Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Efficiency of optimal control for noisy spin qubits in diamond

Hendry M. Lim1,*, Genko T. Genov2,*, Roberto Sailer2, Alfaiz Fahrurrachman1, Muhammad A. Majidi1, Fedor Jelezko2, and Ressa S. Said2,†

  • 1Department of Physics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia
  • 2Institute for Quantum Optics and Center for Integrated Quantum Science and Technology, Universität Ulm, D-89081 Ulm, Germany

  • *These two authors contributed equally.
  • †Contact author: ressa.said@uni-ulm.de

Phys. Rev. Applied 24, 054064 – Published 20 November, 2025

DOI: https://doi.org/10.1103/3hnz-bysr

Abstract

Decoherence is a major challenge for quantum technologies, whose negative effects can be mitigated by quantum optimal control. The decoherence dynamics significantly varies depending on the characteristics of the system’s environment, consequently affecting the optimization outcomes. In this work, we investigate the optimization of a negatively charged nitrogen-vacancy (NV−) center qubit-spin-inversion control pulse, and observe two error-compensation mechanisms, depending on how long the correlation time is compared to the pulse duration. Additionally, we analyze the effects of control space constraints and optimization options to identify a strategy to improve the optimization performance. Finally, we present experimental realizations of the optimized pulses, exhibiting the applicability of the procedure. Our work serves as a generic yet essential guide to implementing optimal control in the presence of realistic noise.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (126)

  1. P. Rembold, N. Oshnik, M. M. Müller, S. Montangero, T. Calarco, and E. Neu, Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond, AVS Quantum Sci. 2, 024701 (2020).
  2. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  3. S. Prawer and I. Aharonovich, Quantum Information Processing with Diamond: Principles and Applications (Woodhead Publishing Limited, Cambridge, 2018), 1st ed.
  4. N. Bar-Gill, L. Pham, A. Jarmola, D. Budker, and R. Walsworth, Solid-state electronic spin coherence time approaching one second, Nat. Commun. 4, 1743 (2013).
  5. E. D. Herbschleb, H. Kato, Y. Maruyama, T. Danjo, T. Makino, S. Yamasaki, I. Ohki, K. Hayashi, H. Morishita, M. Fujiwara, and N. Mizuochi, Ultra-long coherence times amongst room-temperature solid-state spins, Nat. Commun. 10, 3766 (2019).
  6. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  7. N. Oshnik, P. Rembold, T. Calarco, S. Montangero, E. Neu, and M. M. Müller, Robust magnetometry with single nitrogen-vacancy centers via two-step optimization, Phys. Rev. A 106, 013107 (2022).
  8. M. Hirose, C. D. Aiello, and P. Cappellaro, Continuous dynamical decoupling magnetometry, Phys. Rev. A 86, 062320 (2012).
  9. P. J. Vetter, A. Marshall, G. T. Genov, T. F. Weiss, N. Striegler, E. F. Großmann, S. Oviedo-Casado, J. Cerrillo, J. Prior, P. Neumann, and F. Jelezko, Zero- and low-field sensing with nitrogen-vacancy centers, Phys. Rev. Appl. 17, 044028 (2022).
  10. L. Childress and R. Hanson, Diamond NV centers for quantum computing and quantum networks, MRS Bull. 38, 134 (2013).
  11. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature 464, 45 (2010).
  12. K. Nemoto, M. Trupke, S. J. Devitt, B. Scharfenberger, K. Buczak, J. Schmiedmayer, and W. J. Munro, Photonic quantum networks formed from NV-centers, Sci. Rep. 6, 26284 (2016).
  13. M. Ruf, N. H. Wan, H. Choi, D. Englund, and R. Hanson, Quantum networks based on color centers in diamond, J. Appl. Phys. 130, 070901 (2021).
