quantum-computing

Learning thermodynamic master equations for open quantum systems

Peter Sentz, Stanley Nicholson, Yujin Cho, Sohail Reddy, Brendan Keith, and Stefanie Günther

19 min read

0 likes

⚡ Quantum Brief

AI Audio Summary

0:00 / 0:00

Click to play

Quantum News · Media Library

Understand this faster with AI

Ask ChatGPT

Research with Perplexity

Explain with Claude

Discuss on Grok

AbstractThe characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.► BibTeX data@article{Sentz2026learning, doi = {10.22331/q-2026-07-01-2151}, url = {https://doi.org/10.22331/q-2026-07-01-2151}, title = {Learning thermodynamic master equations for open quantum systems}, author = {Sentz, Peter and Nicholson, Stanley and Cho, Yujin and Reddy, Sohail and Keith, Brendan and G{\"{u}}nther, Stefanie}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2151}, month = jul, year = {2026} }► References [1] Antonio D. Córcoles, Abhinav Kandala, Ali Javadi-Abhari, Douglas T. McClure, Andrew W. Cross, Kristan Temme, et al. ``Challenges and opportunities of near-term quantum computing systems''. Proceedings of the IEEE 108, 1338–1352 (2020). https://doi.org/10.1109/JPROC.2019.2954005 [2] David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, and Steven T. Flammia. ``Tailoring surface codes for highly biased noise''. Physical Review X 9, 041031 (2019). https://doi.org/10.1103/PhysRevX.9.041031 [3] Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M. Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, and Cristian Bonato. ``Learning quantum systems''.

Nature Reviews Physics 5, 141–156 (2023). https://doi.org/10.1038/s42254-022-00552-1 [4] Charles H. Baldwin, Amir Kalev, and Ivan H. Deutsch. ``Quantum process tomography of unitary and near-unitary maps''. Physical Review A 90, 012110 (2014). https://doi.org/10.1103/PhysRevA.90.012110 [5] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''.

Cambridge University Press. (2010). 10th Anniversary edition. https://doi.org/10.1017/CBO9780511976667 [6] Javier Cerrillo and Jianshu Cao. ``Non-Markovian dynamical maps: Numerical processing of open quantum trajectories''.

Physical Review Letters 112, 110401 (2014). https://doi.org/10.1103/PhysRevLett.112.110401 [7] Gabriel O. Samach, Ami Greene, Johannes Borregaard, Matthias Christandl, Joseph Barreto, David K. Kim, Christopher M. McNally, Alexander Melville, Bethany M. Niedzielski, et al. ``Lindblad tomography of a superconducting quantum processor''.

Physical Review Applied 18, 064056 (2022). https://doi.org/10.1103/PhysRevApplied.18.064056 [8] Hsin-Yuan Huang, Sitan Chen, and John Preskill. ``Learning to predict arbitrary quantum processes''. PRX Quantum 4, 040337 (2023). https://doi.org/10.1103/PRXQuantum.4.040337 [9] Dominik Hangleiter, Ingo Roth, Jonáš Fuksa, Jens Eisert, and Pedram Roushan. ``Robustly learning the Hamiltonian dynamics of a superconducting quantum processor''. Nature Communications 15, 9595 (2024). https://doi.org/10.1038/s41467-024-52629-3 [10] Naeimeh Mohseni, Junheng Shi, Tim Byrnes, and Michael J. Hartmann. ``Deep learning of many-body observables and quantum information scrambling''. Quantum 8, 1417 (2024). https://doi.org/10.22331/q-2024-07-18-1417 [11] Yan Zhu, Ya-Dong Wu, Qiushi Liu, Yuexuan Wang, and Giulio Chiribella. ``Quantum process learning through neural emulation'' (2023). arXiv:2308.08815. arXiv:2308.08815 [12] Laura Lewis, Hsin-Yuan Huang, Viet T. Tran, Sebastian Lehner, Richard Kueng, and John Preskill. ``Improved machine learning algorithm for predicting ground state properties''. Nature Communications 15, 895 (2024). https://doi.org/10.1038/s41467-024-45014-7 [13] Zidu Liu, L.-M. Duan, and Dong-Ling Deng. ``Solving quantum master equations with deep quantum neural networks''.

