{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:22:14Z","timestamp":1772252534029,"version":"3.50.1"},"reference-count":73,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T00:00:00Z","timestamp":1677801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ANR project","award":["ANR-22-CE47-0008-02"],"award-info":[{"award-number":["ANR-22-CE47-0008-02"]}]},{"name":"ANR project","award":["ANR-17-EURE-0002"],"award-info":[{"award-number":["ANR-17-EURE-0002"]}]},{"name":"EIPHI Graduate School","award":["ANR-22-CE47-0008-02"],"award-info":[{"award-number":["ANR-22-CE47-0008-02"]}]},{"name":"EIPHI Graduate School","award":["ANR-17-EURE-0002"],"award-info":[{"award-number":["ANR-17-EURE-0002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the extent to which a two-level quantum system subjected to an external time-dependent drive can be characterized by supervised learning. We apply this approach to the case of bang-bang control and the estimation of the offset and the final distance to a given target state. For any control protocol, the goal is to find the mapping between the offset and the distance. This mapping is interpolated using a neural network. The estimate is global in the sense that no a priori knowledge is required on the relation to be determined. Different neural network algorithms are tested on a series of data sets. We show that the mapping can be reproduced with very high precision in the direct case when the offset is known, while obstacles appear in the indirect case starting from the distance to the target. We point out the limits of the estimation procedure with respect to the properties of the mapping to be interpolated. We discuss the physical relevance of the different results.<\/jats:p>","DOI":"10.3390\/e25030446","type":"journal-article","created":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T03:22:05Z","timestamp":1678072925000},"page":"446","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Characterization of a Driven Two-Level Quantum System by Supervised Learning"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1490-9592","authenticated-orcid":false,"given":"Rapha\u00ebl","family":"Couturier","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Franche-Comt\u00e9, CNRS, Institut FEMTO-ST, F-90000 Belfort, France"}]},{"given":"Etienne","family":"Dionis","sequence":"additional","affiliation":[{"name":"Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Universit\u00e9 de Bourgogne, 9 Av. A. Savary, BP 47 870, CEDEX, F-21078 Dijon, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9826-9598","authenticated-orcid":false,"given":"St\u00e9phane","family":"Gu\u00e9rin","sequence":"additional","affiliation":[{"name":"Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Universit\u00e9 de Bourgogne, 9 Av. A. Savary, BP 47 870, CEDEX, F-21078 Dijon, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0195-4378","authenticated-orcid":false,"given":"Christophe","family":"Guyeux","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Franche-Comt\u00e9, CNRS, Institut FEMTO-ST, F-90000 Belfort, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1963-333X","authenticated-orcid":false,"given":"Dominique","family":"Sugny","sequence":"additional","affiliation":[{"name":"Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Universit\u00e9 de Bourgogne, 9 Av. A. Savary, BP 47 870, CEDEX, F-21078 Dijon, France"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"045002","DOI":"10.1103\/RevModPhys.91.045002","article-title":"Machine learning and the physical sciences","volume":"91","author":"Carleo","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2019.03.001","article-title":"A high-bias, low-variance introduction to Machine Learning for physicists","volume":"810","author":"Mehta","year":"2019","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1103\/PhysRevLett.68.1500","article-title":"Teaching lasers to control molecules","volume":"68","author":"Judson","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1038\/nature14236","article-title":"Human-level control through deep reinforcement learning","volume":"518","author":"Mnih","year":"2015","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1126\/science.aam6564","article-title":"Machine learning for quantum physics","volume":"355","author":"Hush","year":"2017","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"32","DOI":"10.22331\/qv-2020-03-17-32","article-title":"A non-review of Quantum Machine Learning: Trends and explorations","volume":"4","author":"Dunjko","year":"2020","journal-title":"Quantum Views"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"010101","DOI":"10.1103\/PhysRevA.107.010101","article-title":"Artificial intelligence and machine learning for quantum technologies","volume":"107","author":"Krenn","year":"2023","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural networks","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1038\/nphys4035","article-title":"Machine