From classical to quantum reinforcement learning and its applications in quantum control: A beginner’s tutorial
DOI (Low Temperature Physics):
https://doi.org/10.1063/10.0044783Ключові слова:
reinforcement learning, artificial intelligence, quantum control, Monte Carlo methods, probability theoryАнотація
Посібник розроблено, щоб зробити навчання з підкріпленням (RL) доступнішим для студентів та дослідників, пропонуючи чіткі пояснення на прикладах. Він зосереджений на подоланні розриву між теорією RL та практичним кодуванням, вирішуючи поширені проблеми, з якими стикаються під час переходу від концептуального розуміння до впровадження. За допомогою практичних прикладів та доступних пояснень посібник має на меті надати читачеві базові навички, необхідні для впевненого застосування методів RL у реальних сценаріях.
Доступність коду: https://github.com/asen009/Reinforcement_Tutorial_Undergraduates
Посилання
1. J. Schrittwieser, I. Antonoglou, T. Hubert, K. Simonyan, L. Sifre, S. Schmitt, A. Guez, E. Lockhart, D. Hassabis, T. Graepel, T. Lillicrap, and D. Silver, “Mastering atari, Go, chess and shogi by planning with a learned model,” Nature 588, 604 (2020). https://doi.org/10.1038/s41586-020-03051-4
2. D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, T. Lillicrap, K. Simonyan, and D. Hassabis, “A general reinforcement learning algorithm that masters chess, shogi, and go through self-play,” Science 362, 1140 (2018). https://doi.org/10.1126/science.aar6404
3. M. G. Bellemare, S. Candido, P. Samuel Castro, J. Gong, M. C. Machado, S. Moitra, S. S. Ponda, and Z. Wang, “Autonomous navigation of stratospheric balloons using reinforcement learning,” Nature 588, 77 (2020). https://doi.org/10.1038/s41586-020-2939-8
4. A. Irshayyid, J. Chen, and G. Xiong, “A review on reinforcement learning-based highway autonomous vehicle control,” Green Energy Intell. Transp. 3, 100156 (2024). https://doi.org/10.1016/j.geits.2024.100156
5. Md. Al-M. Khan, Md. R. J. Khan, A. Tooshil, N. Sikder, M. A. P. Mahmud, A. Z. Kouzani, and A.-Al Nahid, “A systematic review on reinforcement learning-based robotics within the last decade,” IEEE Access 8, 176598 (2020). https://doi.org/10.1109/ACCESS.2020.3027152
6. L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. L. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, J. Schulman, J. Hilton, F. Kelton, L. Miller, M. Simens, A. Askell, P. Welinder, P. Christiano, J. Leike, and R. Lowe, “Training language models to follow instructions with human feedback,” NeurIPS 35, 27730 (2022). https://doi.org/10.52202/068431-2011
7. A. Sen and S. Panda, Reinforcement tutorial for undergraduates—booklets, https://github.com/asen009/Reinforcement_Tutorial_Undergraduates/tree/main/Booklets, GitHub repository (2025).
8. B. Jaeger, and A. Geiger, “An invitation to deep reinforcement learning,” Found. Trends Optim. 7, 1 (2024). https://doi.org/10.1561/2400000049
9. A. Gosavi, “Reinforcement learning: A tutorial survey and recent advances,” INFORMS J. Comput. 21, 178 (2009). https://doi.org/10.1287/ijoc.1080.0305
10. V. Heidrich-Meisner, M. Lauer, C. Igel, and M. Riedmiller, Reinforcement learning in a nutshell, in Proceedings of the 15th European Symposium on Artificial Neural Networks (ESANN 2007), Bruges, Belgium (2007), p. 277.
11. M. E. Harmon and S. S. Harmon, Reinforcement Learning: A Tutorial, Technical Report, (Wright Laboratory, WL/AACF, Wright-Patterson Air Force Base, OH (1996). Available at https://sites.socsci.uci.edu/∼lpearl/courses/readings/HarmonHarmon1996.pdf.