  14. M. Block, B. Kobrin, A. Jarmola, S. Hsieh, C. Zu, N. L. Figueroa, V. M. Acosta, J. Minguzzi, J. R. Maze, D. Budker, and N. Y. Yao, Optically enhanced electric field sensing using nitrogen-vacancy ensembles, Phys. Rev. Appl. 16, 024024 (2021).
  15. H. T. Dinani, E. Muñoz, and J. R. Maze, Sensing electrochemical signals using a nitrogen-vacancy center in diamond, Nanomaterials 11, 358 (2021).
  16. M. Chipaux, A. Tallaire, J. Achard, S. Pezzagna, J. Meijer, V. Jacques, J.-F. Roch, and T. Debuisschert, Magnetic imaging with an ensemble of nitrogen-vacancy centers in diamond, Eur. Phys. J. D 69, 166 (2015).
  17. A. Boretti, L. Rosa, J. Blackledge, and S. Castelletto, Nitrogen-vacancy centers in diamond for nanoscale magnetic resonance imaging applications, Beilstein J. Nanotechnol. 10, 2128 (2019).
  18. C. Abeywardana, V. Stepanov, F. H. Cho, and S. Takahashi, in Quantum and Nonlinear Optics III, International Society for Optics and Photonics, edited by Q. Gong, G.-C. Guo, and B. S. Ham (SPIE, Beijing, 2014), Vol. 9269, p. 92690K.
  19. J. D. Breeze, E. Salvadori, J. Sathian, N. M. Alford, and C. W. M. Kay, Continuous-wave room-temperature diamond maser, Nature 555, 493 (2018).
  20. Q. Wu, Y. Zhang, X. Yang, S.-L. Su, C. Shan, and K. Mølmer, A superradiant maser with nitrogen-vacancy center spins, Sci. China Phys. Mech. Astron. 65 (2021).
  21. J.-P. Tetienne, T. Hingant, L. Rondin, A. Cavaillès, L. Mayer, G. Dantelle, T. Gacoin, J. Wrachtrup, J.-F. Roch, and V. Jacques, Spin relaxometry of single nitrogen-vacancy defects in diamond nanocrystals for magnetic noise sensing, Phys. Rev. B 87, 235436 (2013).
  22. A. Mzyk, A. Sigaeva, and R. Schirhagl, Relaxometry with nitrogen vacancy (NV) centers in diamond, Acc. Chem. Res. 55, 3572 (2022).
  23. B. Bürgler, T. F. Sjolander, O. Brinza, A. Tallaire, J. Achard, and P. Maletinsky, All-optical nuclear quantum sensing using nitrogen-vacancy centers in diamond, npj Quantum Inf. 9, 56 (2023).
  24. J. Lau, Color centers in diamond yield potential single photon source for quantum cryptography, MRS Bull. 25, 5 (2000).
  25. R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Nitrogen-vacancy centers in diamond: Nanoscale sensors for physics and biology, Annu. Rev. Phys. Chem. 65, 83 (2014).
  26. T. Zhang, G. Pramanik, K. Zhang, M. Gulka, L. Wang, J. Jing, F. Xu, Z. Li, Q. Wei, P. Cigler, and Z. Chu, Toward quantitative bio-sensing with nitrogen–vacancy center in diamond, ACS Sens. 6, 2077 (2021).
  27. A. Kuwahata, T. Kitaizumi, K. Saichi, T. Sato, R. Igarashi, T. Ohshima, Y. Masuyama, T. Iwasaki, M. Hatano, F. Jelezko, M. Kusakabe, T. Yatsui, and M. Sekino, Magnetometer with nitrogen-vacancy center in a bulk diamond for detecting magnetic nanoparticles in biomedical applications, Sci. Rep. 10, 2483 (2020).
  28. R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, Coherent dynamics of a single spin interacting with an adjustable spin bath, Science 320, 352 (2008).
  29. L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett. 82, 2417 (1999).
  30. G. T. Genov, Y. Ben-Shalom, F. Jelezko, A. Retzker, and N. Bar-Gill, Efficient and robust signal sensing by sequences of adiabatic chirped pulses, Phys. Rev. Res. 2, 033216 (2020).
  31. G. T. Genov, N. Aharon, F. Jelezko, and A. Retzker, Mixed dynamical decoupling, Quantum Sci. Technol. 4, 035010 (2019).
  32. A. Stark, N. Aharon, T. Unden, D. Louzon, A. Huck, A. Retzker, U. L. Andersen, and F. Jelezko, Narrow-bandwidth sensing of high-frequency fields with continuous dynamical decoupling, Nat. Commun. 8, 1105 (2017).
  33. Z.-H. Wang, G. de Lange, D. Ristè, R. Hanson, and V. V. Dobrovitski, Comparison of dynamical decoupling protocols for a nitrogen-vacancy center in diamond, Phys. Rev. B 85, 155204 (2012).
  34. D. Suter and G. A. Álvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys. 88, 041001 (2016).
  35. M. H. Levitt, Composite pulses, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).
  36. S. Wimperis, Broadband, narrowband, and passband composite pulses for use in advanced NMR experiments, J. Magn. Reson., A 109, 221 (1994).
  37. H. K. Cummins, G. Llewellyn, and J. A. Jones, Tackling systematic errors in quantum logic gates with composite rotations, Phys. Rev. A 67, 042308 (2003).
  38. B. T. Torosov and N. V. Vitanov, Smooth composite pulses for high-fidelity quantum information processing, Phys. Rev. A 83, 053420 (2011).
  39. G. T. Genov and N. V. Vitanov, Dynamical suppression of unwanted transitions in multistate quantum systems, Phys. Rev. Lett. 110, 133002 (2013).
  40. G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, Correction of arbitrary field errors in population inversion of quantum systems by universal composite pulses, Phys. Rev. Lett. 113, 043001 (2014).
  41. G. T. Genov, M. Hain, N. V. Vitanov, and T. Halfmann, Universal composite pulses for efficient population inversion with an arbitrary excitation profile, Phys. Rev. A 101, 013827 (2020).
  42. S. S. Ivanov, B. T. Torosov, and N. V. Vitanov, High-fidelity quantum control by polychromatic pulse trains, Phys. Rev. Lett. 129, 240505 (2022).
  43. J.-M. Cai, B. Naydenov, R. Pfeiffer, L. P. McGuinness, K. D. Jahnke, F. Jelezko, M. B. Plenio, and A. Retzker, Robust dynamical decoupling with concatenated continuous driving, New J. Phys. 14, 113023 (2012).
  44. I. Cohen, N. Aharon, and A. Retzker, Continuous dynamical decoupling utilizing time-dependent detuning, Fortschr. Phys. 65, 1600071 (2017).
  45. I. Baumgart, J.-M. Cai, A. Retzker, M. B. Plenio, and C. Wunderlich, Ultrasensitive magnetometer using a single atom, Phys. Rev. Lett. 116, 240801 (2016).
  46. N. Aharon, M. Drewsen, and A. Retzker, General scheme for the construction of a protected qubit subspace, Phys. Rev. Lett. 111, 230507 (2013).
  47. N. Timoney, I. Baumgart, M. Johanning, A. F. Varòn, M. B. Plenio, A. Retzker, and C. Wunderlich, Quantum gates and memory using microwave-dressed states, Nature 476, 185 (2011).
  48. A. Salhov, Q. Cao, J. Cai, A. Retzker, F. Jelezko, and G. Genov, Protecting quantum information via destructive interference of correlated noise, Phys. Rev. Lett. 132, 223601 (2024).
  49. D. Louzon, G. T. Genov, N. Staudenmaier, F. Frank, J. Lang, M. L. Markham, A. Retzker, and F. Jelezko, Robust noise suppression and quantum sensing by continuous phased dynamical decoupling, Phys. Rev. Lett. 134, 120802 (2025).
  50. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).
  51. N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, Laser-induced population transfer by adiabatic passage techniques, Annu. Rev. Phys. Chem. 52, 763 (2001).
