- Open Access
Learning the dynamics of Markovian open quantum systems from experimental data
Phys. Rev. Research 7, 033217 – Published 4 September, 2025
DOI: https://doi.org/10.1103/m66m-lnjh
Abstract
We present a Bayesian algorithm to identify explainable processes generating open quantum system dynamics, described by a Lindblad master equation, that are compatible with measured experimental data. The algorithm, based on a Markov chain Monte Carlo approach, assumes the energy levels of the system are known and outputs a ranked list of interpretable master-equation models that produce predicted measurement traces that closely match experimental data. We benchmark our algorithm on quantum optics experiments performed on single and pairs of quantum emitters. The latter case opens the possibility of cooperative emission effects and additional complexity due to the possible interplay between photon and phonon influences on the dynamics. Our algorithm retrieves various minimal models that are consistent with the experimental data, and which can provide a closer fit to measured data than previously suggested and physically expected approximate models. Our results represent an important step towards automated systems characterization with an approach that is capable of working with diverse and tomographically incomplete input data. This may help with the development of theoretical models for unknown quantum systems as well as providing scientists with alternative interpretations of the data that they might not have originally envisioned and enabling them to challenge their original hypotheses.
Physics Subject Headings (PhySH)
See Also
Learning-agent-based approach to the characterization of open quantum systems
Article Text
References (76)
- K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev, and A. Walsh, Machine learning for molecular and materials science, Nature (London) 559, 547 (2018).
- 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).
- S. M. Moosavi, K. M. Jablonka, and B. Smit, The role of machine learning in the understanding and design of materials, J. Am. Chem. Soc. 142, 20273 (2020).
- M. Krenn, R. Pollice, S. Y. Guo, M. Aldeghi, A. Cervera-Lierta, P. Friederich, G. dos Passos Gomes, F. Höse, A. Jinich, A. Nigam, Z. Yao, and A. Aspuru-Guzik, On scientific understanding with artificial intelligence, Nat. Rev. Phys. 4, 761 (2022).
- V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezzè, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023).
- M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Artificial intelligence and machine learning for quantum technologies, Phys. Rev. A 107, 010101 (2023).
- M. Usman, Y. Z. Wong, C. D. Hill, and L. C. L. Hollenberg, Framework for atomic-level characterisation of quantum computer arrays by machine learning, npj Comput. Mater. 6, 19 (2020).
- H. Moon, D. T. Lennon, J. Kirkpatrick, N. M. van Esbroeck, L. C. Camenzind, L. Yu, F. Vigneau, D. M. Zumbühl, G. A. D. Briggs, M. A. Osborne, D. Sejdinovic, E. A. Laird, and N. Ares, Machine learning enables completely automatic tuning of a quantum device faster than human experts, Nat. Commun. 11, 4161 (2020).
- K. Lee, S. Nam, H. Ji, J. Choi, J.-E. Jin, Y. Kim, J. Na, M.-Y. Ryu, Y.-H. Cho, H. Lee, J. Lee, M.-K. Joo, and G.-T. Kim, Multiple machine learning approach to characterize two-dimensional nanoelectronic devices via featurization of charge fluctuation, npj 2D Mater. Appl. 5, 1 (2021).
- A. Valenti, G. Jin, J. Léonard, S. D. Huber, and E. Greplova, Scalable Hamiltonian learning for large-scale out-of-equilibrium quantum dynamics, Phys. Rev. A 105, 023302 (2022).
- E. J. Thomas, V. N. Ciriano-Tejel, D. F. Wise, D. Prete, M. D. Kruijf, D. J. Ibberson, G. M. Noah, A. Gomez-Saiz, M. F. Gonzalez-Zalba, M. A. I. Johnson, and J. J. L. Morton, Rapid cryogenic characterization of 1,024 integrated silicon quantum dot devices, Nat. Electron. 8, 75 (2025).
- D. Craig, H. Moon, F. Fedele, D. Lennon, B. van Straaten, F. Vigneau, L. Camenzind, D. Zumbühl, G. Briggs, M. Osborne, D. Sejdinovic, and N. Ares, Bridging the reality gap in quantum devices with physics-aware machine learning, Phys. Rev. X 14, 011001 (2024).
- M. Schmidt and H. Lipson, Distilling free-form natural laws from experimental data, Science 324, 81 (2009).
- Z. Liu and M. Tegmark, Machine learning conservation laws from trajectories, Phys. Rev. Lett. 126, 180604 (2021).
- P. Lemos, N. Jeffrey, M. Cranmer, S. Ho, and P. Battaglia, Rediscovering orbital mechanics with machine learning, Mach. Learn.: Sci. Technol. 4, 045002 (2023).
- X. Chen, B. W. Soh, Z.-E. Ooi, E. Vissol-Gaudin, H. Yu, K. S. Novoselov, K. Hippalgaonkar, and Q. Li, Constructing custom thermodynamics using deep learning, Nat. Comput. Sci. 4, 66 (2024).
- G. Torlai and R. G. Melko, Learning thermodynamics with Boltzmann machines, Phys. Rev. B 94, 165134 (2016).
- E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
- J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nat. Mach. Intell. 1, 155 (2019).
- G. Torlai, B. Timar, E. P. van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuletić, M. D. Lukin, R. G. Melko, and M. Endres, Integrating neural networks with a quantum simulator for state reconstruction, Phys. Rev. Lett. 123, 230504 (2019).
- 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).
- A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nat. Phys. 17, 931 (2021).
- D. Flam-Shepherd, T. C. Wu, X. Gu, A. Cervera-Lierta, M. Krenn, and A. Aspuru-Guzik, Learning interpretable representations of entanglement in quantum optics experiments using deep generative models, Nat. Mach. Intell. 4, 544 (2022).
- S. Arlt, C. Ruiz-Gonzalez, and M. Krenn, Emulating multiparticle emitters with pair-sources: digital discovery of a quantum optics building block, Quantum Sci. Technol. 10, 015042 (2025).
- H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems., Science 377, 682 (2022).
- L. Fioroni, I. Rojkov, and F. Reiter, companion paper, Learning-agent-based approach to the characterization of open quantum systems, Phys. Rev. Appl. 24, 034011 (2025).
- H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, 1st ed. (Oxford University Press, Oxford, 2007).
- B. P. Lanyon, C. Maier, M. Holzöpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. Roos, Efficient tomography of a quantum many-body system, Nat. Phys. 13, 1158 (2017).
- C. Granade, J. Combes, and D. G. Cory, Practical Bayesian tomography, New J. Phys. 18, 033024 (2016).
- F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Non-Markovian quantum processes: Complete framework and efficient characterization, Phys. Rev. A 97, 012127 (2018).
- G. White, F. Pollock, L. Hollenberg, K. Modi, and C. Hill, Non-Markovian quantum process tomography, PRX Quantum 3, 020344 (2022).
- F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Operational Markov condition for quantum processes, Phys. Rev. Lett. 120, 040405 (2018).
- M. Cygorek and E. M. Gauger, Understanding and utilizing the inner bonds of process tensors, SciPost Phys. 18, 024 (2025).
- G. O. Samach, A. Greene, J. Borregaard, M. Christandl, J. Barreto, D. K. Kim, C. M. McNally, A. Melville, B. M. Niedzielski, Y. Sung, D. Rosenberg, M. E. Schwartz, J. L. Yoder, T. P. Orlando, J. I.-J. Wang, S. Gustavsson, M. Kjaergaard, and W. D. Oliver, Lindblad tomography of a superconducting quantum processor, Phys. Rev. Appl. 18, 064056 (2022).
- Z. X. Koong, M. Cygorek, D. Scerri, T. S. Santana, S. I. Park, J. D. Song, J. D. Song, E. M. Gauger, and B. D. Gerardot, Coherence in cooperative photon emission from indistinguishable quantum emitters, Sci. Adv. 8, eabm8171 (2022).
- V. Gorini, A. Kossakowski, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of -level systems, J. Math. Phys. 17, 821 (1976).