Physical Review Research 4, 013097 (2022). https://doi.org/10.1103/PhysRevResearch.4.013097 [14] Michael J. Hartmann and Giuseppe Carleo. ``Neural-network approach to dissipative quantum many-body dynamics''.

Physical Review Letters 122, 250502 (2019). https://doi.org/10.1103/PhysRevLett.122.250502 [15] Nima Leclerc. ``Predicting dynamics of transmon qubit-cavity systems with recurrent neural networks'' (2021). arXiv:2109.14471. arXiv:2109.14471 [16] E. Flurin, L. S. Martin, S. Hacohen-Gourgy, and I. Siddiqi. ``Using a recurrent neural network to reconstruct quantum dynamics of a superconducting qubit from physical observations''. Physical Review X 10, 011006 (2020). https://doi.org/10.1103/PhysRevX.10.011006 [17] Akram Youssry, Gerardo A. Paz-Silva, and Christopher Ferrie. ``Characterization and control of open quantum systems beyond quantum noise spectroscopy''. npj Quantum Information 6, 95 (2020). https://doi.org/10.1038/s41534-020-00332-8 [18] Zhilu Lai, Charilaos Mylonas, Satish Nagarajaiah, and Eleni Chatzi. ``Structural identification with physics-informed neural ordinary differential equations''. Journal of Sound and Vibration 508, 116196 (2021). https://doi.org/10.1016/j.jsv.2021.116196 [19] Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. ``Universal differential equations for scientific machine learning'' (2020). arXiv:2001.04385. arXiv:2001.04385 [20] Leonardo Banchi, Edward Grant, Andrea Rocchetto, and Simone Severini. ``Modelling non-Markovian quantum processes with recurrent neural networks''. New Journal of Physics 20, 123030 (2018). https://doi.org/10.1088/1367-2630/aaf749 [21] Stefan Krastanov, Kade Head-Marsden, Sisi Zhou, Steven T. Flammia, Liang Jiang, and Prineha Narang. ``Unboxing quantum black box models: Learning non-Markovian dynamics'' (2020). arXiv:2009.03902. arXiv:2009.03902 [22] Timothy Heightman, Edward Jiang, and Antonio Acín. ``Solving the quantum many-body Hamiltonian learning problem with neural differential equations'' (2024). arXiv:2408.08639. arXiv:2408.08639 [23] Sohail Reddy, Stefanie Günther, and Yujin Cho. ``Data-driven characterization of latent dynamics on quantum testbeds''. AVS Quantum Science 6, 033803 (2024). https://doi.org/10.1116/5.0204409 [24] Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. ``Scientific machine learning through physics–informed neural networks: Where we are and what’s next''. Journal of Scientific Computing 92, 88 (2022). https://doi.org/10.1007/s10915-022-01939-z [25] Salah A. Faroughi, Nikhil Pawar, Celio Fernandes, Maziar Raissi, Subasish Das, Nima K. Kalantari, and Seyed Kourosh Mahjour. ``Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing'' (2022). arXiv:2211.07377. arXiv:2211.07377 [26] Shudong Huang, Wentao Feng, Chenwei Tang, Zhenan He, Caiyang Yu, and Jiancheng Lv. ``Partial differential equations meet deep neural networks: A survey''. IEEE Transactions on Neural Networks and Learning Systems 36, 13649–13669 (2025). https://doi.org/10.1109/TNNLS.2025.3545967 [27] Sung Wook Kim, Iljeok Kim, Jonghwan Lee, and Seungchul Lee. ``Knowledge integration into deep learning in dynamical systems: An overview and taxonomy''. Journal of Mechanical Science and Technology 35, 1331–1342 (2021). https://doi.org/10.1007/s12206-021-0342-5 [28] Elias Cueto and Francisco Chinesta. ``Thermodynamics of learning physical phenomena''. Archives of Computational Methods in Engineering 30, 4653–4666 (2023). https://doi.org/10.1007/s11831-023-09954-5 [29] Philip J. Morrison. ``A paradigm for joined Hamiltonian and dissipative systems''. Physica D: Nonlinear Phenomena 18, 410–419 (1986). https://doi.org/10.1016/0167-2789(86)90209-5 [30] Miroslav Grmela and Hans Christian Öttinger. ``Dynamics and thermodynamics of complex fluids. I. Development of a general formalism''. Physical Review E 56, 6620 (1997). https://doi.org/10.1103/PhysRevE.56.6620 [31] Hans Christian Öttinger and Miroslav Grmela. ``Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism''. Physical Review E 56, 6633 (1997). https://doi.org/10.1103/PhysRevE.56.6633 [32] Hans Christian Öttinger. ``Beyond equilibrium thermodynamics''. John Wiley & Sons. (2005). https://doi.org/10.1002/0471727903 [33] Quercus Hernandez, Alberto Badías, David González, Francisco Chinesta, and Elías Cueto. ``Deep learning of thermodynamics-aware reduced-order models from data''. Computer Methods in Applied Mechanics and Engineering 379, 113763 (2021). https://doi.org/10.1016/j.cma.2021.113763 [34] Francisco Chinesta, Elías Cueto, Miroslav Grmela, Beatriz Moya, Michal Pavelka, and Martin Šípka. ``Learning physics from data: A thermodynamic interpretation''.

In Geometric Structures of Statistical Physics, Information Geometry, and Learning. Pages 276–297.

Springer International Publishing (2021). https://doi.org/10.1007/978-3-030-77957-3_14 [35] Quercus Hernández, Alberto Badías, Francisco Chinesta, and Elías Cueto. ``Thermodynamics-informed graph neural networks''. IEEE Transactions on Artificial Intelligence 5, 967–976 (2024). https://doi.org/10.1109/TAI.2022.3179681 [36] Zhen Zhang, Yeonjong Shin, and George Em Karniadakis. ``GFINNs: GENERIC formalism informed neural networks for deterministic and stochastic dynamical systems''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, 20210207 (2022). https://doi.org/10.1098/rsta.2021.0207 [37] Kookjin Lee, Nathaniel A. Trask, and Panos Stinis. ``Machine learning structure preserving brackets for forecasting irreversible processes''. Advances in Neural Information Processing Systems 34, 5696–5707 (2021). arXiv:2106.12619. arXiv:2106.12619 [38] Anthony Gruber, Kookjin Lee, and Nathaniel Trask. ``Reversible and irreversible bracket-based dynamics for deep graph neural networks''. Advances in Neural Information Processing Systems 36, 38454–38484 (2023). arXiv:2305.15616. arXiv:2305.15616 [39] Quercus Hernández, Alberto Badías, Francisco Chinesta, and Elías Cueto. ``Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems''. Computational Mechanics 72, 553–561 (2023). https://doi.org/10.1007/s00466-023-02296-w [40] Heinz-Peter Breuer and Francesco Petruccione. ``The theory of open quantum systems''.

Oxford University Press. (2007). https://doi.org/10.1093/acprof:oso/9780199213900.001.0001 [41] G. Lindblad. ``On the generators of quantum dynamical semigroups''. Communications in Mathematical Physics 48, 119–130 (1976). https://doi.org/10.1007/BF01608499 [42] H. Grabert. ``Nonlinear relaxation and fluctuations of damped quantum systems''. Zeitschrift für Physik B Condensed Matter 49, 161–172 (1982). https://doi.org/10.1007/BF01314753 [43] Patrick P Potts, Alex Arash Sand Kalaee, and Andreas Wacker. ``A thermodynamically consistent Markovian master equation beyond the secular approximation''. New Journal of Physics 23, 123013 (2021). https://doi.org/10.1088/1367-2630/ac3b2f [44] Anton Trushechkin. ``Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation''. Physical Review A 103, 062226 (2021). https://doi.org/10.1103/PhysRevA.103.062226 [45] Herbert B. Callen and Theodore A. Welton. ``Irreversibility and generalized noise''. Physical Review 83, 34 (1951). https://doi.org/10.1103/PhysRev.83.34 [46] R Kubo. ``The fluctuation-dissipation theorem''. Reports on Progress in Physics 29, 255 (1966). https://doi.org/10.1088/0034-4885/29/1/306 [47] Hans Christian Öttinger. ``Nonlinear thermodynamic quantum master equation: Properties and examples''. Physical Review A 82, 052119 (2010). https://doi.org/10.1103/PhysRevA.82.052119 [48] Alexander Mielke. ``Dissipative quantum mechanics using GENERIC''.

In Recent Trends in Dynamical Systems. Pages 555–585. Springer Basel (2013). https://doi.org/10.1007/978-3-0348-0451-6_21 [49] Hans Christian Öttinger. ``The geometry and thermodynamics of dissipative quantum systems''. Europhysics Letters 94, 10006 (2011). https://doi.org/10.1209/0295-5075/94/10006 [50] Markus Mittnenzweig and Alexander Mielke. ``An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models''. Journal of Statistical Physics 167, 205–233 (2017). https://doi.org/10.1007/s10955-017-1756-4 [51] E. Brüning, H. Mäkelä, A. Messina, and F. Petruccione. ``Parametrizations of density matrices''. Journal of Modern Optics 59, 1–20 (2012). https://doi.org/10.1080/09500340.2011.632097 [52] Gilbert Strang. ``Linear algebra and its applications''. Brooks/Cole. (2006). 4th edition. [53] Matteo Paris and Jaroslav Řeháček, editors. ``Quantum state estimation''. Springer Berlin, Heidelberg. (2004). https://doi.org/10.1007/b98673 [54] Zhichao Peng, Daniel Appelö, N. Anders Petersson, Mohamad Motamed, Fortino Garcia, and Yujin Cho. ``Deterministic and Bayesian characterization of quantum computing devices'' (2023). arXiv:2306.13747. arXiv:2306.13747 [55] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver. ``A quantum engineer's guide to superconducting qubits''.

Applied Physics Reviews 6, 021318 (2019). https://doi.org/10.1063/1.5089550 [56] Norman F. Ramsey. ``A molecular beam resonance method with separated oscillating fields''. Physical Review 78, 695–699 (1950). https://doi.org/10.1103/PhysRev.78.695 [57] Gen Kimura. ``The Bloch vector for N-level systems''. Physics Letters A 314, 339–349 (2003). https://doi.org/10.1016/S0375-9601(03)00941-1 [58] Jorge Nocedal and Stephen J. Wright. ``Numerical optimization''. Springer, New York. (2006). https://doi.org/10.1007/978-0-387-40065-5 [59] Alexander P. M. Place, Lila V. H. Rodgers, Pranav Mundada, Basil M. Smitham, Mattias Fitzpatrick, Zhaoqi Leng, Anjali Premkumar, Jacob Bryon, Andrei Vrajitoarea, et al. ``New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds''. Nature Communications 12, 1779 (2021). https://doi.org/10.1038/s41467-021-22030-5Cited by[1] Bitap Raj Thakuria, Trishna Kalita, Manash Jyoti Sarmah, and Himangshu Prabal Goswami, "Coherence in the Leak and Storage Kurtosis control Ergotropy in Quantum Batteries", arXiv:2511.08063, (2025). [2] Jimmie Adriazola and Katarzyna Roszak, "Learning Volterra Kernels for Non-Markovian Open Quantum Systems", arXiv:2601.09075, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-07-01 13:10:48). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-07-01 13:10:46: Could not fetch cited-by data for 10.22331/q-2026-07-01-2151 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.► BibTeX data@article{Sentz2026learning, doi = {10.22331/q-2026-07-01-2151}, url = {https://doi.org/10.22331/q-2026-07-01-2151}, title = {Learning thermodynamic master equations for open quantum systems}, author = {Sentz, Peter and Nicholson, Stanley and Cho, Yujin and Reddy, Sohail and Keith, Brendan and G{\"{u}}nther, Stefanie}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2151}, month = jul, year = {2026} }► References [1] Antonio D. Córcoles, Abhinav Kandala, Ali Javadi-Abhari, Douglas T. McClure, Andrew W. Cross, Kristan Temme, et al. ``Challenges and opportunities of near-term quantum computing systems''. Proceedings of the IEEE 108, 1338–1352 (2020). https://doi.org/10.1109/JPROC.2019.2954005 [2] David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, and Steven T. Flammia. ``Tailoring surface codes for highly biased noise''. Physical Review X 9, 041031 (2019). https://doi.org/10.1103/PhysRevX.9.041031 [3] Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M. Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, and Cristian Bonato. ``Learning quantum systems''.

In Geometric Structures of Statistical Physics, Information Geometry, and Learning. Pages 276–297.

Read Original

Source Information

Source: Quantum Science and Technology (arXiv overlay)

Website: https://quantum-journal.org/feed

Discussion

0 professional contributions

Be the first to add a constructive contribution.

Learning thermodynamic master equations for open quantum systems

Tags

Source Information