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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 45

Krotov Type Optimization of Coherent and Incoherent Controls for Open Two-Qubit Systems

Author(s)
Oleg V. Morzhin1,2, Alexander N. Pechen1,2

1Steklov Mathematical Institute of RAS, Moscow, Russian Federation

2University of Science and Technology MISIS, Moscow, Russian Federation

Abstract
This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini–Kossakowski– Sudarshan–Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert–Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert–Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous controls with constaints and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap’s values close to their upper bounds. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.
About the Authors

Oleg V. Morzhin, Cand. Sci. (Phys.– Math.), Senior Scientific Researcher at the Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of RAS, project member at MISIS, Moscow, 119991, Russian Federation, https://www.mathnet.ru/eng/person30382

Alexander N. Pechen, Dr. Sci. (Phys.-Math.), RAS Prof., Head of the Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of RAS, Prof. and Chief Researcher at MISIS, Deputy Head of the Department “Methods of Modern Mathematics” of MIPT, Moscow, 119991, Russian Federation, https://www.mathnet.ru/eng/person17991

For citation
Morzhin O. V., Pechen A. N. Krotov Type Optimization of Coherent and Incoherent Controls for Open Two-Qubit Systems. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 45, pp. 3–23. https://doi.org/10.26516/1997-7670.2023.45.3
Keywords
open quantum system, incoherent quantum control, nonlocal improvement, optimization
UDC
530.145 517.97 517.98
MSC
81Q93, 34H05
DOI
https://doi.org/10.26516/1997-7670.2023.45.3
References
  1. Antipin A.S. Minimization of convex functions on convex sets by means of differential equations. Differ. Equat. 1994, vol. 30, no. 9, pp. 1365–1375.
  2. Antonik V.G., Srochko V.A. The projection method in linear-quadratic problems of optimal control. Comput. Math. Math. Phys., 1998, vol. 38, pp. 543–551.
  3. Arguchintsev A.V., Dykhta V.A., Srochko V.A. Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum. Russian Math. (Iz. VUZ), 2009, vol. 53, no. 1, pp. 1–35. https://doi.org/10.3103/S1066369X09010010
  4. Boscain U., Sigalotti M., Sugny D. Introduction to the Pontryagin maximum principle for quantum optimal control. PRX Quantum, 2021, art. no. 030203. https://doi.org/10.1103/PRXQuantum.2.030203
  5. Buldaev A.S. Optimization Methods of Control Systems: Tutorial. Ulan-Ude, ESSTU Publ., 2002. (in Russian) https://search.rsl.ru/ru/record/01002370365
  6. Buldaev A.S., Morzhin O.V. Improvement of controls in nonlinear systems on basis of boundary value problems. The Bull. Irkutsk State Univ. Ser. Math., 2009, vol. 2, no. 1, pp. 94–107. (in Russian)
  7. Butkovskiy A.G., Samoilenko Yu.I. Control of Quantum-Mechanical Processes and Systems. Transl. of the book published in 1984 in Russian, Dordrecht, Kluwer Acad. Publ., 1990.
  8. Caneva T., Calarco T., Montangero S. Chopped random-basis quantum optimization. Phys. Rev. A, 2011, vol. 84, art. no. 022326. https://doi.org/10.1103/PhysRevA.84.022326
  9. de Fouqui`eres P., Schirmer S.G., Glaser S.J., Kuprov I. Second order gradient ascent pulse engineering. J. Magn. Reson., 2011, vol. 212, pp. 412–417. https://doi.org/10.1016/j.jmr.2011.07.023
  10. Dong D.-Y., Chen C.-L., Tarn T.-J., Pechen A., Rabitz H. Incoherent control of quantum systems with wavefunction controllable subspaces via quantum reinforcement learning. IEEE Trans. Syst. Man Cybern., 2008, pp. 957–962. https://doi.org/10.1109/TSMCB.2008.926603
  11. Goerz M.H., Reich D.M., Koch C.P. Optimal control theory for a unitary operation under dissipative evolution. New J. Phys., 2014, vol. 16, art. no. 055012. (Corrigendum: New J. Phys., 2021, art. no. 039501. Also arXiv:1312.0111). https://doi.org/10.1088/1367-2630/16/5/055012
  12. Gough J., Belavkin V.P., Smolyanov O.G. Hamilton–Jacobi–Bellman equations for quantum optimal feedback control. J. Opt. B: Quantum Semiclass. Opt., 2005, vol. 7(10), pp. S237–S244. https://doi.org/10.1088/1464-4266/7/10/006
  13. J¨ager G., Reich D.M., Goerz M.H., Koch C.P., Hohenester U. Optimal quantum control of Bose-Einstein condensates in magnetic microtraps: Comparison of GRAPE and Krotov optimization schemes. Phys. Rev. A, 2014, vol. 90, art. no. 033628. https://doi.org/10.1103/PhysRevA.90.033628
  14. Judson R.S., Rabitz H. Teaching lasers to control molecules. Phys. Rev. Lett., 1992, vol. 68, art. no. 1500. https://doi.org/10.1103/PhysRevLett.68.1500