  52. A. Tannús and M. Garwood, Adiabatic pulses, NMR Biomed. 10, 423 (1997).
  53. M. Garwood and Y. Ke, Symmetric pulses to induce arbitrary flip angles with compensation for rf inhomogeneity and resonance offsets, J. Magn. Reson. (1969) 94, 511 (1991).
  54. K. N. Zlatanov and N. V. Vitanov, Generation of arbitrary qubit states by adiabatic evolution split by a phase jump, Phys. Rev. A 101, 013426 (2020).
  55. R. Lauro, T. Chanelière, and J.-L. Le Gouët, Adiabatic refocusing of nuclear spins in Tm3+:YAG, Phys. Rev. B 83, 035124 (2011).
  56. S. Mieth, D. Schraft, T. Halfmann, and L. P. Yatsenko, Rephasing of optically driven atomic coherences by rapid adiabatic passage in ∖prysolong, Phys. Rev. A 86, 063404 (2012).
  57. M. F. Pascual-Winter, R.-C. Tongning, T. Chanelière, and J.-L. Le Gouët, Spin coherence lifetime extension in Tm3+:YAG through dynamical decoupling, Phys. Rev. B 86, 184301 (2012).
  58. N. V. Vitanov and S. Stenholm, Analytic properties and effective two-level problems in stimulated Raman adiabatic passage, Phys. Rev. A 55, 648 (1997).
  59. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, Stimulated Raman adiabatic passage in physics, chemistry, and beyond, Rev. Mod. Phys. 89, 015006 (2017).
  60. J. Tian, T. Du, Y. Liu, H. Liu, F. Jin, R. S. Said, and J. Cai, Optimal quantum optical control of spin in diamond, Phys. Rev. A 100, 012110 (2019).
  61. X. Laforgue, X. Chen, and S. Guérin, Robust stimulated Raman exact passage using shaped pulses, Phys. Rev. A 100, 023415 (2019).
  62. G. T. Genov, S. Rochester, M. Auzinsh, F. Jelezko, and D. Budker, Robust two-state swap by stimulated Raman adiabatic passage, J. Phys. B 56, 054001 (2023).
  63. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, Optimally robust shortcuts to population inversion in two-level quantum systems, New J. Phys. 14, 093040 (2012).
  64. L. Van-Damme, D. Schraft, G. T. Genov, D. Sugny, T. Halfmann, and S. Guérin, Robust not gate by single-shot-shaped pulses: Demonstration of the efficiency of the pulses in rephasing atomic coherences, Phys. Rev. A 96, 022309 (2017).
  65. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  66. G. Dridi, X. Laforgue, M. Mejatty, and S. Guérin, Optimal ultrarobust quantum gates by inverse optimization, Phys. Rev. A 109, 062613 (2024).
  67. P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. K. Wilhelm, Optimal control of a qubit coupled to a non-Markovian environment, Phys. Rev. Lett. 102, 090401 (2009).
  68. T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, Optimal control for generating quantum gates in open dissipative systems, J. Phys. B 44, 154013 (2011).
  69. C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and 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, 19 (2022).
  70. M. Harutyunyan, F. Holweck, D. Sugny, and S. Guérin, Digital optimal robust control, Phys. Rev. Lett. 131, 200801 (2023).
  71. Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Guérin, D. Guéry-Odelin, and D. Sugny, Introduction to theoretical and experimental aspects of quantum optimal control, J. Phys. B 57, 133001 (2024).
  72. O. E. Alon, V. S. Bagnato, R. Beinke, I. Brouzos, T. Calarco, T. Caneva, L. S. Cederbaum, M. A. Kasevich, S. Klaiman, A. U. J. Lode, S. Montangero, A. Negretti, R. S. Said, K. Sakmann, O. I. Streltsova, M. Theisen, M. C. Tsatsos, S. E. Weiner, T. Wells, and A. I. Streltsov, in High Performance Computing in Science and Engineering ’15, edited by W. E. Nagel, D. H. Kröner, and M. M. Resch (Cham, 2016), p. 23.
