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Power and limitations of learning quantum dynamics incoherently

Sofiene Jerbi1,2,3,*, Joe Gibbs4,5, Manuel S. Rudolph6, Matthias C. Caro7,3, Patrick J. Coles1,8, Hsin-Yuan Huang7,9, and Zoë Holmes1,6

  • *Contact author: sofiene.jerbi@fu-berlin.de
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Phys. Rev. Research 8, 023141 – Published 11 May, 2026

DOI: https://doi.org/10.1103/s5xy-lk91

Abstract

Quantum process learning is emerging as an important tool to study quantum systems. While studied extensively in coherent frameworks, where the target and model system can share quantum information, less attention has been paid to whether the dynamics of quantum systems can be learned without the system and target directly interacting. Such incoherent frameworks are practically appealing since they open up methods of transpiling quantum processes between the different physical platforms without the need for technically challenging hybrid-entanglement schemes. Here, we provide bounds on the sample complexity of learning unitary processes incoherently by analyzing the number of measurements that are required to emulate well-established coherent learning strategies. We prove that if arbitrary measurements are allowed, then any efficiently representable unitary can be efficiently learned within the incoherent framework; however, when restricted to shallow-depth measurements, logarithmic depth unitaries can be learned but there exist linear-depth unitaries that are unlearnable. We demonstrate our incoherent learning algorithm for low-entangling unitaries by successfully learning a 16-qubit unitary on ibmq_kolkata, and further demonstrate the scalability of our proposed algorithm through extensive numerical experiments.

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References (60)