  15. Kazakov V.A., Krotov V.F. Optimal control of resonant interaction between light and matter. Automat. Remote Control, 1987, pp. 430–434.
  16. Kallush S., Dann R., Kosloff R. Controlling the uncontrollable: Quantum control of open system dynamics. Sci. Adv., 2022, vol. 8, art. no. eadd0828. https://doi.org/10.1126/sciadv.add0828
  17. Khaneja N., Reiss T., Kehlet C., Schulte-Herbr¨uggen T., Glaser S.J. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson., 2005, vol. 172, pp. 296–305. https://doi.org/10.1016/j.jmr.2004.11.004
  18. Koch C.P., Boscain U., Calarco T., Dirr G., Filipp S., Glaser S.J., Kosloff R., Montangero S., Schulte-Herbr¨uggen T., Sugny D., Wilhelm F.K. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe. EPJ Quantum Technol., 2022, vol. 9, art. no. 19. https://doi.org/10.1140/epjqt/s40507-022-00138-x
  19. Krotov V.F., Feldman I.N. An iterative method for solving problems of optimal control. Engrg. Cybern., 1983, vol. 21, no. 2, pp. 123–130.
  20. Krotov V.F. Global Methods in Optimal Control Theory. New York, Marcel Dekker, Inc., 1996.
  21. Krotov V.F. Control of the quantum systems and some ideas of the optimal control theory. Autom. Remote Control, 2009, vol. 70, pp. 357–365. https://doi.org/10.1134/S0005117909030035
  22. Krotov V.F., Bulatov A.V., Baturina O.V. Optimization of linear systems with controllable coefficients. Autom. Remote Control, 2011, vol. 72, pp. 1199–1212. https://doi.org/10.1134/S0005117911060063
  23. Krotov V.F., Morzhin O.V., Trushkova E.A. Discontinuous solutions of the optimal control problems. Iterative optimization method. Automat. Remote Control, 2013, vol. 74, pp. 1948–1968. https://doi.org/10.1134/S0005117913120035
  24. Lokutsievskiy L., Pechen A. Reachable sets for two-level open quantum systems driven by coherent and incoherent controls. J. Phys. A, 2021, art. no. 395304. https://doi.org/10.1088/1751-8121/ac19f8
  25. Morzhin O.V. Nonlocal improvement of controlling functions and parameters in nonlinear dynamical systems. Autom. Remote Control, 2012, pp. 1822–1837. https://doi.org/10.1134/S0005117912110057
  26. Morzhin O.V., Pechen A.N. Krotov method for optimal control of closed quantum systems. Russian Math. Surveys, 2019, vol. 74, pp. 851–908. https://doi.org/10.1070/RM9835
  27. Morzhin O.V., Pechen A.N. Maximization of the overlap between density matrices for a two-level open quantum system driven by coherent and incoherent controls. Lobachevskii J. Math., 2019, vol. 40, pp. 1532–1548. https://doi.org/10.1134/S1995080219100202
  28. Morzhin O.V., Pechen A.N. Optimal state manipulation for a two-qubit system driven by coherent and incoherent controls. Quantum Inf. Process., 2023, vol. 22, art. no. 241. https://doi.org/10.1007/s11128-023-03946-x
  29. Oza A., Pechen A., Dominy J., Beltrani V., Moore K., Rabitz H. Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution. J. Phys. A, 2009, vol. 42, art. no. 205305. https://doi.org/10.1088/1751-8113/42/20/205305
  30. Pechen A., Rabitz H. Teaching the environment to control quantum systems. Phys. Rev. A, 2006, vol. 73, art. no. 062102. https://doi.org/10.1103/PhysRevA.73.062102
  31. Pechen A. Engineering arbitrary pure and mixed quantum states. Phys. Rev. A, 2011, vol. 84, art. no. 042106. https://doi.org/10.1103/PhysRevA.84.042106
  32. Pechen A.N., Borisenok S., Fradkov A.L. Energy control in a quantum oscillator using coherent control and engineered environment. Chaos, Solitons & Fractals, 2022, vol. 164, art. no. 112687. https://doi.org/10.1016/j.chaos.2022.112687
  33. 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, 2023, vol. 10, art. no. 220. https://doi.org/10.3390/photonics10020220
  34. Saff E.B., Kuijlaars A.B.J. Distributing many points on a sphere. Math. Intell., 1997, vol. 19, no. 1, pp. 5–11. https://doi.org/10.1007/BF03024331
  35. Srochko V.A. Iterative Methods for Solving Optimal Control Problems. Moscow, Fizmatlit Publ., 2000. (in Russian) https://search.rsl.ru/ru/record/01000686861
  36. Srochko V.A., Ushakova S.N. The method of complete quadratic approximation in optimal control problems. Russian Math. (Iz. VUZ), 2004, no. 1, pp. 84–90.
  37. Tannor D.J., Kazakov V., Orlov V. Control of photochemical branching: Novel procedures for finding optimal pulses and global upper bounds. Time-Dependent Quantum Molecular Dynamics. Boston, Springer, 1992, pp. 347–360. https://doi.org/10.1007/978-1-4899-2326-4_24
  38. Vasiliev O.V., Arguchintsev A.V. Optimization Methods in Tasks and Exercises. Moscow, Fizmatlit Publ., 1999. (in Russian) https://search.rsl.ru/ru/record/01000641549
  39. Zhu W., Rabitz H. A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator. J. Chem. Phys., 1998, vol. 109, pp. 385–391. https://doi.org/10.1063/1.476575

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