Skip to main content
SpringerLink
Log in

Quantum Optimal Control: Practical Aspects and Diverse Methods

  • Review Article
  • Published:
Journal of the Indian Institute of Science Aims and scope

Abstract

Quantum controls realize the unitary or nonunitary operations employed in quantum computers, quantum simulators, quantum communications, and other quantum information devices. They implement the desired quantum dynamics with the help of electric, magnetic, or electromagnetic control fields. Quantum optimal control (QOC) deals with designing an optimal control field modulation that most precisely implements a desired quantum operation with minimum energy consumption and maximum robustness against hardware imperfections as well as external noise. Over the last 2 decades, numerous QOC methods have been proposed. They include asymptotic methods, direct search, gradient methods, variational methods, machine learning methods, etc. In this review, we shall introduce the basic ideas of QOC, discuss practical challenges, and then take an overview of the diverse QOC methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure: 7

Similar content being viewed by others

References

  1. Peirce AP, Dahleh MA, Rabitz H (1988) Optimal control of quantum-mechanical systems: existence, numerical approximation, and applications. Phys Rev A 37:4950. https://doi.org/10.1103/PhysRevA.37.4950

    Article  CAS  Google Scholar 

  2. Kosloff R, Rice S, Gaspard P, Tersigni S, Tannor D (1989) Wavepacket dancing: achieving chemical selectivity by shaping light pulses. Chem Phys 139:201. https://doi.org/10.1016/0301-0104(89)90012-8

    Article  CAS  Google Scholar 

  3. Zare RN (1998) Laser control of chemical reactions. Science 279:1875. https://doi.org/10.1126/science.279.5358.1875

    Article  CAS  Google Scholar 

  4. Rabitz H, de Vivie-Riedle R, Motzkus M, Kompa K (2000) Whither the Future of Controlling Quantum Phenomena? Science 288:824. https://doi.org/10.1126/science.288.5467.824

    Article  CAS  Google Scholar 

  5. Dowling JP, Milburn GJ (2003) Quantum technology: the second quantum revolution. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 361:1655

    Google Scholar 

  6. Cavanagh J, Fairbrother WJ, Palmer III AG, Skelton NJ, Protein NMR spectroscopy: principles and practice. In: Protein NMR spectroscopy: principles and practice. Academic press

  7. Dorai K, Mahesh T, Arvind Kumar A (2000) Quantum computation using NMR. Curr Sci 1447

  8. Kopp RE (1962) Pontryagin maximum principle. In: Mathematics in Science and Engineering, Vol. 5, pp. 255–279. Elsevier

  9. Pontryagin LS (1987) Mathematical theory of optimal processes. In: Mathematical theory of optimal processes. CRC press

  10. Kirk DE (2004) Optimal control theory: an introduction. In: Optimal control theory: an introduction. Courier Corporation

  11. Boscain U, Sigalotti M, Sugny D (2021) Introduction to the pontryagin maximum principle for quantum optimal control. PRX Quant 2:030203

    Google Scholar 

  12. Boscain U, Sigalotti M, Sugny D (2021) Introduction to the Pontryagin maximum principle for quantum optimal control. PRX Quant 2:030203. https://doi.org/10.1103/PRXQuantum.2.030203

    Article  Google Scholar 

  13. Werschnik J, Gross E (2007) Quantum optimal control theory. J Phys B Atom Mol Opt Phys 40:R175

    CAS  Google Scholar 

  14. Cong S (2014) Control of quantum systems: theory and methods. In: Control of quantum systems: theory and methods. Wiley

  15. Glaser SJ, Boscain U, Calarco T, Koch CP, Köckenberger W, Kosloff R, Kuprov I, Luy B, Schirmer S, Schulte-Herbrüggen T et al (2015) Training Schrödinger’s cat: quantum optimal control. Eur Phys J D 69:1

    CAS  Google Scholar 

  16. d’ Alessandro D (2021) Introduction to quantum control and dynamics. In: Introduction to quantum control and dynamics (Chapman and hall/CRC)

  17. Schäfer F, Fukuhara T, Sugawa S, Takasu Y, Takahashi Y (2020) Tools for quantum simulation with ultracold atoms in optical lattices. Nat Rev Phys 2:411

    Google Scholar 

  18. Häffner H, Roos CF, Blatt R (2008) Quantum computing with trapped ions. Phys Rep 469:155

    Google Scholar 

  19. Veldhorst M, Yang C, Hwang J, Huang W, Dehollain J, Muhonen J, Simmons S, Laucht A, Hudson F, Itoh KM et al (2015) A two-qubit logic gate in silicon. Nature 526:410

    CAS  Google Scholar 

  20. Watson T, Philips S, Kawakami E, Ward D, Scarlino P, Veldhorst M, Savage D, Lagally M, Friesen M, Coppersmith S et al (2018) A programmable two-qubit quantum processor in silicon. Nature 555:633

    CAS  Google Scholar 

  21. Krantz P, Kjaergaard M, Yan F, Orlando TP, Gustavsson S, Oliver WD (2019) A quantum engineer’s guide to superconducting qubits. Appl Phys Rev 6:021318

    Google Scholar 

  22. Bucher DB, Aude Craik DP, Backlund MP, Turner MJ, Ben Dor O, Glenn DR, Walsworth RL (2019) Quantum diamond spectrometer for nanoscale NMR and ESR spectroscopy. Nat Protoc 14:2707

    CAS  Google Scholar 

  23. Fortunato EM, Pravia MA, Boulant N, Teklemariam G, Havel TF, Cory DG (2002) Design of strongly modulating pulses to implement precise effective Hamiltonians for quantum information processing. J Chem Phys 116:7599

    CAS  Google Scholar 

  24. Khaneja N, Reiss T, Kehlet C, Schulte-Herbrüggen T, Glaser SJ (2005) Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J Magn Reson 172:296

    CAS  Google Scholar 

  25. Levitt MH (2013) Spin dynamics: basics of nuclear magnetic resonance. In: Spin dynamics: basics of nuclear magnetic resonance. Wiley

  26. Bayer M, Hawrylak P, Hinzer K, Fafard S, Korkusinski M, Wasilewski Z, Stern O, Forchel A (2001) Coupling and entangling of quantum states in quantum dot molecules. Science 291:451

    CAS  Google Scholar 

  27. Tiwari Y, Poonia VS (2021) Universal quantum gates based on quantum dots. arXiv preprint arXiv:2105.07021

  28. Jelezko F, Gaebel T, Popa I, Domhan M, Gruber A, Wrachtrup J (2004) Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate. Phys Rev Lett 93:130501

    CAS  Google Scholar 

  29. Pfender M, Aslam N, Simon P, Antonov D, Thiering G, Burk S, Fávaro de Oliveira F, Denisenko A, Fedder H, Meijer J et al (2017) Protecting a diamond quantum memory by charge state control. Nano Lett 17:5931

    CAS  Google Scholar 

  30. Doherty MW, Manson NB, Delaney P, Jelezko F, Wrachtrup J, Hollenberg LC (2013) The nitrogen-vacancy colour centre in diamond. Phys Rep 528:1

    CAS  Google Scholar 

  31. Anwar M, Xiao L, Short A, Jones J, Blazina D, Duckett S, Carteret H (2005) Practical implementations of twirl operations. Phys Rev A 71:032327

    Google Scholar 

  32. Bhole G, Anjusha V, Mahesh T (2016) Steering quantum dynamics via bang-bang control: implementing optimal fixed-point quantum search algorithm. Phys Rev A 93:042339

    Google Scholar 

  33. Nielsen MA, Chuang I (2002) Quantum computation and quantum information, “Quantum computation and quantum information,”

  34. Tannús A, Garwood M (1997) Adiabatic pulses, NMR in Biomedicine: an international journal devoted to the development and application of magnetic resonance. In Vivo 10:423

    Google Scholar 

  35. Manu V, Kumar A (2012) Singlet-state creation and universal quantum computation in NMR using a genetic algorithm. Phys Rev A 86:022324

    Google Scholar 

  36. Khurana D, Mahesh T (2017) Bang-bang optimal control of large spin systems: enhancement of 13C–13C singlet-order at natural abundance. J Magn Reson 284:8

    CAS  Google Scholar 

  37. Cory DG, Fahmy AF, Havel TF (1997) Ensemble quantum computing by NMR spectroscopy. Proc Natl Acad Sci 94:1634

    CAS  Google Scholar 

  38. Suter D, Mahesh T (2008) Spins as qubits: quantum information processing by nuclear magnetic resonance. J Chem Phys 128:052206

    Google Scholar 

  39. Wu R-B, Rabitz H (2012) Control landscapes for open system quantum operations. J Phys A Math Theor 45:485303

    Google Scholar 

  40. Pravia MA, Boulant N, Emerson J, Farid A, Fortunato EM, Havel TF, Martinez R, Cory DG (2003) Robust control of quantum information. J Chem Phys 119:9993

    CAS  Google Scholar 

  41. Fletcher R (1983) Penalty functions. Math Program State Art 87

  42. Lucarelli D (2018) Quantum optimal control via gradient ascent in function space and the time-bandwidth quantum speed limit. Phys Rev A 97:062346

    CAS  Google Scholar 

  43. Feng G, Cho FH, Katiyar H, Li J, Lu D, Baugh J, Laflamme R (2018) Gradient-based closed-loop quantum optimal control in a solid-state two-qubit system. Phys Rev A 98:052341

    Google Scholar 

  44. Boulant N, Edmonds K, Yang J, Pravia M, Cory D (2003) Experimental demonstration of an entanglement swapping operation and improved control in NMR quantum-information processing. Phys Rev A 68:032305

    Google Scholar 

  45. Möttönen M, de Sousa R, Zhang J, Whaley KB (2006) High-fidelity one-qubit operations under random telegraph noise. Phys Rev A 73:022332

    Google Scholar 

  46. Zhang Y, Lapert M, Sugny D, Braun M, Glaser S (2011) Time-optimal control of spin 1/2 particles in the presence of radiation damping and relaxation. J Chem Phys 134:054103

    CAS  Google Scholar 

  47. Tibbetts KWM, Brif C, Grace MD, Donovan A, Hocker DL, Ho T-S, Wu R-B, Rabitz H (2012) Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations. Phys Rev A 86:062309

    Google Scholar 

  48. Xu X, Wang Z, Duan C, Huang P, Wang P, Wang Y, Xu N, Kong X, Shi F, Rong X et al (2012) Coherence-protected quantum gate by continuous dynamical decoupling in diamond. Phys Rev Lett 109:070502

    Google Scholar 

  49. Zhang J, Souza AM, Brandao FD, Suter D (2014) Protected quantum computing: interleaving gate operations with dynamical decoupling sequences. Phys Rev Lett 112:050502

    Google Scholar 

  50. Viola L, Knill E, Lloyd S (1999) Dynamical decoupling of open quantum systems. Phys Rev Lett 82:2417. https://doi.org/10.1103/PhysRevLett.82.2417

    Article  CAS  Google Scholar 

  51. Ram MH, Krithika V, Batra P, Mahesh T (2022) Robust quantum control using hybrid pulse engineering. Phys Rev A 105:042437

    CAS  Google Scholar 

  52. