghigliazza, and K. Musaelian, A scale-dependent notion of effective dimension, arXiv:2001.10872.
  123. R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput. 16, 1190 (1995).
  124. M. J. Powell, Direct search algorithms for optimization calculations, Acta Numer. 7, 287 (1998).
  125. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  126. T. Hastie, R. Tibshirani, and J. H. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Vol. 2 (Springer, New York, 2009).

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Phys. Rev. Research 8, 023247– Published 5 June, 2026
  • Received 7 October 2025
  • Revised 2 March 2026
  • Accepted 9 May 2026
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