learning phases of matter","volume":"13","author":"Carrasquilla","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"130501","DOI":"10.1103\/PhysRevLett.117.130501","article-title":"Quantum-Enhanced Machine Learning","volume":"117","author":"Dunjko","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"140504","DOI":"10.1103\/PhysRevLett.124.140504","article-title":"Retrieving Quantum Information with Active Learning","volume":"124","author":"Ding","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"042403","DOI":"10.1103\/PhysRevA.105.042403","article-title":"Learning quantum dynamics with latent neural ordinary differential equations","volume":"105","author":"Choi","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"062612","DOI":"10.1103\/PhysRevA.106.062612","article-title":"Stochastic learning control of adiabatic speedup in a non-Markovian open qutrit system","volume":"106","author":"Xie","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"063019","DOI":"10.1088\/1367-2630\/ac66f9","article-title":"Machine learning for continuous quantum error correction on superconducting qubits","volume":"24","author":"Convy","year":"2022","journal-title":"New J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1140\/epjd\/e2015-60464-1","article-title":"Training Schr\u00f6dinger\u2019s cat: Quantum optimal control. Strategic report on current status, visions and goals for research in Europe","volume":"69","author":"Glaser","year":"2015","journal-title":"Eur. Phys. J. D"},{"key":"ref_16","unstructured":"D\u2019Alessandro, D. (2008). Introduction to Quantum Control and Dynamics, Chapman, Hall\/CRC."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"075008","DOI":"10.1088\/1367-2630\/12\/7\/075008","article-title":"Control of quantum phenomena: Past, present and future","volume":"12","author":"Brif","year":"2010","journal-title":"New J. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/B978-0-12-408090-4.00002-5","article-title":"Chapter 2\u2014Shortcuts to Adiabaticity","volume":"Volume 62","author":"Arimondo","year":"2013","journal-title":"Advances in Atomic, Molecular, and Optical Physics"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1103\/RevModPhys.91.045001","article-title":"Shortcuts to adiabaticity: Concepts, methods, and applications","volume":"91","author":"Ruschhaupt","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"035005","DOI":"10.1103\/RevModPhys.91.035005","article-title":"Quantum control of molecular rotation","volume":"91","author":"Koch","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"080201","DOI":"10.1088\/1367-2630\/aad1ea","article-title":"The quantum technologies roadmap: A European community view","volume":"20","author":"Bloch","year":"2018","journal-title":"New J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1140\/epjqt\/s40507-022-00138-x","article-title":"Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe","volume":"9","author":"Koch","year":"2022","journal-title":"EPJ Quantum Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"128054","DOI":"10.1016\/j.physleta.2022.128054","article-title":"A tutorial on optimal control and reinforcement learning methods for quantum technologies","volume":"434","author":"Giannelli","year":"2022","journal-title":"Phys. Lett. A"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mart\u00edn-Guerrero, J.D., and Lamata, L. (2021). Reinforcement Learning and Physics. Appl. Sci., 11.","DOI":"10.3390\/app11188589"},{"key":"ref_25","first-page":"031086","article-title":"Reinforcement Learning in Different Phases of Quantum Control","volume":"8","author":"Bukov","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"020601","DOI":"10.1103\/PhysRevLett.122.020601","article-title":"Glassy Phase of Optimal Quantum Control","volume":"122","author":"Day","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"052333","DOI":"10.1103\/PhysRevA.97.052333","article-title":"Automatic spin-chain learning to explore the quantum speed limit","volume":"97","author":"Zhang","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1038\/s42005-022-00837-y","article-title":"Quantum imaginary time evolution steered by reinforcement learning","volume":"5","author":"Cao","year":"2022","journal-title":"Comm. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"60002","DOI":"10.1209\/0295-5075\/126\/60002","article-title":"Deep reinforcement learning for quantum gate control","volume":"126","author":"An","year":"2019","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1038\/s42005-019-0169-x","article-title":"Coherent transport of quantum states by deep reinforcement learning","volume":"2","author":"Porotti","year":"2019","journal-title":"Commun. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1038\/s41534-019-0141-3","article-title":"Universal quantum control through deep reinforcement learning","volume":"5","author":"Niu","year":"2019","journal-title":"NPJ Quantum Inf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"100401","DOI":"10.1103\/PhysRevLett.125.100401","article-title":"Deep Reinforcement Learning Control of Quantum Cartpoles","volume":"125","author":"Wang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"L040401","DOI":"10.1103\/PhysRevA.103.L040401","article-title":"Breaking adiabatic quantum control with deep learning","volume":"103","author":"Ding","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"190403","DOI":"10.1103\/PhysRevLett.127.190403","article-title":"Measurement-Based Feedback Quantum Control with Deep Reinforcement Learning for a Double-Well Nonlinear Potential","volume":"127","author":"Borah","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","first-page":"031070","article-title":"Reinforcement Learning for Many-Body Ground-State Preparation Inspired by Counterdiabatic Driving","volume":"11","author":"Yao","year":"2021","journal-title":"Phys. Rev. X"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"062443","DOI":"10.1103\/PhysRevA.105.062443","article-title":"Robust optimization for quantum reinforcement learning control using partial observations","volume":"105","author":"Jiang","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"020601","DOI":"10.1103\/PhysRevLett.126.020601","article-title":"Reinforcement Learning Approach to Nonequilibrium Quantum Thermodynamics","volume":"126","author":"Sgroi","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"093035","DOI":"10.1088\/1367-2630\/ac2393","article-title":"Reinforcement learning-enhanced protocols for coherent population-transfer in three-level quantum systems","volume":"23","author":"Brown","year":"2021","journal-title":"New J. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41534-021-00512-0","article-title":"Identifying optimal cycles in quantum thermal machines with reinforcement-learning","volume":"8","author":"Erdman","year":"2022","journal-title":"NPJ Quantum Inf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"042327","DOI":"10.1103\/PhysRevA.99.042327","article-title":"Learning robust and high-precision quantum controls","volume":"99","author":"Wu","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"042324","DOI":"10.1103\/PhysRevA.97.042324","article-title":"Neural-network-designed pulse sequences for robust control of singlet-triplet qubits","volume":"97","author":"Yang","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"023155","DOI":"10.1103\/PhysRevResearch.4.023155","article-title":"Robust quantum gates using smooth pulses and physics-informed neural networks","volume":"4","author":"Kestner","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"024040","DOI":"10.1103\/PhysRevApplied.17.024040","article-title":"Machine-Learning-Assisted Quantum Control in a Random Environment","volume":"17","author":"Huand","year":"2022","journal-title":"Phys. Rev. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"034080","DOI":"10.1103\/PhysRevApplied.15.034080","article-title":"Integrated Tool Set for Control, Calibration, and Characterization of Quantum Devices Applied to Superconducting Qubits","volume":"15","author":"Wittler","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1051\/cocv:2007013","article-title":"Hamiltonian identification for quantum systems: Well-posedness and numerical approaches","volume":"13","author":"Mirrahimi","year":"2007","journal-title":"ESAIM COCV"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5369","DOI":"10.1063\/1.1538242","article-title":"Optimal Hamiltonian identification: The synthesis of quantum optimal control and quantum inversion","volume":"118","author":"Geremia","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"12315","DOI":"10.1021\/jp021762e","article-title":"Nonlinear Kinetic Parameter Identification through Map Inversion","volume":"106","author":"Shenvi","year":"2002","journal-title":"J. Phys. Chem. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"265302","DOI":"10.1088\/1751-8113\/47\/26\/265302","article-title":"Newton algorithm for Hamiltonian characterization in quantum control","volume":"47","author":"Ndong","year":"2014","journal-title":"J. Phys. Math. Theor."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"022125","DOI":"10.1103\/PhysRevA.91.022125","article-title":"Ubiquitous problem of learning system parameters for dissipative two-level quantum systems: Fourier analysis versus Bayesian estimation","volume":"91","author":"Schirmer","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"080401","DOI":"10.1103\/PhysRevLett.113.080401","article-title":"Quantum Hamiltonian Identification from Measurement Time Traces","volume":"113","author":"Zhang","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"022335","DOI":"10.1103\/PhysRevA.95.022335","article-title":"Hamiltonian identifiability assisted by a single-probe measurement","volume":"95","author":"Sone","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"030402","DOI":"10.1103\/PhysRevLett.119.030402","article-title":"Evolution-Free Hamiltonian