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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 5(449), Pages 83–144 (Mi umn9835)  

This article is cited in 5 scientific papers (total in 5 papers)

Krotov method for optimal control of closed quantum systems

O. V. Morzhina, A. N. Pechenab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b National University of Science and Technology "MISIS"

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.
Bibliography: 158 titles.

Keywords: quantum control, coherent control, Krotov method, closed quantum systems, quantum technology.

Funding Agency Grant Number
Russian Science Foundation 17-11-01388
Ministry of Science and Higher Education of the Russian Federation 1.669.2016/1.4
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.


DOI: https://doi.org/10.4213/rm9835

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English version:
Russian Mathematical Surveys, 2019, 74:5, 851–908

Bibliographic databases:

UDC: 517.958
MSC: Primary 81Q93; Secondary 49Mxx, 35Q40, 93C15
Received: 29.05.2018
Revised: 20.04.2019

Citation: O. V. Morzhin, A. N. Pechen, “Krotov method for optimal control of closed quantum systems”, Uspekhi Mat. Nauk, 74:5(449) (2019), 83–144; Russian Math. Surveys, 74:5 (2019), 851–908

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Morzhin O.V., Pechen A.N., “Maximization of the Overlap Between Density Matrices For a Two-Level Open Quantum System Driven By Coherent and Incoherent Controls”, Lobachevskii J. Math., 40:10, SI (2019), 1532–1548  mathnet  crossref  mathscinet  zmath  isi
    2. O. V. Morzhin, A. N. Pechen, “Maximization of the uhlmann-jozsa fidelity for an open two-level quantum system with coherent and incoherent controls”, Phys. Part. Nuclei, 51:4 (2020), 464–469  crossref  isi
    3. Sh. Cong, L. Zhou, F. Meng, “Lyapunov-based unified control method for closed quantum systems”, J. Frankl. Inst.-Eng. Appl. Math., 357:14 (2020), 9220–9247  crossref  mathscinet  zmath  isi
    4. A. N. Pechen, O. V. Morzhin, “Machine Learning For Finding Suboptimal Final Times and Coherent and Incoherent Controls For An Open Two-Level Quantum System”, Lobachevskii J. Math., 41:12, SI (2020), 2353–2368  mathnet  crossref  zmath  isi
    5. Oleg V. Morzhin, Alexander N. Pechen, “On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System”, Proc. Steklov Inst. Math., 313 (2021), 149–164  mathnet  crossref  crossref  isi
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