- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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
  • Invited
  • Open Access

Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information Matrix

Hyukgun Kwon1,*, Seok Hyung Lie2,†, and Liang Jiang3,‡

  • *Contact author: kwon37hg@sejong.ac.kr
  • †Contact author: seokhyung@unist.ac.kr
  • ‡Contact author: liangjiang@uchicago.edu
PDF

PRX Quantum 7, 033063 – Published 22 September, 2026

DOI: https://doi.org/10.1103/8k5v-ddtw

Abstract

In this work, we show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix, differing only by a logarithmic factor in the number of parameters to be estimated. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Characterizing quantum systems is essential for the development of quantum science and technology. In this context, quantum learning theory aims to estimate parameters of quantum systems within a desired accuracy and with high confidence while using as few experimental repetitions as possible, a cost known as the sample complexity. Despite its importance, a systematic framework for determining the sample complexity in general learning problems has not yet been established. In this work, we develop such a framework by deriving upper and lower bounds on the sample complexity for general parametric quantum learning settings governed by the Fisher information matrix. Our results address the important open problem of establishing task-independent sample complexity bounds, thereby providing a unified analytical framework for quantum learning protocols. This framework provides a crucial foundation for tasks in quantum information science that rely on characterizing quantum systems, including hardware benchmarking, noise modeling, and quantum error correction.

Article Text

References (75)

  1. A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Characterizing large-scale quantum computers via cycle benchmarking, Nat. Commun. 10, 5347 (2019).
  2. A. Hashim, L. B. Nguyen, N. Goss, B. Marinelli, R. K. Naik, T. Chistolini, J. Hines, J. P. Marceaux, Y. Kim, P. Gokhale, T. Tomesh, S. Chen, L. Jiang, S. Ferracin, K. Rudinger, T. Proctor, K. C. Young, I. Siddiqi, and R. Blume-Kohout, Practical introduction to benchmarking and characterization of quantum computers, PRX Quantum 6, 030202 (2025).
  3. R. Harper, S. T. Flammia, and J. J. Wallman, Efficient learning of quantum noise, Nat. Phys. 16, 1184 (2020).
  4. S. Chen, Z. Zhang, L. Jiang, and S. T. Flammia, Efficient self-consistent learning of gate set Pauli noise, PRX Quantum 7, 010305 (2026).
  5. D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultrahigh error threshold for surface codes with biased noise, Phys. Rev. Lett. 120, 050505 (2018).
  6. E. H. Chen, T. J. Yoder, Y. Kim, N. Sundaresan, S. Srinivasan, M. Li, A. D. Córcoles, A. W. Cross, and M. Takita, Calibrated decoders for experimental quantum error correction, Phys. Rev. Lett. 128, 110504 (2022).
  7. E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Phys. 19, 1116 (2023).
  8. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  9. Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  10. K. Tsubouchi, H. Kwon, L. Jiang, and N. Yoshioka, Quantum advantages for syndrome-aware noisy logical observable estimation, arXiv:2603.05145.
  11. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  12. S. Chen, J. Cotler, H.-Y. Huang, and J. Li, Exponential separations between learning with and without quantum memory, in Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022), pp. 574–585.
  13. J. Haah, R. Kothari, R. O’Donnell, and E. Tang, Query-optimal estimation of unitary channels in diamond distance, in Proceedings of the 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2023), pp. 363–390.
  14. H.-Y. Hu, M. Ma, W. Gong, Q. Ye, Y. Tong, S. T. Flammia, and S. F. Yelin, Ansatz-free Hamiltonian learning with Heisenberg-limited scaling, PRX Quantum 6, 040315 (2025).
  15. H.-Y. Huang, Y. Tong, D. Fang, and Y. Su, Learning many-body Hamiltonians with Heisenberg-limited scaling, Phys. Rev. Lett. 130, 200403 (2023).
  16. Y. Wu, Y. Zhang, C. Wang, and X. Yuan, Hamiltonian dynamics learning: A scalable approach to quantum process characterization, arXiv:2503.24171.
  17. H.-Y. Huang, R. Kueng, and J. Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021).
  18. Z.-H. Liu, R. Brunel, E. E. B. Østergaard, O. Cordero, S. Chen, Y. Wong, J. A. H. Nielsen, A. B. Bregnsbo, S. Zhou, H.-Y. Huang, C. Oh, L. Jiang, J. Preskill, J. S. Neergaard-Nielsen, and U. L. Andersen, Quantum learning advantage on a scalable photonic platform, Science 389, 1332 (2025).
  19. S. Chen, S. Zhou, A. Seif, and L. Jiang, Quantum advantages for Pauli channel estimation, Phys. Rev. A 105, 032435 (2022).
  20. M. Subramanian, H. Kwon, and L. Jiang, Quantum channel learning with limited parallel access, arXiv:2608.05307.
  21. H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Quantum advantage in learning from experiments, Science 376, 1182 (2022).
  22. S. Chen, W. Gong, and Q. Ye, Optimal tradeoffs for estimating Pauli observables, in Proceedings of the 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2024), pp. 1086–1105.
  23. E. Coroi and C. Oh, Exponential advantage in continuous-variable quantum state learning, arXiv:2501.17633.
  24. M. C. Caro, Learning quantum processes and Hamiltonians via the Pauli transfer matrix, ACM Trans. Quantum Comput. 5, 1 (2024).
  25. S. Chen, C. Oh, S. Zhou, H.-Y. Huang, and L. Jiang, Tight bounds on Pauli channel learning without entanglement, Phys. Rev. Lett. 132, 180805 (2024).
  26. A. Seif, S. Chen, S. Majumder, H. Liao, D. S. Wang, M. Malekakhlagh, A. Javadi-Abhari, L. Jiang, and Z. K. Minev, Entanglement-enhanced learning of quantum processes at scale, arXiv:2408.03376.
  27. M. Kim and C. Oh, On the fundamental resource for exponential advantage in quantum channel learning, arXiv:2507.11089.
  28. C. Oh, S. Chen, Y. Wong, S. Zhou, H.-Y. Huang, J. A. H. Nielsen, Z.-H. Liu, J. S. Neergaard-Nielsen, U. L. Andersen, L. Jiang, and J. Preskill, Entanglement-enabled advantage for learning a bosonic random displacement channel, Phys. Rev. Lett. 133, 230604 (2024).
  29. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  30. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  31. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photon. 5, 222 (2011).
  32. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Quantum circuit architecture, Phys. Rev. Lett. 101, 060401 (2008).
  33. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009).
  34. G. Chiribella, Optimal networks for quantum metrology: Semidefinite programs and product rules, New J. Phys. 14, 125008 (2012).
  35. Y. Yang, Memory effects in quantum metrology, Phys. Rev. Lett. 123, 110501 (2019).
