Control of Quantum Systems
September 14–December 28, 2020, MIPT - MI RAS, Moscow

This one-semester course is one of the compulsory courses of the "quantum" track of the Department of "Methods of Modern Mathematics" at FPMI MIPT. The course considers the foundations of the mathematical methods for solving problems of controlling quantum systems (individual atoms, molecules, etc.). Control problems are considered for both closed, i.e. isolated from interaction with the environment (excluding the control), and open quantum systems. For closed systems: (a) an overview of the main classes of optimal control problems for the case when the Hamiltonian contains coherent control is provided, as well as main controllability theorems; (b) analytical results are discussed for the problems of generating quantum logic gates and maximizing the transition probability; (c) various numerical optimization methods are described, including (quasi) gradient methods, GRAPE, Krotov's method, reduction to finite-dimensional optimization using stochastic global optimization methods. Such key concepts as Pontryagin maximum principle, landscape of the control problem, the expansion of the target functional to the first / second order, the gradient and the Hessian of the target functional, special control, traps in control problems, etc. are considered. For open quantum systems, control of systems with master equation of the Gorini – Kossakowski – Sudarshan – Lindblad form is studied. Coherent and / or incoherent control (the latter is included in the dissipation superoperator), Kraus maps are considered, and some numerical optimization methods are described. The course requires basic knowledge of quantum mechanics, functional analysis, linear algebra, differential equations theory, optimal control theory, and uses the results of parallel courses on quantum information theory and the theory of open quantum systems.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

RSS: Forthcoming seminars

Seminar organizers
Pechen Alexander Nikolaevich
Morzhin Oleg Vasilevich

Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center

Control of Quantum Systems, Moscow, September 14–December 28, 2020

December 14, 2020 (Mon)
1. Lecture 13. Coherent and incoherent control of closed and open quantum systems. Comparison of optimization methods
A. N. Pechen, O. V. Morzhin
December 14, 2020 15:05, Moscow, MIPT - MI RAS

December 7, 2020 (Mon)
2. Lecture 12. Control of open quantum systems
A. N. Pechen, O. V. Morzhin
December 7, 2020 15:05, Moscow, MIPT - MI RAS

November 30, 2020 (Mon)
3. Lecture 11. Krotov method. II
A. N. Pechen, O. V. Morzhin
November 30, 2020 15:05, Moscow, MIPT - MI RAS

November 23, 2020 (Mon)
4. Lecture 10. Gradient optimization methods (part II) and Krotov method (part I)
A. N. Pechen, O. V. Morzhin
November 23, 2020 15:05, Moscow, MIPT - MI RAS

November 16, 2020 (Mon)
5. Lecture 9. Gradient optimization methods. I
A. N. Pechen, O. V. Morzhin
November 16, 2020 15:05, Moscow, MIPT - MI RAS

November 9, 2020 (Mon)
6. Lecture 8. Landscapes of quantum control problems. III
A. N. Pechen, O. V. Morzhin
November 9, 2020 15:05, Moscow, MIPT - MI RAS

October 26, 2020 (Mon)
7. Lecture 7. Landscapes of quantum control problems. II
A. N. Pechen, O. V. Morzhin
October 26, 2020 15:05, Moscow, MIPT - MI RAS

October 19, 2020 (Mon)
8. Lecture 6. Landscapes of quantum control problems
A. N. Pechen, O. V. Morzhin
October 19, 2020 15:05, Moscow, MIPT - MI RAS

October 12, 2020 (Mon)
9. Lecture 5. Proof of the main theorem on the controllability conditions
A. N. Pechen, O. V. Morzhin
October 12, 2020 15:05, Moscow, MIPT - MI RAS

October 5, 2020 (Mon)
10. Lecture 4. Theorem on the equivalence of the definitions of controllability for pure states. Examples of checking the controllability conditions
A. N. Pechen, O. V. Morzhin
October 5, 2020 15:05, Moscow, MIPT - MI RAS

September 28, 2020 (Mon)
11. Lecture 3. Definitions of controllability. Main theorem about controllability
A. N. Pechen, O. V. Morzhin
September 28, 2020 15:05, Moscow, MIPT - MI RAS

September 21, 2020 (Mon)
12. Lecture 2. Schrödinger equation with control. Theorem on the existence and uniqueness of the solution. The problems for maximizing mean and population transfer, for generation of quantum gates
A. N. Pechen, O. V. Morzhin
September 21, 2020 15:05, Moscow, MIPT - MI RAS

September 14, 2020 (Mon)
13. Lecture 1. Introduction to control of quantum systems
A. N. Pechen, O. V. Morzhin
September 14, 2020 15:05, Moscow, MIPT - MI RAS
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