On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

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Дәйексөз келтіру

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Аннотация

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem 3 in B.O. Volkov, O.V. Morzhin, A.N. Pechen, J. Phys. A: Math. Theor. 54, 215303 (2021)].

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Авторлар туралы

Boris Volkov

Steklov Mathematical Institute of Russian Academy of Sciences; National University of Science and Technology «MISIS»

Email: boris--volkov@yandex.ru
Candidate of physico-mathematical sciences, no status

Alexander Pechen

Steklov Mathematical Institute of Russian Academy of Sciences; National University of Science and Technology «MISIS»

Email: apechen@gmail.com
Doctor of physico-mathematical sciences, no status

Әдебиет тізімі

  1. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, F. K. Wilhelm, “Training Schrödinger's cat: quantum optimal control. Strategic report on current status, visions and goals for research in Europe”, Eur. Phys. J. D, 69:12 (2015), 279, 24 pp.
  2. C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, F. K. Wilhelm, “Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe”, EPJ Quantum Technol., 9 (2022), 19, 60 pp.
  3. А. Г. Бутковский, Ю. И. Самойленко, Управление квантовомеханическими процессами, Наука, М., 1984, 256 с.
  4. D. J. Tannor, Introduction to quantum mechanics: a time dependent perspective, Univ. Science Books, Sausalito, CA, 2007, 662 pp.
  5. V. S. Letokhov, Laser control of atoms and molecules, Oxford Univ. Press, Oxford, 2007, 328 pp.
  6. K. W. Moore, A. Pechen, Xiao-Jiang Feng, J. Dominy, V. J. Beltrani, H. Rabitz, “Why is chemical synthesis and property optimization easier than expected?”, Phys. Chem. Chem. Phys., 13:21 (2011), 10048–10070
  7. C. P. Koch, “Controlling open quantum systems: tools, achievements, and limitations”, J. Phys. Condens. Matter, 28:21 (2016), 213001
  8. D. D'Alessandro, Introduction to quantum control and dynamics, Adv. Appl. Math., 2nd ed., CRC Press, Boca Raton, FL, 2021, xvi+400 pp.
  9. H. A. Rabitz, M. M. Hsieh, C. M. Rosenthal, “Quantum optimally controlled transition landscapes”, Science, 303:5666 (2004), 1998–2001
  10. Tak-San Ho, H. Rabitz, “Why do effective quantum controls appear easy to find?”, J. Photochem. Photobiol. A, 180:3 (2006), 226–240
  11. K. W. Moore, R. Chakrabarti, G. Riviello, H. Rabitz, “Search complexity and resource scaling for the quantum optimal control of unitary transformations”, Phys. Rev. A, 83:1 (2011), 012326, 15 pp.
  12. A. N. Pechen, D. J. Tannor, “Are there traps in quantum control landscapes?”, Phys. Rev. Lett., 106:12 (2011), 120402, 3 pp.
  13. A. Pechen, N. Il'in, “Trap-free manipulation in the Landau–Zener system”, Phys. Rev. A, 86:5 (2012), 052117, 6 pp.
  14. A. N. Pechen, D. J. Tannor, “Quantum control landscape for a $Lambda$-atom in the vicinity of second-order traps”, Israel J. Chem., 52:5 (2012), 467–472
  15. P. de Fouquieres, S. G. Schirmer, “A closer look at quantum control landscapes and their implication for control optimization”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16:3 (2013), 1350021, 24 pp.
  16. А. Н. Печень, Н. Б. Ильин, “Когерентное управление кубитом свободно от ловушек”, Избранные вопросы математической физики и анализа, Сборник статей. К 90-летию со дня рождения академика Василия Сергеевича Владимирова, Труды МИАН, 285, МАИК «Наука/Интерпериодика», М., 2014, 244–252
  17. A. N. Pechen, D. J. Tannor, “Control of quantum transmission is trap-free”, Can. J. Chem., 92:2 (2014), 157–159
  18. M. Larocca, P. M. Poggi, D. A. Wisniacki, “Quantum control landscape for a two-level system near the quantum speed limit”, J. Phys. A, 51:38 (2018), 385305, 14 pp.