  1. A. Bisio, G. Chiribella, G. M. D’Ariano, S. Facchini, and P. Perinotti, Optimal quantum learning of a unitary transformation, Phys. Rev. A 81, 032324 (2010).
  2. K. Poland, K. Beer, and T. J. Osborne, No free lunch for quantum machine learning, arXiv:2003.14103.
  3. K. Sharma, M. Cerezo, Z. Holmes, L. Cincio, A. Sornborger, and P. J. Coles, Reformulation of the no-free-lunch theorem for entangled datasets, Phys. Rev. Lett. 128, 070501 (2022).
  4. S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, Quantum-assisted quantum compiling, Quantum 3, 140 (2019).
  5. T. Jones and S. C. Benjamin, Robust quantum compilation and circuit optimisation via energy minimisation, Quantum 6, 628 (2022).
  6. K. Heya, Y. Suzuki, Y. Nakamura, and K. Fujii, Variational quantum gate optimization, arXiv:1810.12745.
  7. C. Crstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time, npj Quantum Inf. 6, 82 (2020).
  8. J. Gibbs, K. Gili, Z. Holmes, B. Commeau, A. Arrasmith, L. Cincio, P. J. Coles, and A. Sornborger, Long-time simulations for fixed input states on quantum hardware, npj Quantum Inf. 8, 135 (2022).
  9. J. Gibbs, Z. Holmes, M. C. Caro, N. Ezzell, H.-Y. Huang, L. Cincio, A. T. Sornborger, and P. J. Coles, Dynamical simulation via quantum machine learning with provable generalization, Phys. Rev. Res. 6, 013241 (2024).
  10. H.-Y. Huang, R. Kueng, and J. Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021).
  11. 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).
  12. 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).
  13. M. C. Caro, H.-Y. Huang, N. Ezzell, J. Gibbs, A. T. Sornborger, L. Cincio, P. J. Coles, and Z. Holmes, Out-of-distribution generalization for learning quantum dynamics, Nat. Commun. 14, 3751 (2023).
  14. H.-Y. Huang, S. Chen, and J. Preskill, Learning to predict arbitrary quantum processes, PRX Quantum 4, 040337 (2023).
  15. M. C. Caro, Learning quantum processes and Hamiltonians via the Pauli transfer matrix, ACM Trans. Quantum Comput. 5, 1 (2024).
  16. H. Cai, Q. Ye, and D.-L. Deng, Sample complexity of learning parametric quantum circuits, Quantum Sci. Technol. 7, 025014 (2022).
  17. Y. Mao, M. Gong, K. Nemoto, W. J. Munro, and J. Majer, Perspective on witnessing entanglement in hybrid quantum systems, Appl. Phys. Lett. 119, 110501 (2021).
  18. E. Waks and C. Monroe, Protocol for hybrid entanglement between a trapped atom and a quantum dot, Phys. Rev. A 80, 062330 (2009).
  19. C. Eichler, C. Lang, J. M. Fink, J. Govenius, S. Filipp, and A. Wallraff, Observation of entanglement between itinerant microwave photons and a superconducting qubit, Phys. Rev. Lett. 109, 240501 (2012).
  20. A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, Tunable ion–photon entanglement in an optical cavity, Nature (London) 485, 482 (2012).
  21. A. N. Craddock, J. Hannegan, D. P. Ornelas-Huerta, J. D. Siverns, A. J. Hachtel, E. A. Goldschmidt, J. V. Porto, Q. Quraishi, and S. L. Rolston, Quantum interference between photons from an atomic ensemble and a remote atomic ion, Phys. Rev. Lett. 123, 213601 (2019).
  22. P. Scarlino, D. J. Van Woerkom, U. C. Mendes, J. V. Koski, A. J. Landig, C. K. Andersen, S. Gasparinetti, C. Reichl, W. Wegscheider, K. Ensslin, et al., Coherent microwave-photon-mediated coupling between a semiconductor and a superconducting qubit, Nat. Commun. 10, 3011 (2019).
  23. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  24. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  25. A. Anshu, Z. Landau, and Y. Liu, Distributed quantum inner product estimation, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing Machinery, New York, 2022), pp. 44–51.
  26. J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, and M. G. Thompson, Experimental quantum Hamiltonian learning, Nat. Phys. 13, 551 (2017).
  27. N. Wiebe, C. Granade, C. Ferrie, and D. Cory, Quantum Hamiltonian learning using imperfect quantum resources, Phys. Rev. A 89, 042314 (2014).
  28. A. A. Gentile, B. Flynn, S. Knauer, N. Wiebe, S. Paesani, C. E. Granade, J. G. Rarity, R. Santagati, and A. Laing, Learning models of quantum systems from experiments, Nat. Phys. 17, 837 (2021).
  29. F. Wilde, A. Kshetrimayum, I. Roth, D. Hangleiter, R. Sweke, and J. Eisert, Scalably learning quantum many-body Hamiltonians from dynamical data, arXiv:2209.14328.
  30. V. Gebhart, R. Santagati, A. A. Gentile, E. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezze, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023).
  31. D. Stilck França, L. A. Markovich, V. Dobrovitski, A. H. Werner, and J. Borregaard, Efficient and robust estimation of many-qubit Hamiltonians, Nat. Commun. 15, 311 (2024).
  32. J. Haah, R. Kothari, and E. Tang, Optimal learning of quantum Hamiltonians from high-temperature Gibbs states, Nat. Phys. 20, 1027 (2024).
  33. A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nat. Phys. 17, 931 (2021).
  34. L. Pastori, T. Olsacher, C. Kokail, and P. Zoller, Characterization and verification of Trotterized digital quantum simulation via Hamiltonian and Liouvillian learning, PRX Quantum 3, 030324 (2022).
  35. D. Hangleiter, I. Roth, J. Fuksa, J. Eisert, and P. Roushan, Robustly learning the Hamiltonian dynamics of a superconducting quantum processor, Nat. Commun. 15, 9595 (2024).
  36. D. K. Mark, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Benchmarking quantum simulators using ergodic quantum dynamics, Phys. Rev. Lett. 131, 110601 (2023).
  37. M. Fanizza, Y. Quek, and M. Rosati, Learning quantum processes without input control, PRX Quantum 5, 020367 (2024).
  38. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  39. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  40. 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).
  41. We note that a similar argument was made in passing in Ref. [13], but the argument was not formalized and its implications for incoherent learning were not appreciated.
  42. D. Maslov and M. Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, IEEE Trans. Inform. Theory 64, 4729 (2018).
  43. One way around this would be to build prior knowledge about the target unitary into the guess ansatz. It has been shown, for example, that highly symmetrized ansätze can avoid globality-induced barren plateaus [60]. Nonetheless, such approaches are unlikely to work in all cases.
  44. M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Practical characterization of quantum devices without tomography, Phys. Rev. Lett. 107, 210404 (2011).
  45. A. Basheer, Y. Feng, C. Ferrie, and S. Li, Alternating layered variational quantum circuits can be classically optimized efficiently using classical shadows, in Proceedings of the AAAI Conference on Artificial Intelligence (AAAI Press, Washington, DC, 2023), Vol. 37, pp. 6770–6778.
  46. M. Cerezo, M. Larocca, D. Garcia-Martin, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, et al., Does provable absence of barren plateaus imply classical simulability, Nat. Commun. 16, 7907 (2025).
  47. 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).
  48. Random product states enjoy the same out-of-distribution generalization guarantees as products of random single-qubit stabilizer states [13].
  49. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: Composable transformations of Python + NumPy programs, GitHub (2018), https://github.com/jax-ml/jax.
  50. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  51. K. Mizuta, Y. O. Nakagawa, K. Mitarai, and K. Fujii, Local variational quantum compilation of large-scale Hamiltonian dynamics, PRX Quantum 3, 040302 (2022).
  52. E. Younis, K. Sen, K. Yelick, and C. Iancu, QFAST: Conflating search and numerical optimization for scalable quantum circuit synthesis, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2021), p. 232.
  53. T. Patel, E. Younis, C. Iancu, W. de Jong, and D. Tiwari, QUEST: Systematically approximating quantum circuits for higher output fidelity, in Proceedings of the 27th ACM International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS’22) (Association for Computing Machinery, New York, 2022), p. 514.
  54. F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Phys. Rev. A 69, 032315 (2004).
  55. M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, 2017).
  56. H. Zhao, L. Lewis, I. Kannan, Y. Quek, H.-Y. Huang, and M. C. Caro, Learning quantum states and unitaries of bounded gate complexity, PRX Quantum 5, 040306 (2024).
  57. B. Yu, Assouad, Fano, and Le Cam, in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, edited by P. David, T. Erik, and L. Y. Grace (Springer, New York, 1997), p. 423.
  58. M. StackExchange, Total variation inequality for the product measure (2011), https://math.stackexchange.com/questions/72315/total-variation-inequality-for-the-product-measure.
  59. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energy Phys. 04 (2017) 121.
  60. 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).

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Phys. Rev. Research 8, 023141– Published 11 May, 2026
  • Received 29 June 2023
  • Accepted 5 February 2026
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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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