- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Unraveling quantum environments: Transformer-assisted learning in Lindblad dynamics

Chi-Sheng Chen1 and En-Jui Kuo1,2

PDF

Phys. Rev. A 112, 042227 – Published 28 October, 2025

DOI: https://doi.org/10.1103/gsxk-45mk

Abstract

Understanding dissipation in open quantum systems is crucial for the development of robust quantum technologies. In this work, we introduce a transformer-based machine learning framework to infer time-dependent dissipation rates in systems governed by the Lindblad master equation. Our method requires only time series of observable quantities, such as 〈σx(t)〉, 〈σy(t)〉, and 〈σz(t)〉, and does not require prior knowledge of the initial quantum state or the system Hamiltonian during inference. We demonstrate the effectiveness of our approach across a hierarchy of open quantum models, including single-qubit systems with time-independent and time-dependent jump rates, two-qubit interacting systems (e.g., Heisenberg and transverse Ising models), and the Jaynes-Cummings model with cavity loss and time-dependent decay. Our method accurately reconstructs both fixed and time-varying dissipation profiles from observable trajectories. We further show that under reasonable assumptions, the jump rates in these models are uniquely determined by a small set of measurable observables. The results suggest that transformer-based learning provides a scalable and data-driven route for identifying environmental effects in open quantum systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).
  2. Á. Rivas and S. F. Huelga, Open Quantum Systems, Vol. 10 (Springer, New York, 2012).
  3. M. S. Sarandy and D. A. Lidar, Adiabatic quantum computation in open systems, Phys. Rev. Lett. 95, 250503 (2005).
  4. L. Banchi, E. Grant, A. Rocchetto, and S. Severini, Modelling non-Markovian quantum processes with recurrent neural networks, New J. Phys. 20, 123030 (2018).
  5. K. Fujimoto, R. Hamazaki, and Y. Kawaguchi, Impact of dissipation on universal fluctuation dynamics in open quantum systems, Phys. Rev. Lett. 129, 110403 (2022).
  6. S. Yin, P. Mai, and F. Zhong, Nonequilibrium quantum criticality in open systems: The dissipation rate as an additional indispensable scaling variable, Phys. Rev. B 89, 094108 (2014).
  7. G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera, and H. Ott, Controlling the dynamics of an open many-body quantum system with localized dissipation, Phys. Rev. Lett. 110, 035302 (2013).
  8. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  9. J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Noise spectroscopy through dynamical decoupling with a superconducting flux qubit, Nat. Phys. 7, 565 (2011).
  10. J. Wang, L. Davidovich, and G. S. Agarwal, Quantum sensing of open systems: Estimation of damping constants and temperature, Phys. Rev. Res. 2, 033389 (2020).
  11. F. Wolf and P. O. Schmidt, Quantum sensing of oscillating electric fields with trapped ions, Measur. Sensors 18, 100271 (2021).
  12. A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, Attention is all you need, in Proceedings of the 31st International Conference on Neural Information Processing Systems (Curran Associates Inc., Red Hook, NY, USA, 2017), pp. 6000–6010.
  13. S. Li, X. Jin, Y. Xuan, X. Zhou, W. Chen, Y. Wang, and X. Yan, Enhancing the locality and breaking the memory bottleneck of transformer on time series forecasting, in Proceedings of the 33rd International Conference on Neural Information Processing Systems (Curran Associates Inc., Red Hook, NY, USA, 2019), pp. 5243–5253.
  14. G. Zerveas, S. Jayaraman, D. Patel, A. Bhamidipaty, and C. Eickhoff, A transformer-based framework for multivariate time series representation learning, in Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining (KDD '21) (unpublished).
  15. B. Lim and S. Zohren, Nicolas Loeff, and Tomas Pfister. Temporal fusion transformers” for interpretable multi-horizon time series forecasting, Intl. J. Forecast. 37, 1748 (2021).
  16. H. Qiao, Y. P. Kandel, K. Deng, S. Fallahi, G. C. Gardner, M. J. Manfra, E. Barnes, and J. M. Nichol, Coherent multispin exchange coupling in a quantum-dot spin chain, Phys. Rev. X 10, 031006 (2020).
  17. Z. Zhou, Y. Li, Z. Wu, X. Ma, S. Fan, and S. Huang, The exchange interaction between neighboring quantum dots: Physics and applications in quantum information processing, J. Semicond. 45, 101701 (2024).
  18. X. Hu and S. Das Sarma, Overview of spin-based quantum dot quantum computation, Phys. Status Solidi B 238, 360 (2003).
  19. K. W. Chan, H. Sahasrabudhe, W. Huang, Y. Wang, H. C. Yang, M. Veldhorst, J. C. C. Hwang, F. A. Mohiyaddin, F. E. Hudson, K. M. Itoh, and A. Morello, Exchange coupling in a linear chain of three quantum-dot spin qubits in silicon, Nano Lett. 21, 1517 (2021).
  20. Y. P. Kandel, H. Qiao, S. Fallahi, G. C. Gardner, M. J. Manfra, and J. M. Nichol, Coherent spin-state transfer via Heisenberg exchange, Nature (London) 573, 553 (2019).
  21. B. W. Shore and P. L. Knight, The Jaynes-Cummings model, J. Mod. Opt. 40, 1195 (1993).
  22. A. R. Morgillo, M. F. Sacchi, and C. Macchiavello, Detecting markovianity of quantum processes via recurrent neural networks, Quantum Mach. Intell. 7, 47 (2025).
  23. S. Martina, S. Gherardini, and F. Caruso, Machine learning classification of non-markovian noise disturbing quantum dynamics, Physica Scr. 98, 035104 (2023).
  24. P. P. Mazza, D. Zietlow, F. Carollo, S. Andergassen, G. Martius, and I. Lesanovsky, Machine learning time-local generators of open quantum dynamics, Phys. Rev. Res. 3, 023084 (2021).
  25. C. Guo, K. Modi, and D. Poletti, Tensor-network-based machine learning of non-Markovian quantum processes, Phys. Rev. A 102, 062414 (2020).
  26. I. A. Luchnikov, E. O. Kiktenko, M. A. Gavreev, H. Ouerdane, S. N. Filippov, and A. K. Fedorov, Probing non-Markovian quantum dynamics with data-driven analysis: Beyond “black-box” machine-learning models, Phys. Rev. Res. 4, 043002 (2022).
  27. E. Bairey, C. Guo, D. Poletti, N. H. Lindner, and I. Arad, Learning the dynamics of open quantum systems from their steady states, New J. Phys. 22, 032001 (2020).
  28. 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).
  29. S. J. Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders, Phys. Rev. E 96, 022140 (2017).
  30. E. P. L. Van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
  31. E.-J. Kuo, A. Seif, R. Lundgren, S. Whitsitt, and M. Hafezi, Decoding conformal field theories: From supervised to unsupervised learning, Phys. Rev. Res. 4, 043031 (2022).
  32. E.-J. Kuo and H. Dehghani, Unsupervised learning of interacting topological and symmetry-breaking phase transitions, Phys. Rev. B 105, 235136 (2022).
  33. E. van Nieuwenburg, E. Bairey, and G. Refael, Learning phase transitions from dynamics, Phys. Rev. B 98, 060301(R) (2018).
  34. F. Schindler, N. Regnault, and T. Neupert, Probing many-body localization with neural networks, Phys. Rev. B 95, 245134 (2017).
  35. J. Carifio, J. Halverson, D. Krioukov, and B. D. Nelson, Machine learning in the string landscape, J. High Energy Phys. 09 (2017) 157.
  36. K. Hashimoto, S. Sugishita, A. Tanaka, and A. Tomiya, Deep learning and the AdS/CFT correspondence, Phys. Rev. D 98, 046019 (2018).
  37. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  38. G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Neural-network quantum state tomography, Nat. Phys. 14, 447 (2018).
  39. J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nat. Mach. Intell. 1, 155 (2019).
  40. G. Torlai, B. Timar, E. P. L. Van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuletić, M. D. Lukin et al., Integrating neural networks with a quantum simulator for state reconstruction, Phys. Rev. Lett. 123, 230504 (2019).
  41. C.-Y. Liu, E.-J. Kuo, C.-H. A. Lin, S. Chen, J. G. Young, Y.-J. Chang, and M.-H. Hsieh, Training classical neural networks by quantum machine learning, in IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. 2 (IEEE, 2024), pp. 34–38.
  42. C.-Y. Liu, E.-J. Kuo, C.-H. A. Lin, J. G. Young, Y.-J. Chang, M.-H. Hsieh, and H.-S. Goan, Quantum-train: Rethinking hybrid quantum-classical machine learning in the model compression perspective, Quantum Mach. Intell. 7, 80 (2025).
  43. S.-T. Tsai, E.-J. Kuo, and P. Tiwary, Learning molecular dynamics with simple language model built upon long short-term memory neural network, Nat. Commun. 11, 5115 (2020).
  44. S.-T. Tsai, E. Fields, Y. Xu, E.-J. Kuo, and P. Tiwary, Path sampling of recurrent neural networks by incorporating known physics, Nat. Commun. 13, 7231 (2022).
  45. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Nature (London) 323, 533 (1986).
  46. K. Cho, B. van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, Learning phrase representations using RNN encoder-decoder for statistical machine translation, arXiv:1406.1078.
  47. S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput. 9, 1735 (1997).
  48. J. Kaplan, S. McCandlish, T. Henighan et al., Scaling laws for neural language models, arXiv:2001.08361.
  49. Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, and A. Anandkumar, Physics-informed neural operator for learning partial differential equations, ACM/IMS J. Data Science 1, 1 (2024).
  50. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  51. P. Pearle, Simple derivation of the Lindblad equation, Eur. J. Phys. 33, 805 (2012).
  52. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2002).
  53. J. R. Johansson, P. D. Nation, and F. Nori, Qutip: An open-source python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  54. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011).
  55. R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, and C. Monroe, Onset of a quantum phase transition with a trapped ion quantum simulator, Nat. Commun. 2, 377 (2011).
  56. J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (London) 551, 601 (2017).
  57. I. Georgescu, The Divincenzo criteria 20 years on, Nat. Rev. Phys. 2, 666 (2020).
  58. E.-J. Kuo and C.-S. Chen, Data for “Unraveling quantum environments: Transformer-assisted learning in Lindblad dynamics” (v1.0) [Data set], Zenodo (2025), doi:10.5281/zenodo.17139233.
  59. M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21, 237 (1948).
  60. R. O'Donnell and J. Wright, Efficient quantum tomography, in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2016), pp. 899–912.
  61. M.-C. Hsu, E.-J. Kuo, W.-H. Yu, J.-F. Cai, and M.-H. Hsieh, Quantum state tomography via nonconvex Riemannian gradient descent, Phys. Rev. Lett. 132, 240804 (2024).

Outline

Information

Phys. Rev. A 112, 042227– Published 28 October, 2025
  • Received 13 May 2025
  • Revised 4 August 2025
  • Accepted 16 September 2025
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