Some topics in control of quantum systems

We will overview progress in some directions in the mathematical theory of quantum control. Quantum control deals with problems of controlling atomic and molecular scale systems which exhibit quantum features in their dynamics. High interest to this topic is motivated by rich mathematical theory together with numerous existing and prospective applications ranging from quantum information and computing to laser control of chemical reactions, that are commonly known as quantum technologies. We will start with mathematical introduction to basics of quantum mechanics, formulate typical quantum control problems, and then outline progress in one of interesting topics in quantum control-analysis of quantum control landscapes. Quantum control landscapes are graphs of quantum control objective functionals. Analysis of local but not global extrema (i.e., traps) in quantum control landscapes is a highly attrative topic in modern quantum control. We will discuss the original hypothesis of general absence of traps, proof of this hypothesis for important quantum systems with two states (such as qubit) and for systems with infinite number of states, discuss existence of trapping features for various systems with number of states more than two and relation of the analysis of control landscapes for open quantum systems to optimization on complex Stiefel manifolds.

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