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COMMON PART

Project Number20-11-18044

Project titleMathematical methods for problems of quantum technologies and dynamics of open quantum systems

Project LeadPechen Alexander

AffiliationSteklov Mathematical Institute of Russian Academy of Sciences,

Implementation period | 2020 - 2021 |

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

Keywordscontrol of open quantum systems, quantum technologies, quantum decoherence, many-body systems, thermalization, quantum information theory, quantum entanglement, superadditivity of quantum information

PROJECT CONTENT

Annotation

Quantum technologies are technologies based on the use of individual quantum systems of atomic, molecular and nanoscale sizes. This area is of high importance motivated by a wide range of applications, from tasks of information processing, cryptography, metrology to the synthesis of materials with desired properties and control of chemical reactions. At present, quantum technologies are actively developing in the Russian Federation and in the world. In Russia, the national competence centers of the NTI "Quantum Communications" and the NTI "Quantum Technologies" were created, the roadmap "Quantum Technologies" was developed. Many leading world scientific centers have laboratories in the fields of quantum technology.
The development of quantum technologies requires the development of mathematical methods for studying dynamics and controlling open, that is, interacting with the environment, quantum systems, the transmission and processing of information using quantum channels with the almost inevitable presence of noise requires the development of methods for constructing error correction codes. The actuality of solving this scientific problem is determined by the fact that the presence of noise and interaction with the environment are inevitable factors in many experimental works on quantum technologies. The development of appropriate mathematical methods is required.
During the implementation of the project 2017, the work was carried out to solve the fundamental scientific problem of developing mathematical methods for quantum technologies, including theory of control of open quantum systems, the derivation and analysis of master equations for open quantum systems, the study of decoherence of quantum systems, establishing a relationship between quantum uncertainty, quantum coherence, applying the results to quantum cryptography and for controlling quantum systems with similar spectra. Project 2020 is aimed at further solving novel important scientific problems in this area, including analysis of reachability and controllability sets and development of methods for constructing optimal controls for open quantum systems, applying machine learning to constructing new dynamic master equations for open quantum systems, studying systems with non-markovian noise, determining non-uniqueness master equations taking into account given restrictions, the analysis of the control problem of quantum systems with similar spectra, analysis of error correction subspaces, the analysis of the characteristics of quantum ensembles which are divided among several participants.

Expected results

In the course of the project, it is planned to conduct research in the areas of mathematical methods of dynamics and control of open quantum systems, including the study of reachability and controllability sets and the development of methods for constructing optimal controls for open quantum systems, the use of machine learning to build new dynamic master equations for open quantum systems, the study of systems with non-markovian noises, the analysis of the non-uniqueness of master equations under given constraints, analysis of control of quantum systems with similar spectra for laser-assisted isotope separation, analysis of error correction subspaces, the study of characteristics of quantum ensembles divided between several participants.
Expected results include the construction of reachability and controllability sets for open two-level quantum systems under the influence of coherent and incoherent piecewise constant controls depending on the allowed variations of these controls; adaptation of second-order methods for constructing optimal controls for controlling open quantum systems depending on piecewise continuous coherent and incoherent controls; investigation of control problems for quantum systems with similar spectra; investigation of the ambiguity of master equations describing the dynamics of markovian and non-markovian open quantum systems under given constraints, including for a model of a quantum oscillator with eliminated quantum harmonics; the construction of a theory of control of quantum systems in the presence of non-markovian noises, which will increase the accuracy of quantum operations on noisy quantum processors from a small number of qubits and will open up new possibilities for their application; study of the characteristics of quantum ensembles when they are divided between several participants. Detailed work plan and expected results are formulated in the relevant forms of the application.
The high significance of the expected results is ensured by the fact that in most situations the physical systems used for tasks of quantum technologies are open, that is, they interact with the environment. This circumstance requires the development of methods for the analysis of controlled open quantum systems. The results will be at the high world scientific level in the field of mathematical problems of quantum technologies and are planned for publication in leading Russian and world scientific journals. The results of the project in the economy can be applied to the development of quantum technologies, including various aspects of the tasks of creating quantum simulators and protocols of quantum key distribution, including in connection with the work on the Quantum Technologies roadmap. The methods planned for development can be applied to study various aspects of atomic and molecular dynamics induced by external laser fields, problems of controlling quantum systems, and decoherence dynamics in quantum systems. Methods of analysis of open quantum systems can be applied to the construction of error correction protocols. The results in the field of establishing the connection of the quantitative characteristics of various quantum phenomena can be applied in problems of quantum communications.

REPORTS

Annotation of the results obtained in 2021

In the course of implementation of the project in the reporting year, the work was carried out to solve the tasks set for this year on the control of quantum systems, including the adaptation of the second-order Krotov method for the optimal generation of the final density matrix of an open one-qubit quantum system under the influence of coherent and incoherent controls, the development of methods for controlling quantum systems with similar spectra, study of non-Markov open quantum systems, study of the relationship between the coherence of an ensemble of quantum states and the uncertainty of a quantum observable, quantum anticliques. In the reporting year, the results were published in a number of articles in journals indexed in the Web of Science / Scopus and presented in 18 presentations at conferences and seminars, including video recordings of a number of reports posted in the public domain on the All-Russian mathematical portal MathNet.Ru.
For an open single-qubit quantum system evolving under the influence of coherent and incoherent controls, the problem of optimal generation of a final density matrix, which for a given final time and control constraints has the minimum Hilbert – Schmidt distance to a given target density matrix is considered and solved. To numerically solve this problem in terms of Bloch vectors, two variants of Krotov's method of the second order are adapted, and regularization with respect to controls is used. For this optimal control problem, numerical experiments with these versions of the Krotov method for different initial and target points in the Bloch ball, constraints on the amplitudes of controls, and final times have shown the operability of both versions of the Krotov method with an appropriate setting. The problem of controlling open quantum systems under the influence of coherent and incoherent controls was considered in the model of a quantum oscillator interacting with an environment with ejected quantum harmonics, for which the degree of non-uniqueness of the obtained equations was established. The problem of applying control methods for quantum systems in a mathematical model of separation of quantum systems with similar spectra is investigated, for which the objective function was constructed, constraints on the controls were found, the evolution of the objective function in this model was calculated depending on the parameters of the model. A quantum channel is constructed that generates the operator graph as a linear span of positive definite operator-valued measures. In the considered model, the corresponding quantum anti-clique is infinite-dimensional.
The problem of propagation of quantum states through quantum channels with memory was posed and solved. The mathematical description of non-Markovian memory effects was carried out in the Markovian embedding model. The properties of solutions of equations describing the effect of non-Markovian memory effects on the dynamics of composite quantum systems and the degree of their entanglement are studied. In particular, it is shown that a two-qubit state at the output of a channel with memory can contain quantum correlations (i.e., be entangled), even if the effective environment is initially in the maximally mixed state, and the channel input is a factorized state of two qubits, which do not interact with each other directly when they propagate through the channel. A model of non-Markovian noise for qubit registers is constructed based on the Markovian embedding theory. We have calculated the non-Markovian quantum dynamics of two qubits under non-Markovian noise and external control used to implement an entangling quantum operation on qubits. In the process-tensor formalism, we have considered some methods for controlling quantum systems in the presence of non-Markov noises.
An explicit relation between a pure entangled state, an observable applied to one part of it, and an ensemble of states obtained in another part of it, for an arbitrary pure entangled quantum state, and an arbitrary observable (or arbitrary ensemble) is obtained. It is shown that this relation coincides with the quantum ensemble-observable duality which has already been considered earlier. An estimate on the coherence (minimum relative entropy of coherence over all bases) of the ensemble of states through the uncertainty of the observable and the entanglement measure of the pure quantum state is obtained. The result suggests that the higher the uncertainty of the applied observable and the higher the entanglement measure of the initial pure two-particle state, the greater will be the coherence of the resulting ensemble.

Publications

**1.** *Amosov G.G., Mokeev A.S., Pechen A.N.* **On the construction of a quantum channel corresponding to non-commutative graph for a qubit interacting with quantum oscillator** Lobachevskii Journal of Mathematics, Vol. 42, No. 10, P. 2280—2284 (year - 2021) https://doi.org/10.1134/S1995080221100036

**2.** *Kodukhov A.D., Kronberg D.A.* **Measuring entangled state: On connection between observable uncertainty and ensemble coherence** AIP Conference Proceedings, Vol. 2362, P. 050003 (year - 2021) https://doi.org/10.1063/5.0055362

**3.** *Lyakhov K.A., and Pechen A.N.* **Constrained optimization criterion for zirconium isotope separation by the method of laser-assisted retardation of condensation** Proceedings of the Steklov Institute of Mathematics, Vol. 313, No. 1, P. 131—141 (year - 2021) https://doi.org/10.1134/S0081543821020139

**4.** *Lyakhov K.A., and Pechen A.N.* **Selective multi-line excitation of isotopologues with similar quantum spectra as a function of gas flow pressure, temperature, and laser pulse spectrum** AIP Conference Proceedings, Vol. 2362, P. 040008 (year - 2021) https://doi.org/10.1063/5.0054950

**5.** *Morzhin O.V., Pechen A.N.* **On reachable and controllability sets for minimum-time control of an open two-level quantum system** Proceedings of the Steklov Institute of Mathematics, Vol. 313, No. 1, P. 149—164 (year - 2021) https://doi.org/10.1134/S0081543821020152

**6.** *Morzhin O.V., Pechen A.N.* **Numerical estimation of reachable and controllability sets for a two-level open quantum system driven by coherent and incoherent controls** AIP Conference Proceedings, Vol. 2362, P. 060003 (year - 2021) https://doi.org/10.1063/5.0055004

Annotation of the results obtained in 2020

In 2020, the research was carried out for the first year of this project, which is a continuation of the RSF project performed in 2017-2019. In the reporting year, the work was carried out to solve the problems posed for this year on control of quantum systems, including the study of reachability and controllability sets for open quantum systems, development of numerical methods for optimal quantum control, study of master equations and non-Markov dynamics for open quantum systems, study of the effect of entanglement on capabilities of quantum communication protocols, application of theory of control of quantum systems with similar spectra to zirconium isotope separation, development of quantum error correction theory. Some of the results were published in articles in journals indexed in the Web of Science, Scopus. Some articles are under preparation and submission to journals. The results of the work were presented in 16 talks at conferences and seminars.
For an open two-level quantum system with dynamics determined by the Gorini – Kossakowski – Sudarshan – Lindblad (GKSL) master equation, which includes coherent and incoherent controls, new results were obtained on the numerical estimation and analytical description of reachable and controllability sets. New numerical results were obtained on the analysis of reachable points in the Bloch ball for the case with zero incoherent control and coherent control equal to zero before some switching time moment and the cosine function after it. Controllability sets for zero coherent control and incoherent control from the class of constant functions for certain target states are described analytically. The section method was adapted, taking into account additional restrictions on the variations of piecewise constant controls, to numerical estimation of the reachability and controllability sets in the Bloch ball, and the corresponding computational experiments were carried out. By analyzing the possibilities of using machine learning for a class of optimal performance problems for a given two-level quantum system with piecewise constant controls, the variations of which are constrained, the machine learning algorithm for finding suboptimal final times and controls was improved, based on a combination of the kNN method and neural network training. A statistical analysis of the performance of this algorithm was obtained numerically, including dependence on the size of the training data set.
Non-Markovian dynamics of a qubit subjected to an external coherent control was considered within the model of Markovian embedding that reflects the physical idea of dividing the environment into the near one and the distant one. A formula was derived which relates the time-local dissipator for a qubit with the GKSL generator for the system «qubit + near environment». To find the GKSL generator, the maximum likelihood estimation for a series of single-shot projective measurements on the qubit was used. To maximize the likelihood function, its gradient with respect to the quantum channel describing the evolution of the qubit and its near environment for a unit time was computed. The process tensor for the qubit, i.e., a multi-time input-output tensor for the qubit density operator that takes into account local in time operations on the qubit (measurements, unitary gates) was constructed. The non-Markovian dynamics in the collision model with composed environment without and with external driving (in the latter case the adiabatic approach is not applicable) was analyzed.
The uncertainty in the definition of master equations was investigated, which does not allow unambiguously identify the master equation for the evolution of an open quantum system controlled by coherent and incoherent controls. Differences in the dynamics are obtained for higher-order moments of various quantum observables and for equilibrium states. It is shown that the equilibrium states for the master equations in the considered class are different.
The use of the entanglement resource in a number of quantum communication protocols, both with one and several recipients, was considered. These protocols include the transfer of public information to one or more recipients, as well as the distribution of independent secret keys with multiple recipients, including in situations of incomplete trust between participants. The estimates were obtained on mutual information for individual and collective measurements by the recipients, as well as on the secret key length, depending on the number of trusted participants. The Also the situation of application of quantum observables to a part of the entangled state was considered. Such a measurement generates the ensemble for the other part of the entangled state. Relations were established between the ensemble and the observable. The analysis was carried out to generalize the relationship between the entropic observable uncertainty and the coherence of the dual ensemble.
A model of noncommutative graph of errors was constructed, including a unitary evolution that determines the dynamic transformation of errors in time. In this model, the concept of maximum anticlique was considered, that is, the maximum possible set of quantum states that can be transmitted with zero error for a given set of errors. The constructions were illustrated for the case of unitary dynamics.

Publications

**1.** *Kronberg D.A.* **Role of collective preparation and measurement of states in some quantum communication protocols** Quantum Electronics, Т. 50, Вып. 5, С. 461-468 (year - 2020) https://doi.org/10.1070/QEL17200

**2.** *Luchnikov I.A., Vintskevich S.V., Grigoriev D.A., Filippov S.N.* **Machine Learning Non-Markovian Quantum Dynamics** Physical Review Letters, Том 124, Стр. 140502 (year - 2020) https://doi.org/10.1103/PhysRevLett.124.140502

**3.** *Lyakhov K.A., Pechen A.N.* **CO2 laser system design for efficient boron isotope separation by the method of selective laser-assisted retardation of condensation** Applied Physics B, Том 126, Стр. 141 (year - 2020) https://doi.org/10.1007/s00340-020-07445-0

**4.** *Lyakhov K.A., Pechen A.N.* **Evolution of the Enrichment Factor for an Iterative Scheme of Zirconium Isotopes Separation** Lobachevskii Journal of Mathematics, Том 41, Вып. 12, Стр. 2345-2352 (year - 2020) https://doi.org/10.1134/S1995080220120252

**5.** *Morzhin O.V., Pechen A.N.* **Machine Learning for Finding Suboptimal Final Times and Coherent and Incoherent Controls for an Open Two-Level Quantum System** Lobachevskii Journal of Mathematics, Том 41, Вып. 12, Стр. 2353–2368 (year - 2020) https://doi.org/10.1134/S199508022012029X