The article considers a two-level open quantum system whose dynamics is driven by a combination of coherent and incoherent controls. Coherent control enters into the Hamiltonian part of the dynamics whereas incoherent control enters into the dissipative part. The goal is to find controls which move the system from an initial density matrix to a given target density matrix as fast as possible. To achieve this goal, we reformulate the optimal control problem in terms of controlled evolution in the Bloch ball and then apply Pontryagin maximum principle and gradient projection method to numerically find minimal time and optimal coherent and incoherent controls. General method is provided and several examples of initial and target states are explicitly considered.

Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as gradient ascent pulse engineering, differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.

Mathematical problems of optimal control in quantum systems attract high interest in connection with fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method initially proposed beyond quantum control in the articles by V.F. I.N. Feldman (1978, 1983). The method was used to develop a novel approach for finding optimal controls for quantum systems in [D.J. Tannor, V. Kazakov, V. Orlov, In: Time-Dependent Quantum Molecular Dynamics, Boston, Springer, 347-360 (1992)] and [J. Somlói, V.A. Kazakov, D.J. Tannor, Chem. Phys., 172:1, 85-98 (1993)], and in many works of various scientists, as described in details in this review. The review discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with respect to defining the improvement function (which in most cases is linear or linear-quadratic), constraints on control spectrum and on the states of a quantum system, regularizers, etc. The review describes applications of the Krotov method to control of molecular dynamics, manipulation of Bose-Einstein condensate, quantum gate generation. We also discuss comparison with GRAPE (GRadient Ascent Pulse Engineering), CRAB (Chopped Random-Basis), the Zhu Rabitz and the Maday Turinici methods.Bibliography: 154 titles.3 quantum tomography [53,54]. Quantum machine learning is considered [55]. One of the commonly used methods for constructing program controls for quantum systems is the Krotov 5 method. This method was initially proposed beyond quantum control by V.F. I.N. Feldman [56, 57] (1978, 1983) based on the Krotov optimality principle [58,59] and further developed by A.I. Konnov and V.F. Krotov [60] (1999). An example with control of an open (i.e. interacting with the environment) quantum system was analyzed by V.A. Kazakov and V.F. Krotov in 1987 in the article [61] (also in [62]). Crucial step in development to quantum systems was done in 1992-1993, when D.J. Tannor and coauthors used the 1st order Krotov method to develop a general approach for finding optimal controls for quantum systems [63,64]. In 2002, the 2nd order Krotov method [57,60] was adapted by S.E. Sklarz and D.J. Tannor for modeling of optimal control for Bose-Einstein condensate, whose dynamics is defined in terms of a controlled Gross-Pitaevskii equation [65]. In 2008, J.P. Palao, R. Kosloff, and C.P. Koch developed the method to optimal control for the problem of obtaining an objective in a subspace of the Hilbert space while avoiding population transfer to other subspaces [70]. The Krotov method was applied with various modifications and taking into account specific details of quantum optimal control problems, for atomic and molecular dynamics [9,19,42,[66][67][68][69][70][71][72][73][74][75][76][77]; qubits, quantum gates, quantum networks [78-94]; manipulation of Bose-Einstein condensate [39,65,95,96]; nuclear magnetic reson...

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