12. M. Naeem, G. De Pietro, and A. Coronato, “Application of reinforcement learning and deep learning in multiple-input and multiple-output (MIMO) systems,” Sensors 22, 309 (2022). https://doi.org/10.3390/s22010309
13. Please consult the official documentation https://docs.python.org/3/tutorial/datastructures.html#dictionaries for details about Python dictionaries.
14. Note there is a separate data type in Python for ordered dictionaries https://docs.python.org/3/library/collections.html#ordereddict-objects.
15. R. S. Sutton, and A. G. Barto, Reinforcement Learning, Adaptive Computation and Machine Learning Series (Bradford Books, Cambridge, MA, 2018).
16. https://docs.python.org/3/tutorial/datastructures.html #more-on-lists.
17. https://docs.python.org/3/tutorial/datastructures.html #tuples-and-sequences.
18. M. Ghasemi, A. H. Moosavi, I. Sorkhoh, A. Agrawal, F. Alzhouri, and D. Ebrahimi, An introduction to reinforcement learning: Fundamental concepts and practical applications, arXiv.2408.07712 (2024).
19. R. S. Sutton, “Learning to predict by the methods of temporal differences,” Mach. Learn. 3, 9 (1988). https://doi.org/10.1023/A:1022633531479
20. R. J. Williams, “Simple statistical gradient-following algorithms for connectionist reinforcement learning,” Mach. Learn. 8, 229 (1992). https://doi.org/10.1023/A:1022672621406
21. E. Greensmith, P. L. Bartlett, and J. Baxter, “Variance reduction techniques for gradient estimates in reinforcement learning,” J. Mach. Learn. Res. 5, 1471 (2004).
22. W. van Heeswijk, Three Fundamental Flaws in Common Reinforcement Learning Algorithms (And How To Fix Them), Towards Data Science (2023). Available at https://towardsdatascience.com/three-fundamental-flaws-in-common-reinforcement-learning-algorithms-and-how-to-fix-them-951160b7a207
23. J. Peters and S. Schaal, “Natural actor-critic, proceedings of the 25th international conference on machine learning,” Neurocomputing. 71, 1180 (2008). https://doi.org/10.1016/j.neucom.2007.11.026
24. R. Wang, S. S. Du, L. F. Yang, and S. M. Kakade, “Is long horizon reinforcement learning more difficult than short horizon reinforcement learning?,” in Advances in Neural Information Processing Systems (NeurIPS 2020), edited by H. Larochelle et al (Curran Associates, Inc., 2020), Vol. 33. Available at https://proceedings.neurips.cc/paper/2020/file/6734fa703f6633ab896eecbdfad8953a-Paper.pdf.
25. J. Zhang, J. Kim, B. O’Donoghue, and S. Boyd, “Sample efficient reinforcement learning with REINFORCE,” Proceedings of the AAAI Conference on Artificial Intelligence 35, 10887 (2021). https://doi.org/10.1609/aaai.v35i12.17300
26. A. G. Barto, R. S. Sutton, and C. W. Anderson, “Neuronlike adaptive elements that can solve difficult learning control problems,” IEEE Transactions on Systems, Man, and Cybernetics SMC-13, 834 (1983). https://doi.org/10.1109/TSMC.1983.6313077
27. D. J. Tannor and S. A. Rice, “Control of selectivity of chemical reaction via control of wave packet evolution,” J. Chem. Phys. 83, 5013 (1985).
28. P. Brumer, and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541 (1986). https://doi.org/10.1016/S0009-2614(86)80171-3
29. R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500 (1992). https://doi.org/10.1103/PhysRevLett.68.1500
30. W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics,” Science 259, 1581 (1993). https://doi.org/10.1126/science.259.5101.1581
31. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998). https://doi.org/10.1126/science.282.5390.919
32. J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, “Quantum control of energy flow in light-harvesting complexes,” Nature 417, 533 (2002). https://doi.org/10.1038/417533a
33. A. P. Peirce, M. A. Dahleh, and H. Rabitz, “Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications,” Phys. Rev. A 37, 4950 (1988). https://doi.org/10.1103/PhysRevA.37.4950
34. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms,” J. Magn. Reson. 172, 296 (2005). https://doi.org/10.1016/j.jmr.2004.11.004
35. V. F. Krotov, Global Methods in Optimal Control Theory (Marcel Dekker, New York, 1996).
36. T. Caneva, T. Calarco, and S. Montangero, “Chopped random-basis quantum optimization,” Phys. Rev. A 84, 022326 (2011). https://doi.org/10.1103/PhysRevA.84.022326
37. D. J. Egger, and F. K. Wilhelm, “Adaptive hybrid optimal quantum control for imprecisely characterized systems,” Phys. Rev. Lett. 112, 240503 (2014). https://doi.org/10.1103/PhysRevLett.112.240503
38. S. Machnes, U. Sander, S. J. Glaser, P. de Fouquieres, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, “Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework,” Phys. Rev. A 84, 022305 (2011). https://doi.org/10.1103/PhysRevA.84.022305
39. D. I. Bondar, L. Balada Gaggioli, G. Korpas, J. Mareček, J. Vala, and K. Jacobs, “Globally optimal control of quantum dynamics,” Phys. Rev. Res. 7, 043202 (2025). https://doi.org/10.1103/g4fb-xm13
40. L. B. Gaggioli, D. I. Bondar, J. Vala, R. Ovsiannikov, and J. Mareček, Unitary gate synthesis via polynomial optimization, arXiv:2508.01356 (2025).
41. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, “Training schrödinger’s cat: Quantum optimal control. strategic report on current status, visions and goals for research in Europe,” Eur. Phys. J. D 69, 279 (2015). https://doi.org/10.1140/epjd/e2015-60464-1
42. É Genois, N. J. Stevenson, N. Goss, I. Siddiqi, and A. Blais, “Quantum optimal control of superconducting qubits based on machine-learning characterization,” Phys. Rev. Appl. 24, 034073 (2025). https://doi.org/10.1103/d9yg-d3qr
43. D. Bukov, A. G. R. Day, P. Weinberg, A. Polkovnikov, and P. Mehta, “Reinforcement learning in different phases of quantum control,” Phys. Rev. X 8, 031086 (2018). https://doi.org/10.1103/PhysRevX.8.031086
44. Z. An, and D. L. Zhou, “Deep reinforcement learning for quantum gate control,” Europhys. Lett. 126, 60002 (2019). https://doi.org/10.1209/0295-5075/126/60002
45. L. Giannelli, S. Sgroi, J. Brown, G. S. Paraoanu, M. Paternostro, E. Paladino, and G. Falci, “A tutorial on optimal control and reinforcement learning methods for quantum technologies,” Phys. Lett. A 434, 128054 (2022). https://doi.org/10.1016/j.physleta.2022.128054
46. Y. Gao, X. Wang, N. Yu, and B. M. Wong, “Harnessing deep reinforcement learning to construct time-dependent optimal fields for quantum control dynamics,” Phys. Chem. Chem. Phys. 24, 24208 (2022).
47. D. Koutromanos, D. Stefanatos, and E. Paspalakis, “Control of qubit dynamics using reinforcement learning,” Information 15, 272 (2024). https://doi.org/10.3390/info15050272
48. J. O. Ernst, A. Chatterjee, T. Franzmeyer, and A. Kuhn, Reinforcement learning for quantum control under physical constraints, arXiv:2501.14372 (2025).