  73. A. F. L. Poulsen, J. D. Clement, J. L. Webb, R. H. Jensen, L. Troise, K. Berg-Sørensen, A. Huck, and U. L. Andersen, Optimal control of a nitrogen-vacancy spin ensemble in diamond for sensing in the pulsed domain, Phys. Rev. B 106, 014202 (2022).
  74. S. Hernández-Gómez, F. Poggiali, P. Cappellaro, and N. Fabbri, in Quantum Nanophotonic Materials, Devices, and Systems 2019, edited by C. Soci, M. T. Sheldon, and M. Agio (San Diego, 2019), Vol. 11091, p. 110910L.
  75. R. S. Said and J. Twamley, Robust control of entanglement in a nitrogen-vacancy center coupled to a 13C nuclear spin in diamond, Phys. Rev. A 80, 032303 (2009).
  76. J. Tian, H. Liu, R. Sailer, L. Xiao, F. Jelezko, and R. S. Said, Rapid transform optimization strategy for decoherence-protected quantum register in diamond, Phys. Rev. A 109, 022614 (2024).
  77. T. Nöbauer, A. Angerer, B. Bartels, M. Trupke, S. Rotter, J. Schmiedmayer, F. Mintert, and J. Majer, Smooth optimal quantum control for robust solid-state spin magnetometry, Phys. Rev. Lett. 115, 190801 (2015).
  78. J. Tian, H. Liu, Y. Liu, P. Yang, R. Betzholz, R. S. Said, F. Jelezko, and J. Cai, Quantum optimal control using phase-modulated driving fields, Phys. Rev. A 102, 043707 (2020).
  79. R. D. Iradat, M. A. Majidi, and R. S. Said, Quantum dynamics of two nitrogen-vacancy center ensembles coupled to a driven superconducting quantum circuit, J. Phys. Conf. Ser. 2377, 012050 (2022).
  80. T. Joas, F. Ferlemann, R. Sailer, P. J. Vetter, J. Zhang, R. S. Said, T. Teraji, S. Onoda, T. Calarco, G. Genov, M. M. Müller, and F. Jelezko, High-fidelity electron spin gates for scaling diamond quantum registers, Phys. Rev. X 15, 021069 (2025).
  81. J. Tian, R. S. Said, F. Jelezko, J. Cai, and L. Xiao, Bayesian-based hybrid method for rapid optimization of NV center sensors, Sensors 23, 3244 (2023).
  82. G. de Lange, T. van der Sar, M. Blok, Z.-H. Wang, V. Dobrovitski, and R. Hanson, Controlling the quantum dynamics of a mesoscopic spin bath in diamond, Sci. Rep. 2, 382 (2012).
  83. P. Jamonneau, M. Lesik, J. P. Tetienne, I. Alvizu, L. Mayer, A. Dréau, S. Kosen, J.-F. Roch, S. Pezzagna, J. Meijer, T. Teraji, Y. Kubo, P. Bertet, J. R. Maze, and V. Jacques, Competition between electric field and magnetic field noise in the decoherence of a single spin in diamond, Phys. Rev. B 93, 024305 (2016).
  84. M. Kim, H. J. Mamin, M. H. Sherwood, K. Ohno, D. D. Awschalom, and D. Rugar, Decoherence of near-surface nitrogen-vacancy centers due to electric field noise, Phys. Rev. Lett. 115, 087602 (2015).
  85. J. Meinel, M. Kwon, R. Maier, D. Dasari, H. Sumiya, S. Onoda, J. Isoya, V. Vorobyov, and J. Wrachtrup, High-resolution nanoscale NMR for arbitrary magnetic fields, Commun. Phys. 6, 302 (2023).
  86. D. T. Gillespie, Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral, Phys. Rev. E 54, 2084 (1996).
  87. G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36, 823 (1930).
  88. M. C. Wang and G. E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Mod. Phys. 17, 323 (1945).
  89. K. Senkalla, G. Genov, M. H. Metsch, P. Siyushev, and F. Jelezko, Germanium vacancy in diamond quantum memory exceeding 20 ms, Phys. Rev. Lett. 132, 026901 (2024).
