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Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-113 - Mathematical physics

Keywordsquantum control, quantum control landscape, traps, coherent control, incoherent control, Krotov method, open quantum systems, mathematical models of controlled quantum systems

Annotation
Quantum technologies are technologies based on the use of the properties of individual quantum systems of atomic, molecular and nanoscopic size. This area is currently of high interest, both for fundamental and practical reasons, associated with a wide range of applications, ranging from problems of quantum information, quantum computing, cryptography and metrology to synthesis of materials with desired properties and laser control of chemical reactions and processes. The development of quantum technologies requires the development of mathematical methods for controlling closed and open, that is, interacting with the environment, quantum systems. One of the most important tasks required for the development of quantum technologies is the development of methods for controlling quantum systems using modulated laser pulses or other external influence. The project is aimed at solving mathematical problems that are relevant for quantum technologies, namely, the development and application of control methods for closed and open quantum systems, the study of new local properties of target functionals, that is, quantum control landscapes, control problems for closed multilevel quantum systems, the development of methods for constructing controls for previously unexplored control problems for open quantum systems in the presence of noise, coherent and incoherent controls, including the use of gradient projection methods and the Krotov method, including, in the presence of phase constraints, the application of these methods to control problems of quantum systems on examples of various quantum systems, for which previously the simultaneous application of coherent and incoherent control has not been studied, and for mathematical modeling of the control of molecular dissociation using laser pulses. The following tasks will be solved within the framework of the project. The first task is to study the local properties of quantum control objective functionals. The quantum control problem can be formulated as the problem of maximizing a certain objective quantum functional depending on the state of the quantum system. A trap is a point of local, but not global, maximum of the target functional. The n-th order trap is determined by the Taylor expansion of the objective functional up to the n-th order of smallness. The question of the existence of traps is important for the theory of quantum control, since they can significantly complicate the process of searching for the global maximum in the laboratory. The problem of the existence of traps has many open questions. It is planned to investigate higher-order traps in the landscape of the problem of maximizing the expectation of a quantum mechanical observable for some previously unexplored multilevel quantum systems. The second task is the development of methods for constructing controls for optimal control problems for open finite-level quantum systems, the dynamics of which is described by a master equation of the Gorini – Kossakovsky – Sudarshan – Lindblad type with coherent and incoherent controls, based on two-step and three-step gradient projection methods and the Krotov method, taking into account various restrictions on the controls and on the evolution of the density matrix. In this model, the environment (reservoir) of an open quantum system is a source of noise that affects the dynamics of the system, which, however, is considered as a useful resource for controlling the quantum system. The third problem is the application of methods of coherent and incoherent control of quantum systems to the problem of laser assisted and incoherent control of molecular dissociation. In the example planned for consideration, to achieve this goal, incoherent control methods are usually used via the selection of materials and surface structures. During the implementation of the project, it is also planned to consider the possibility of coherent control using laser radiation and apply methods of simultaneous coherent and incoherent control to this problem. The high importance and actuality of solving these problems is determined by the growing needs to control quantum systems, including open ones, for problems of quantum technologies. The actuality of the study of control landscapes is determined by the need to select the most effective methods for constructing controls, which depends on the properties of the landscape. The relevance of the development of methods for constructing controls for open quantum systems under the influence of coherent and incoherent controls is determined by the fact that in real conditions controlled quantum systems usually interact with the environment, and this interaction cannot be neglected. Despite the high interest, in these areas there are open important problems that form the basis of the proposed project, the implementation of which will make it to contribute to the development of mathematical methods for problems of quantum technologies.