  36. A. Altherr and Y. Yang, Quantum metrology for non-Markovian processes, Phys. Rev. Lett. 127, 060501 (2021).
  37. S. Kurdziałek, P. Dulian, J. Majsak, S. Chakraborty, and R. Demkowicz-Dobrzański, Quantum metrology using quantum combs and tensor network formalism, New J. Phys. 27, 013019 (2025).
  38. T. J. Proctor, P. A. Knott, and J. A. Dunningham, Multiparameter estimation in networked quantum sensors, Phys. Rev. Lett. 120, 080501 (2018).
  39. M. Gessner, L. Pezzè, and A. Smerzi, Sensitivity bounds for multiparameter quantum metrology, Phys. Rev. Lett. 121, 130503 (2018).
  40. F. Albarelli, J. F. Friel, and A. Datta, Evaluating the Holevo Cramér-Rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).
  41. C. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  42. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  43. M. G. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 7, 125 (2009).
  44. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A 53, 023001 (2020).
  45. S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).
  46. J. S. Sidhu, Y. Ouyang, E. T. Campbell, and P. Kok, Tight bounds on the simultaneous estimation of incompatible parameters, Phys. Rev. X 11, 011028 (2021).
  47. F. Albarelli and R. Demkowicz-Dobrzański, Probe incompatibility in multiparameter noisy quantum metrology, Phys. Rev. X 12, 011039 (2022).
  48. Y. Watanabe, T. Sagawa, and M. Ueda, Optimal measurement on noisy quantum systems, Phys. Rev. Lett. 104, 020401 (2010).
  49. R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5, 329 (1996).
  50. F. W. Olver, NIST Handbook of Mathematical Functions Hardback and CD-ROM (Cambridge university press, Cambridge, England, 2010).
  51. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, England, 1985).
  52. D.-J. Zhang and D. M. Tong, Inferring physical properties of symmetric states from the fewest copies, Phys. Rev. Lett. 133, 040202 (2024).
  53. S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achieving the Heisenberg limit in quantum metrology using quantum error correction, Nat. Commun. 9, 78 (2018).
  54. W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, Improved quantum metrology using quantum error correction, Phys. Rev. Lett. 112, 080801 (2014).
  55. G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Increasing sensing resolution with error correction, Phys. Rev. Lett. 112, 150801 (2014).
  56. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Quantum error correction for metrology, Phys. Rev. Lett. 112, 150802 (2014).
  57. D. Layden, S. Zhou, P. Cappellaro, and L. Jiang, Ancilla-free quantum error correction codes for quantum metrology, Phys. Rev. Lett. 122, 040502 (2019).
  58. R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general Markovian noise, Phys. Rev. X 7, 041009 (2017).
  59. S. Zhou and L. Jiang, Asymptotic theory of quantum channel estimation, PRX Quantum 2, 010343 (2021).
  60. I. Rojkov, D. Layden, P. Cappellaro, J. Home, and F. Reiter, Bias in error-corrected quantum sensing, Phys. Rev. Lett. 128, 140503 (2022).
  61. Q. Zhuang, J. Preskill, and L. Jiang, Distributed quantum sensing enhanced by continuous-variable error correction, New J. Phys. 22, 022001 (2020).
  62. H. Kwon, U. R. Fischer, S.-W. Lee, and L. Jiang, Restoring Heisenberg scaling in time via autonomous quantum error correction, arXiv:2504.13168.
  63. H. Kwon, K. Tsubouchi, C.-T. Chu, and L. Jiang, Criteria for unbiased estimation: Applications to noise-agnostic sensing and learnability of quantum channel, arXiv:2503.17362.
  64. M. Tsang, F. Albarelli, and A. Datta, Quantum semiparametric estimation, Phys. Rev. X 10, 031023 (2020).
  65. S. Chen, W. Gong, and S. Zhou, Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach, arXiv:2602.04952.
  66. A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Am. Math. Soc. 49, 122 (1941).
  67. R. Durrett, Probability: Theory and Examples (Cambridge university press, Cambridge, England, 2019), Vol. 49.
  68. W. Feller, An Introduction to Probability Theory and its Applications (John Wiley & Sons, New York, 1991), Vol. 2.
  69. Z. W. Birnbaum, An inequality for mill’s ratio, Ann. Math. Stat. 13, 245 (1942).
  70. R. D. Gordon, Values of mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument, Ann. Math. Stat. 12, 364 (1941).
  71. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, Oxford, 2001).
  72. J. P. Mills, Table of the ratio: Area to bounding ordinate, for any portion of normal curve, Biometrika 18, 395 (1926).
  73. V. Spokoiny, Parametric estimation. Finite sample theory, Ann. Stat. 40, 2877 (2012).
  74. A. W. Van der Vaart, Asymptotic Statistics (Cambridge university press, Cambridge, England, 2000), Vol. 3.
  75. I. A. Ibragimov and R. Z. Has’ Minskii, Statistical Estimation: Asymptotic Theory (Springer Science & Business Media, New York, 2013).

Outline

Information

PRX Quantum 7, 033063– Published 22 September, 2026
  • Received 9 March 2026
  • Revised 27 May 2026
  • Accepted 10 August 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