49. S. Li, Y. Fan, X. Li, X. Ruan, Q. Zhao, Z. Peng, R.-B. Wu, J. Zhang, and P. Song, “Robust quantum control using reinforcement learning from demonstration,” npj Quantum Inf. 11, 124 (2025). https://doi.org/10.1038/s41534-025-01065-2
50. R. Porotti, A. Essig, B. Huard, and F. Marquardt, “Deep reinforcement learning for quantum state preparation with weak nonlinear measurements,” Quantum 6, 747 (2022). https://doi.org/10.22331/q-2022-06-28-747
51. M. Yuezhen Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, “Universal quantum control through deep reinforcement learning,” npj quantum Inf. 5, 33 (2019). https://doi.org/10.1038/s41534-019-0141-3
52. T. Fösel, P. Tighineanu, T. Weiss, and F. Marquardt, “Reinforcement learning with neural networks for quantum feedback,” Phys. Rev. X 8, 031084 (2018). https://doi.org/10.1103/PhysRevX.8.031084
53. F. Shuang and H. Rabitz, “Control of quantum observables by tracking control fields,” J. Chem. Phys. 121, 9270 (2004). https://doi.org/10.1063/1.1799591
54. D. Dong and I. R. Petersen, “Quantum control theory and applications: A survey,” IET Control Theory Appl. 4, 2651 (2010). https://doi.org/10.1049/iet-cta.2009.0508
55. A. Isidori, Nonlinear Control Systems, Springer, 1995).
56. R. Marino, and P. Tomei, Nonlinear Control Design: Geometric, Adaptive And Robust (Prentice Hall, London, New York, 1995).
57. A. G. Campos, D. I. Bondar, R. Cabrera, and H. A. Rabitz, “How to make distinct dynamical systems appear spectrally identical,” Phys. Rev. Lett. 118, 083201 (2017). https://doi.org/10.1103/PhysRevLett.118.083201
58. G. McCaul, C. Orthodoxou, K. Jacobs, G. H. Booth, and D. I. Bondar, “Driven imposters: Controlling expectations in many-body systems,” Phys. Rev. Lett. 124, 183201 (2020). https://doi.org/10.1103/PhysRevLett.124.183201
59. G. McCaul, A. F. King, and D. I. Bondar, “Optical indistinguishability via twinning fields,” Phys. Rev. Lett. 127, 113201 (2021). https://doi.org/10.1103/PhysRevLett.127.113201
60. A. B. Magann, T.-S. Ho, C. Arenz, and H. A. Rabitz, “Quantum tracking control of the orientation of symmetric-top molecules,” Phys. Rev. A 108, 033106 (2023). https://doi.org/10.1103/PhysRevA.108.033106
61. R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour, Policy gradient methods for reinforcement learning with function approximation, in Advances in Neural Information Processing Systems, edited by S. A. Solla, T. K. Leen, and K.-R. Müller (MIT Press, Cambridge, MA, 2000), Vol. 12, p. 1057.
62. J. Schulman, S. Levine, P. Abbeel, M. Jordan, and P. Moritz, “Trust region policy optimization, in: Proceedings of the 32nd international conference on machine learning (ICML 2015),” Proc. Mach. Learn. Res. 37, 1889 (2015).
63. J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, Proximal policy optimization algorithms, arXiv:1707.06347 (2017).
64. T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, Continuous control with deep reinforcement learning, arXiv:1509.02971 (2015).
65. S. Fujimoto, H. van Hoof, and D. Meger, Addressing function approximation error in actor-critic methods, arXiv:1802.09477 (2018).
66. T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, Soft actor-critic algorithms and applications, arXiv:1812.05905 (2018).
67. K. Doya, “Reinforcement learning: Computational theory and biological mechanisms,” HFSP Journal 1, 30 (2007). https://doi.org/10.2976/1.2732246/10.2976/1
68. S. Y.-C. Chen, An introduction to quantum reinforcement learning (QRL), arXiv.2409.05846 (2024).
69. T. H. Su, S. Shresthamali, and M. Kondo, “Quantum framework for reinforcement learning: Integrating the markov decision process, quantum arithmetic, and trajectory search,” Phys. Rev. A 111, 062421 (2025). https://doi.org/10.1103/5lfr-xb8m
70. S. hua Wang and H. Lin, Quantum reinforcement learning (2023).
71. M. Ismail, M. Shaban, and S. Y.-C. Chen, “Hands-on introduction to quantum machine learning,” The International FLAIRS Conference Proceedings 37 (2024). https://doi.org/10.32473/flairs.37.1.135478
72. M. Schuld and F. Petruccione, Machine Learning with Quantum Computers (Springer International Publishing, 2021). https://doi.org/10.1007/978-3-030-83098-4