Expected results
During the implementation of the project, it is planned to obtain the following results. It is planned to construct a landscape of quantum control of the problem of maximizing the expectation of a quantum mechanical observable for a special class of quantum systems, taking into account higher-order traps. For the class of quantum systems under consideration, it is planned to obtain sufficient conditions for the existence of traps of an arbitrary finite order, perhaps for uncontrollable systems. It is supposed to estimate the order of the trap depending on the dimension of the space of quantum states. The expected results will be at a high world scientific level. Despite the great efforts of the scientific community, in the general case, the problem of the existence or absence of traps in quantum control problems remains unsolved. The study of existence or absence of traps is important for applications of quantum control, since their presence can significantly complicate the process of searching for the global maximum by numerical methods or in laboratory conditions. This circumstance ensures the high significance of the expected results. It is planned to develop methods for constructing controls for optimal control problems for open finite-level quantum systems, the dynamics of which is governed by the Gorini – Kossakovsky – Sudarshan – Lindblad master equation with coherent and incoherent controls, using optimization methods such as two-step and three-step gradient projection methods and the Krotov method, in the presence of restrictions on the evolution of the density matrix in the form of terminal and integral terms in the objective Mayer – Bolza functional. It is planned to consider examples of previously unexplored tasks for three-level systems, to describe the structure of controls that will be built using these methods, taking into account the constraints on controls. It is planned to apply the methods of coherent and incoherent control of quantum systems to the problem of laser and incoherent control of molecular dissociation. In the example planned for consideration, to achieve this goal, incoherent control methods are usually used, via the selection of materials and surface structures. In the course of the project, it is also planned to consider the possibility of coherent control using laser radiation, apply to this problem simultaneous coherent and incoherent control and find the corresponding controls. The high significance of the expected results in the study of control landscapes is determined by the fact that the properties of control landscapes are important for the making effective choice of the correct methods for constructing optimal controls. The high significance of the expected results in the development of methods for constructing controls for open quantum systems under the influence of coherent and incoherent controls is provided by the fact that in most situations the physical systems used in problems of quantum technologies are open and noisy, that is, they interact with the environment. This circumstance requires the development of methods for constructing controls for controlled open quantum systems. The results obtained will be at a high world scientific level in the field of mathematical problems of quantum technologies and are planned to be published in leading scientific journals. The methods planned for development can be applied to study various aspects of atomic and molecular dynamics induced by external laser fields, problems of control of quantum systems, dynamics and control of decoherence in quantum systems. The proposed mathematical research will have a significant impact on the actively developing field of quantum technologies.

Annotation of the results obtained in 2022
Quantum technologies exploit properties of individual quantum systems of atomic, molecular and nanoscopic dimensions. This area is of high fundamental and practical interest associated with a wide range of applications, from the problems of quantum information, quantum computing, cryptography and metrology to the problems of synthesizing materials with desired properties and laser control of chemical reactions and processes. The development of quantum technologies requires the development of mathematical methods for controlling closed and open, that is, interacting with the environment, quantum systems. One of the most important tasks necessary for the development of quantum technologies is the development of methods for controlling quantum systems using modulated laser pulses or other external influences, and the study of the effectiveness of such methods. In the current year, work was carried out to solve the tasks set for this year on the control of quantum systems, including the study of local properties of the objective functionals of quantum control, the development of methods for constructing controls for optimal control problems of open finite-level quantum systems, the application of methods of coherent and incoherent control of quantum systems to problem of laser coherent and incoherent control of molecular dissociation. In the course of the project, within the framework of the study of local methods for constructing optimal controls, a rigorous derivation of analytical expressions for the highest variations of the objective functional in the problem of maximizing the average value of the observable for coherent control of a certain class of multilevel closed quantum systems was carried out. The presence of traps of arbitrary order is shown for a number of systems. Controllability for a certain class of quantum systems arising from this analysis is investigated, it is proved that some systems are completely controllable, some systems, on the contrary, are not completely controllable. Analytical and numerical results on controllability are obtained for four-level quantum systems with a three-fold degenerate one of the levels for any complex values of the Hamiltonian of interaction with the control, and for three four-level systems with a two-fold degenerate upper level and with a forbidden transition between the lowest and intermediate levels. As a part of the study of control for open quantum systems, the optimal control problems for open N-level quantum systems (for various N>2) were considered, which are under the simultaneous influence of coherent and incoherent controls, with restrictions on controls. For such quantum systems, we considered optimal control problems with terminal objective functionals describing the minimization of the Hilbert–Schmidt distance between the final and given target density matrices and the maximization of the Hilbert–Schmidt scalar product between the final and target density matrices without taking into account restrictions on the evolution of the density matrix over the entire time. Nonlocal methods for improving controls for a state-linear problem were constructed based on the two-step and three-step gradient projection methods and the Krotov method, taking into account various restrictions on controls. For these problems, taking into account constraints on controls, adaptations of two- and three-step gradient projection methods were constructed for a qutrit; for the problems with the linear functional, non-local methods for improving controls based on the Krotov method were constructed. Analytical expressions were constructed for the objects appearing in these methods. The effectiveness of the methods was evaluated and numerical analysis was carried out for various quantum systems, including three-level open quantum systems. As part of the application of control methods, a mathematical model was constructed that describes the dissociation of the bond of molecules on the surface using the simultaneous use of coherent control methods based on a specially selected series of laser pulses and incoherent control methods based on selection of conditions on the surface, including the calculation of the proton desorption rate. The Kohn-Sham equations and the time-dependent density functional method were used to find the rate of desorption of protons from the surface under the influence of a specially selected sequence of laser pulses of a given shape in the approximation of a harmonic transition state. The importance of solving these problems is determined by the growing need to develop methods for controlling quantum systems, including open ones, that is, interacting with the environment. The relevance of the study of control landscapes is determined by the need to evaluate the effectiveness of local search methods for constructing controls, which depends on the properties of the landscape. The relevance of developing methods for constructing controls for open quantum systems under the influence of coherent and incoherent controls is determined by the fact that in real conditions controlled quantum systems usually interact with the environment, and this interaction cannot be neglected, and moreover, in some cases it can be used as a useful resource. There are open problems in these areas, which form the basis of the project, solution of which will contribute to the development of mathematical methods for problems of quantum technologies.