  90. N. Grimm, K. Senkalla, P. J. Vetter, J. Frey, P. Gundlapalli, T. Calarco, G. Genov, M. M. Müller, and F. Jelezko, Coherent control of a long-lived nuclear memory spin in a germanium-vacancy multi-qubit node, Phys. Rev. Lett. 134, 043603 (2025).
  91. I. Arrazola, J. Casanova, J. S. Pedernales, Z.-Y. Wang, E. Solano, and M. B. Plenio, Pulsed dynamical decoupling for fast and robust two-qubit gates on trapped ions, Phys. Rev. A 97, 052312 (2018).
  92. M. M. Müller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, One decade of quantum optimal control in the chopped random basis, Rep. Prog. Phys. 85, 076001 (2022).
  93. J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, Sensitivity optimization for NV-diamond magnetometry, Rev. Mod. Phys. 92, 015004 (2020).
  94. Y. B. Band and Y. Japha, Tuning the adiabaticity of spin dynamics in diamond nitrogen vacancy centers, J. Phys. Condens. Matter 34, 255503 (2022).
  95. F. Bloch and A. Siegert, Magnetic resonance for nonrotating fields, Phys. Rev. 57, 522 (1940).
  96. Y.-C. Yang, S. N. Coppersmith, and M. Friesen, Achieving high-fidelity single-qubit gates in a strongly driven charge qubit with 1/f charge noise, npj Quantum Inf. 5, 12 (2019).
  97. D. Burgarth, P. Facchi, G. Gramegna, and K. Yuasa, One bound to rule them all: From adiabatic to Zeno, Quantum 6, 737 (2022).
  98. G. de Lange, Z. H. Wang, D. Ristè, V. V. Dobrovitski, and R. Hanson, Universal dynamical decoupling of a single solid-state spin from a spin bath, Science 330, 60 (2010).
  99. D. Gillespie, Markov Processes: An Introduction for Physical Scientists (Elsevier Science, San Diego, 1992).
  100. D. T. Gillespie, The mathematics of Brownian motion and Johnson noise, Am. J. Phys. 64, 225 (1996).
  101. A. Fahrurrachman, M. A. Majidi, G. T. Genov, and R. S. Said, Simulating decoherence of a nitrogen—vacancy centre spin qubit via Ornstein—Uhlenbeck process, J. Phys. Conf. Ser. 2498, 012025 (2023).
  102. N. Aharon, I. Cohen, F. Jelezko, and A. Retzker, Fully robust qubit in atomic and molecular three-level systems, New J. Phys. 18, 123012 (2016).
  103. C. Lei, S. Peng, C. Ju, M.-H. Yung, and J. Du, Decoherence control of nitrogen-vacancy centers, Sci. Rep. 7, 11937 (2017).
  104. S.-H. Wu, E. Turner, and H. Wang, Continuous real-time sensing with a nitrogen-vacancy center via coherent population trapping, Phys. Rev. A 103, 042607 (2021).
  105. Z.-H. Wang and S. Takahashi, Spin decoherence and electron spin bath noise of a nitrogen-vacancy center in diamond, Phys. Rev. B 87, 115122 (2013).
  106. N. Staudenmaier, A. Vijayakumar-Sreeja, S. Oviedo-Casado, G. Genov, D. Cohen, D. Dulog, T. Unden, N. Striegler, A. Marshall, J. Scheuer, C. Findler, J. Lang, I. Schwartz, P. Neumann, A. Retzker, and F. Jelezko, Power-law scaling of correlations in statistically polarised nano-NMR, npj Quantum Inf. 8, 120 (2022).