Parameter Estimation through Zeeman Markers","volume":"119","author":"Burgarth","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"022604","DOI":"10.1103\/PhysRevA.103.022604","article-title":"Gradient algorithm for Hamiltonian identification of open quantum systems","volume":"103","author":"Xue","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"063112","DOI":"10.1103\/PhysRevA.104.063112","article-title":"Greedy reconstruction algorithm for the identification of spin distribution","volume":"104","author":"Buchwald","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"110401","DOI":"10.1103\/PhysRevLett.115.110401","article-title":"Optimal Feedback Scheme and Universal Time Scaling for Hamiltonian Parameter Estimation","volume":"115","author":"Yuan","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"012117","DOI":"10.1103\/PhysRevA.96.012117","article-title":"Quantum parameter estimation with optimal control","volume":"96","author":"Liu","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"052607","DOI":"10.1103\/PhysRevA.103.052607","article-title":"Optimal control for quantum metrology via Pontryagin\u2019s principle","volume":"103","author":"Lin","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"042621","DOI":"10.1103\/PhysRevA.105.042621","article-title":"Application of Pontryagin\u2019s maximum principle to quantum metrology in dissipative systems","volume":"105","author":"Lin","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"160505","DOI":"10.1103\/PhysRevLett.128.160505","article-title":"Variational Principle for Optimal Quantum Controls in Quantum Metrology","volume":"128","author":"Yang","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1038\/nature11971","article-title":"Magnetic resonance fingerprinting","volume":"495","author":"Ma","year":"2013","journal-title":"Nature"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"053419","DOI":"10.1103\/PhysRevA.96.053419","article-title":"Optimizing fingerprinting experiments for parameter identification: Application to spin systems","volume":"96","author":"Ansel","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"044033","DOI":"10.1103\/PhysRevApplied.10.044033","article-title":"Experimental Phase Estimation Enhanced by Machine Learning","volume":"10","author":"Lumino","year":"2018","journal-title":"Phys. Rev. Appl."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1038\/s41534-021-00513-z","article-title":"Parameter estimation in quantum sensing based on deep reinforcement learning","volume":"8","author":"Xiao","year":"2022","journal-title":"NPJ Quantum Inf."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/101.8118","article-title":"Artificial neural networks","volume":"4","author":"Hopfield","year":"1988","journal-title":"IEEE Circuits Devices Mag."},{"key":"ref_65","first-page":"2825","article-title":"Scikit-learn: Machine learning in Python","volume":"12","author":"Pedregosa","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"022605","DOI":"10.1103\/PhysRevA.105.022605","article-title":"Neural-network-based qubit-environment characterization","volume":"105","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"010316","DOI":"10.1103\/PRXQuantum.2.010316","article-title":"Using Deep Learning to Understand and Mitigate the Qubit Noise Environment","volume":"2","author":"Wise","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"022425","DOI":"10.1103\/PhysRevA.103.022425","article-title":"Estimating the degree of non-Markovianity using machine learning","volume":"103","author":"Fanchini","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"263902","DOI":"10.1103\/PhysRevLett.89.263902","article-title":"Optimal Identification of Hamiltonian Information by Closed-Loop Laser Control of Quantum Systems","volume":"89","author":"Geremia","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"062101","DOI":"10.1063\/1.2203236","article-title":"Time minimal trajectories for a spin 1\/2 particle in a magnetic field","volume":"47","author":"Boscain","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"013415","DOI":"10.1103\/PhysRevA.82.013415","article-title":"Simultaneous time-optimal control of the inversion of two spin-12 particles","volume":"82","author":"Lapert","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"030203","DOI":"10.1103\/PRXQuantum.2.030203","article-title":"Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control","volume":"2","author":"Boscain","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/446\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:46:56Z","timestamp":1760122016000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/446"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,3]]},"references-count":73,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["e25030446"],"URL":"https:\/\/doi.org\/10.3390\/e25030446","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202212.0433.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,3]]}}}