Publications

1. Kuznetsov S.A., Pechen A.N. On Controllability of a Highly Degenerate Four-Level Quantum System with a “Chained” Coupling Hamiltonian Lobachevskii J. Math., Vol. 43, No. 7, pp. 1683-1692 (year - 2022) https://doi.org/10.1134/S1995080222100225

2. Myachkova A.A., Pechen A.N. Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems Proceedings of the Steklov Institute of Mathematics, - (year - 2023)

3. Volkov B.O., Pechen A.N. Higher order traps for some strongly degenerate quantum control systems Russian Mathematical Surveys, - (year - 2023)

Annotation of the results obtained in 2023
Quantum technologies are based on the use of the properties of individual quantum systems of atomic, molecular and nanoscopic sizes. This area is of high fundamental and practical interest associated with a wide range of applications, from problems of information transmission, quantum computing, cryptography and metrology to problems of synthesis of materials with specified properties and laser control of chemical reactions and processes. The development of quantum technologies requires the development of mathematical methods for controlling closed and open quantum systems. One of the important tasks is the development, analysis and application of various optimization methods for control of quantum systems. In this year, the work was carried out to solve the problems planned for this year for control of quantum systems, including the study of local properties of objective functionals for quantum control problems, the development of methods for constructing controls for optimal control of open finite-level quantum systems, applications of methods for coherent and incoherent control of quantum systems. As part of the study of local methods for constructing optimal controls, the local properties of objective functionals of control problems for finite-level quantum systems were studied. A trap is a point of local, but not global extremum of the objective functional of the quantum control problem. More generally, a trap in the quantum control landscape is a critical control which slows down local search algorithms. A finite-order trap is determined by the Taylor expansion of the objective functional in a neighborhood of this control. The landscape of the problem of maximizing the average of a quantum mechanical observable was studied for traps in the general sense. We considered a class of quantum systems for which the free Hamiltonian, the initial state, and the quantum observable pairwise commute. For controllable systems from this class, sufficient conditions are obtained for the case when a zero constant control becomes a trap of finite order. For three-level and four-level quantum systems, traps of various order were studied using analytical methods. A numerical analysis of the influence of higher-order traps on the difficulty of the optimization problem was carried out for a number of three-level and four-level quantum systems with traps of various orders. In addition, the order of the trap for an N-level quantum system with a special symmetry in the free Hamiltonian and a “chain” interaction Hamiltonian is investigated. The presence of second-order traps has been established for the special case of a non-fully controllable three-level system with one forbidden transition. All controllable three-level quantum systems such as the lambda-atom, which arise when constructing quantum landscapes with higher-order traps, are fully described. As part of the study of control methods for open quantum systems, the problems of optimal control of open finite-level quantum systems under the simultaneous influence of coherent and incoherent controls, with restrictions on controls, with Mayer-type objective functionals representing the scalar product and the Hilbert–Schmidt distance between the final and given target density matrices, as well as similar problems of the Mayer–Boltz type were considered. For Mayer-type problems, analytical and numerical results were obtained for various local optimization methods, including those using a gradient. For