Abstract:
The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods.
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Work on Sections 1, 3, 4, and 5 was undertaken by both authors in the Steklov Mathematical Institute of the Russian Academy of Sciences and supported by the Russian Science Foundation under grant no. 17-11-01388,
work on Subsections 2.1, 2.2, and 2.3 was undertaken by both authors and supported in the framework of the state programme assigned to the Steklov Mathematical Institute of the Russian Academy of Sciences, work on Subsection 2.4 and Section 6 was undertaken by the first author in the framework of the state programme assigned to the Steklov Mathematical Institute of the Russian Academy of Sciences and by the second author in MISIS in the framework of project no. 1.669.2016/1.4 of the Ministry of Science and Higher Education of the Russian Federation, and work on Subsections 2.5 and 2.6 was undertaken by the second author in MISIS also in the framework of project no. 1.669.2016/1.4.
\Bibitem{MorPec19}
\by O.~V.~Morzhin, A.~N.~Pechen
\paper Krotov method for optimal control of closed quantum systems
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 5
\pages 851--908
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Vincent Hardel, Giovanni Manfredi, Paul-Antoine Hervieux, Rémi Goerlich, “Shortcuts to adiabaticity in harmonic traps: A quantum-classical analog”, Phys. Rev. E, 110:5 (2024)
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Oleg V. Morzhin, Alexander N. Pechen, “Krotov type optimization of coherent and incoherent controls for open two-qubit systems”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 45 (2023), 3–23
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A. N. Pechen, V. N. Petruhanov, “GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls”, J. Phys. A, 56:30 (2023), 305303, 26 pp.
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