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).

Seminar organizers
Pechen Alexander Nikolaevich
Morzhin Oleg Vasilevich

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