  19. D. V. Zhdanov, “Comment on 'Control landscapes are almost always trap free: a geometric assessment'”, J. Phys. A, 51:50 (2018), 508001, 8 pp.
  20. B. Russell, Rebing Wu, H. Rabitz, “Reply to comment on 'Control landscapes are almost always trap free: a geometric assessment'”, J. Phys. A, 51:50 (2018), 508002, 7 pp.
  21. А. Н. Печень, Н. Б. Ильин, “Об экстремумах целевого функционала в задаче генерации однокубитных квантовых вентилей на малых временах”, Изв. РАН. Сер. матем., 80:6 (2016), 217–229
  22. B. O. Volkov, O. V. Morzhin, A. N. Pechen, “Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates”, J. Phys. A, 54:21 (2021), 215303, 23 pp.
  23. M. Dalgaard, F. Motzoi, J. Sherson, “Predicting quantum dynamical cost landscapes with deep learning”, Phys. Rev. A, 105:1 (2022), 012402, 12 pp.
  24. A. Pechen, D. Prokhorenko, Rebing Wu, H. Rabitz, “Control landscapes for two-level open quantum systems”, J. Phys. A, 41:4 (2008), 045205, 18 pp.
  25. A. Oza, A. Pechen, J. Dominy, V. Beltrani, K. Moore, H. Rabitz, “Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution”, J. Phys. A, 42:20 (2009), 205305, 22 pp.
  26. A. Pechen, C. Brif, Rebing Wu, R. Chakrabarti, H. Rabitz, “General unifying features of controlled quantum phenomena”, Phys. Rev. A, 82:3 (2010), 030101, 4 pp.
  27. A. Pechen, H. Rabitz, “Unified analysis of terminal-time control in classical and quantum systems”, Europhys. Lett. EPL, 91:6 (2010), 60005, 6 pp.
  28. Л. С. Понтрягин, В. Г. Болтянский, Р. В. Гамкрелидзе, Е. Ф. Мищенко, Математическая теория оптимальных процессов, Физматгиз, М., 1961, 391 с.
  29. U. Boscain, M. Sigalotti, D. Sugny, “Introduction to the Pontryagin maximum principle for quantum optimal control”, PRX Quantum, 2:3 (2021), 030203, 31 pp.
  30. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms”, J. Magn. Reson., 172:2 (2005), 296–305
  31. T. Schulte-Herbrüggen, S. J. Glaser, G. Dirr, U. Helmke, “Gradient flows for optimization in quantum information and quantum dynamics: foundations and applications”, Rev. Math. Phys., 22:6 (2010), 597–667
  32. D. J. Tannor, V. Kazakov, V. Orlov, “Control of photochemical branching: novel procedures for finding optimal pulses and global upper bounds”, Time-dependent quantum molecular dynamics, NATO ASI Ser. B, 299, Springer, New York, 1992, 347–360
  33. О. В. Моржин, А. Н. Печень, “Метод Кротова в задачах оптимального управления замкнутыми квантовыми системами”, УМН, 74:5(449) (2019), 83–144
  34. T. Caneva, T. Calarco, S. Montangero, “Chopped random-basis quantum optimization”, Phys. Rev. A, 84:2 (2011), 022326, 9 pp.
  35. R. Eitan, M. Mundt, D. J. Tannor, “Optimal control with accelerated convergence: combining the Krotov and quasi-Newton methods”, Phys. Rev. A, 83:5 (2011), 053426, 10 pp.
  36. А. А. Аграчев, Ю. Л. Сачков, Геометрическая теория управления, Физматлит, М., 2005, 392 с.
  37. R. S. Judson, H. Rabitz, “Teaching lasers to control molecules”, Phys. Rev. Lett., 68:10 (1992), 1500–1503
  38. Daoyi Dong, Chunlin Chen, Tzyh-Jong Tarn, A. Pechen, H. Rabitz, “Incoherent control of quantum systems with wavefunction-controllable subspaces via quantum reinforcement learning”, IEEE Trans. Systems Man Cybernet. B, 38:4 (2008), 957–962
  39. O. V. Morzhin, A. N. Pechen, “Generation of density matrices for two qubits using coherent and incoherent controls”, Lobachevskii J. Math., 42:10 (2021), 2401–2412
  40. O. V. Morzhin, A. N. Pechen, “On optimization of coherent and incoherent controls for two-level quantum systems”, Изв. РАН. Сер. матем., 87:5 (2023) (to appear)
  41. В. И. Богачев, О. Г. Смолянов, В. И. Соболев, Топологические векторные пространства и их приложения, НИЦ “Регулярная и хаотическая динамика”, М.–Ижевск, 2012, 584 с.
  42. B. C. Владимиров, Уравнения математической физики, 4-е изд., Наука, М., 1981, 512 с.

© Волков Б.O., Печень А.N., 2023

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