Mayer–Boltz type problems, one-, two-, and three-step gradient projection methods and the regularized Krotov method were modified, and numerical results were obtained, including for the Werner–Holevo qutrit channel. A method for modifying Krotov's method through a combination of maximizing and projection mappings is proposed, and key mathematical constructions are constructed. As part of the study of the application of quantum control methods to models of dissotiation of molecules, a system of partial differential equations with respect to time and normal to the electrode surface with boundary conditions of the third kind was solved in order to find the distribution of the electric field and the concentration of hydrogen both on the surface and in the pore space of the electrode in the model of dissociation of a hydrogen molecule on the surface of dimolybdenum carbide in an external field of coherent radiation of various forms, for example, in the form of a sequence of instantaneous pulses. The dissociation rate was studied in the linear response theory approximation. The importance of solving these problems is determined by the growing need to develop methods for controlling quantum systems, including open ones, that is, those interacting with the environment. The relevance of studying control landscapes, that is, the local properties of target functionals of control problems for quantum systems, is determined by the need to evaluate the effectiveness of local search methods for constructing controls, which depends on the properties of the landscape. The relevance of developing methods for constructing controls for open quantum systems under the influence of coherent and incoherent controls is determined by the fact that in real experimental conditions controlled quantum systems, as a rule, cannot be neglected by the interaction of controlled quantum systems with the environment. Additionally, in some cases the environment can be used as a useful resource. There are open current theoretical problems in these areas, some of which form the basis of the proposed project, the implementation of which will make it possible to contribute to the development of mathematical methods for problems of quantum technologies.

Publications

1. Elovenkova M.A. Pechen A.N. Control landscape of measurement-assisted transition probability for a three-level quantum system with dynamical symmetry Quantum Reports, Vol. 5, No. 3, Pp. 526-545 (year - 2023) https://doi.org/10.3390/quantum5030035

2. Kuznetsov S.A., Pechen A.N. On controllability of Λ- and V-atoms and other three-level systems with two allowed transitions Lobachevskii Journal of Mathematics, Vol. 44, No. 6, Pp. 2101-2108 (year - 2023) https://doi.org/10.1134/S199508022306029X

3. Lyakhov K.A., Pechen A.N. Mathematical model of hydrogen dissociation on Mo2C surface in the presence of a laser field Lobachevskii Journal of Mathematics, Vol. 44, No. 6, pp. 2125–2134 (year - 2023) https://doi.org/10.1134/S1995080223060331

4. Morzhin O.V., Pechen A.N. Krotov type optimization of coherent and incoherent controls for open two-qubit systems Bulletin of Irkutsk State University-Series Mathematics, Vol. 45, Pp. 3–23 (year - 2023) https://doi.org/10.26516/1997-7670.2023.45.3

5. Morzhin O.V., Pechen A.N. Optimal state manipulation for a two-qubit system driven by coherent and incoherent controls Quantum Information Processing, Vol. 22. No. 6. Art. no. 241 (year - 2023) https://doi.org/10.1007/s11128-023-03946-x

6. Morzhin O.V., Pechen A.N. Using and optimizing time-dependent decoherence rates and coherent control for a qutrit system Proceedings of the Steklov Institute of Mathematics. Noncommutative Analysis and Quantum Information Theory Collected papers. Dedicated to Alexander Semenovich Holevo on the occasion of his 80th birthday, - (year - 2024)

7. Petruhanov V.N., Pechen A.N. Quantum gate generation in two-level open quantum systems by coherent and incoherent photons found with gradient search Photonics, Vol. 10. No. 2. Art. no. 220 (year - 2023) https://doi.org/10.3390/photonics10020220