- G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
- S. Gammelmark and K. Mølmer, Bayesian parameter inference from continuously monitored quantum systems, Phys. Rev. A 87, 032115 (2013).
- M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution, Opt. Express 18, 20229 (2010).
- J.-H. Kim, S. Aghaeimeibodi, C. J. K. Richardson, R. P. Leavitt, and E. Waks, Super-radiant emission from quantum dots in a nanophotonic waveguide, Nano Lett. 18, 4734 (2018).
- D. Levonian, R. Riedinger, B. Machielse, E. Knall, M. Bhaskar, C. Knaut, R. Bekenstein, H. Park, M. Lončar, and M. Lukin, Optical entanglement of distinguishable quantum emitters, Phys. Rev. Lett. 128, 213602 (2021).
- R. E. K. Fishman, R. N. Patel, D. A. Hopper, T.-Y. Huang, and L. C. Bassett, Photon-emission-correlation spectroscopy as an analytical tool for solid-state quantum defects, PRX Quantum 4, 010202 (2023).
- M. Cygorek, E. D. Scerri, T. S. Santana, Z. X. Koong, B. D. Gerardot, and E. M. Gauger, Signatures of cooperative emission in photon coincidence: Superradiance versus measurement-induced cooperativity, Phys. Rev. A 107, 023718 (2023).
- D. M. Lukin, M. A. Guidry, J. Yang, M. Ghezellou, S. D. Mishra, H. Abe, T. Ohshima, J. Ul-Hassan, and J. Vučković, Two-emitter multimode cavity quantum electrodynamics in thin-film silicon carbide photonics, Phys. Rev. X 13, 011005 (2023).
- A. Tiranov, V. Angelopoulou, C. J. Van Diepen, B. Schrinski, O. A. D. Sandberg, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, Collective super- and subradiant dynamics between distant optical quantum emitters, Science 379, 389 (2023).
- L. Banchi, E. Grant, A. Rocchetto, and S. Severini, Modelling non-Markovian quantum processes with recurrent neural networks, New J. Phys. 20, 123030 (2018).
- M. J. Hartmann and G. Carleo, Neural-network approach to dissipative quantum many-body dynamics, Phys. Rev. Lett. 122, 250502 (2019).
- I. Luchnikov, S. Vintskevich, D. Grigoriev, and S. Filippov, Machine learning non-Markovian quantum dynamics, Phys. Rev. Lett. 124, 140502 (2020).
- 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).
- S. Krastanov, K. Head-Marsden, S. Zhou, S. T. Flammia, L. Jiang, and P. Narang, Unboxing quantum black box models: Learning non-Markovian dynamics, arXiv:2009.03902.
- F. Carnazza, F. Carollo, D. Zietlow, S. Andergassen, G. Martius, and I. Lesanovsky, Inferring Markovian quantum master equations of few-body observables in interacting spin chains, New J. Phys. 24, 073033 (2022).
- G. Cemin, F. Carnazza, S. Andergassen, G. Martius, F. Carollo, and I. Lesanovsky, Inferring interpretable dynamical generators of local quantum observables from projective measurements through machine learning, Phys. Rev. Appl. 21, L041001 (2024).
- M. Cranmer, A. Sanchez-Gonzalez, P. W. Battaglia, R. Xu, R. Xu, K. Cranmer, D. N. Spergel, and S. Ho, Discovering symbolic models from deep learning with inductive biases, Adv. Neural Inf. Process. Syst. 33, 17429 (2020).
- H. P. Nautrup, T. Metger, R. Iten, S. Jerbi, L. M. Trenkwalder, H. Wilming, H. J. Briegel, and R. Renner, Operationally meaningful representations of physical systems in neural networks, Mach. Learn.: Sci. Technol. 3, 045025 (2022).
- D. L. Craig, N. Ares, and E. M. Gauger, Differentiable master equation solver for quantum device characterization, Phys. Rev. Res. 6, 043175 (2024).
- J. Wiercinski, E. M. Gauger, and M. Cygorek, Phonon coupling versus pure dephasing in the photon statistics of cooperative emitters, Phys. Rev. Res. 5, 013176 (2023).
- A. Auffèves, D. Gerace, S. Portolan, A. Drezet, and M. F. Santos, Few emitters in a cavity: From cooperative emission to individualization, New J. Phys. 13, 093020 (2011).
- P. J. Green and P. H. Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika 82, 711 (1995).
- J. W. Rosenberg, M. Kuffer, I. Zohar, R. Stöhr, A. Denisenko, A. Zwick, G. A. Álvarez, and A. Finkler, Witnessing non-stationary and non-Markovian environments with a quantum sensor, arXiv:2501.05814.
- C. Bonato, M. S. Blok, H. T. Dinani, D. W. Berry, M. L. Markham, D. J. Twitchen, and R. Hanson, Optimized quantum sensing with a single electron spin using real-time adaptive measurements, Nat. Nanotechnol. 11, 247 (2016).
- M. J. Arshad, C. Bekker, B. Haylock, K. Skrzypczak, D. White, B. Griffiths, J. Gore, G. W. Morley, P. Salter, J. Smith, I. Zohar, A. Finkler, Y. Altmann, E. M. Gauger, and C. Bonato, Real-time adaptive estimation of decoherence timescales for a single qubit, Phys. Rev. Appl. 21, 024026 (2024).
- F. Berritta, J. A. Krzywda, J. Benestad, J. van der Heijden, F. Fedele, S. Fallahi, G. C. Gardner, M. J. Manfra, E. van Nieuwenburg, J. Danon, A. Chatterjee, and F. Kuemmeth, Physics-informed tracking of qubit fluctuations, Phys. Rev. Appl. 22, 014033 (2024).
- F. Belliardo, F. Zoratti, and V. Giovannetti, Application of machine learning to experimental design in quantum mechanics, Intl. J. Quantum Inf. 22, 2450002 (2024).
- L. Sarra and F. Marquardt, Deep Bayesian experimental design for quantum many-body systems, Mach. Learn.: Sci. Technol. 4, 045022 (2023).
- A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Quantum simulation of the Lindblad equation using a unitary decomposition of operators, Phys. Rev. Res. 4, 023216 (2022).
- M. Pocrnic, D. Segal, and N. Wiebe, Quantum simulation of Lindbladian dynamics via repeated interactions, arXiv:2312.05371.
- M. S. Moss, S. Ebadi, T. T. Wang, G. Semeghini, A. Bohrdt, M. D. Lukin, and R. G. Melko, Enhancing variational Monte Carlo simulations using a programmable quantum simulator, Phys. Rev. A 109, 032410 (2024).
- J. Luo, K. Lin, and X. Gao, Variational quantum simulation of Lindblad dynamics via quantum state diffusion, J. Phys. Chem. Lett. 15, 3516 (2024).
- Z. Ding, X. Li, and L. Lin, Simulating open quantum systems using Hamiltonian simulations, PRX Quantum 5, 020332 (2024).
- V. C. Olaya-Agudelo, B. Stewart, C. H. Valahu, R. J. MacDonell, M. J. Millican, V. G. Matsos, F. Scuccimarra, T. R. Tan, and I. Kassal, Simulating open-system molecular dynamics on analog quantum computers, Phys. Rev. Res. 7, 023215 (2025).
- R. Trivedi, A. Franco Rubio, and J. I. Cirac, Quantum advantage and stability to errors in analogue quantum simulators, Nat. Commun. 15, 6507 (2024).
- D. Layden, G. Mazzola, R. V. Mishmash, M. Motta, P. Wocjan, J.-S. Kim, and S. Sheldon, Quantum-enhanced Markov chain Monte Carlo, Nature (London) 619, 282 (2023).
- F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
- K. Pearson, LIII. On lines and planes of closest fit to systems of points in space, Philos. Mag. 2, 559 (1901).
- S. Lloyd, Least squares quantization in PCM, IEEE Trans. Inf. Theory 28, 129 (1982).
- D. J. Ketchen and C. L. Shook, The application of cluster analysis in strategic management research: An analysis and critique, Strategic Manage. J. 17, 441 (1996).