  107. Robust spin inversion is generally sufficient for robust dynamical coupling if the sequence contains an even number of π pulses, since any additional, unwanted phase, acquired during a spin inversion, is compensated by the subsequent inversion, given that the operation is done perfectly [126]. For example, this allows the application of robust rapid adiabatic passage (RAP) pulses even though they perform only spin inversions and each pulse typically induces an unknown dynamic phase [30].
  108. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  109. P. J. Vetter, T. Reisser, M. G. Hirsch, T. Calarco, F. Motzoi, F. Jelezko, and M. M. Müller, Gate-set evaluation metrics for closed-loop optimal control on nitrogen-vacancy center ensembles in diamond, npj Quantum Inf. 10, 96 (2024).
  110. M. Rossignolo, T. Reisser, A. Marshall, P. Rembold, A. Pagano, P. J. Vetter, R. S. Said, M. M. Müller, F. Motzoi, T. Calarco, F. Jelezko, and S. Montangero, QuOCS: The quantum optimal control suite, Comput. Phys. Commun. 291, 108782 (2023).
  111. F. Hoeb, F. Angaroni, J. Zoller, T. Calarco, G. Strini, S. Montangero, and G. Benenti, Amplification of the parametric dynamical Casimir effect via optimal control, Phys. Rev. A 96, 033851 (2017).
  112. J. J. W. H. Sørensen, M. O. Aranburu, T. Heinzel, and J. F. Sherson, Quantum optimal control in a chopped basis: Applications in control of Bose-Einstein condensates, Phys. Rev. A 98, 022119 (2018).
  113. A. Pagano, M. M. Müller, T. Calarco, S. Montangero, and P. Rembold, Role of bases in quantum optimal control, Phys. Rev. A 110, 062608 (2024).
  114. J. Scheuer, X. Kong, R. S. Said, J. Chen, A. Kurz, L. Marseglia, J. Du, P. R. Hemmer, S. Montangero, T. Calarco, B. Naydenov, and F. Jelezko, Precise qubit control beyond the rotating wave approximation, New J. Phys. 16, 093022 (2014).
  115. F. Frank, T. Unden, J. Zoller, R. S. Said, T. Calarco, S. Montangero, B. Naydenov, and F. Jelezko, Autonomous calibration of single spin qubit operations, npj Quantum Inf. 3, 48 (2017).
  116. By limited computational resource, we mean that, given a computational setup, one may only afford to allocate a given amount of time to perform the optimization. To illustrate this, with our choice of Niter=2000, each optimization run takes about 2 h to run in our HPC system with eight allocated compute threads. This is arguably too restrictive for a modern setup, but makes for a good proof of concept.
  117. F. Gao and L. Han, Implementing the Nelder-Mead simplex algorithm with adaptive parameters, Comput. Optim. Appl. 51, 259 (2010).
  118. J. M. Binder, A. Stark, N. Tomek, J. Scheuer, F. Frank, K. D. Jahnke, C. Müller, S. Schmitt, M. H. Metsch, T. Unden, T. Gehring, A. Huck, U. L. Andersen, L. J. Rogers, and F. Jelezko, Qudi: A modular python suite for experiment control and data processing, SoftwareX 6, 85 (2017).
  119. Ulm-IQO. Qudi: A modular laboratory experiment management suite, https://github.com/Ulm-IQO/qudi-core.
  120. Hendry. Optimization result and plotting resources for: Efficiency of optimal control for noisy spin qubits in diamond, https://doi.org/10.5281/zenodo.17456493 (2025).
  121. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2018), 3rd ed.
  122. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2004).
  123. M. Schlosshauer, Decoherence: And the Quantum-To-Classical Transition, The Frontiers Collection (Springer, 2007).
  124. D. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, Cambridge, 2008).
  125. H.-B. Chen and Y.-N. Chen, Canonical Hamiltonian ensemble representation of dephasing dynamics and the impact of thermal fluctuations on quantum-to-classical transition, Sci. Rep. 11, 10046 (2021).
  126. G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, Arbitrarily accurate pulse sequences for robust dynamical decoupling, Phys. Rev. Lett. 118, 133202 (2017).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation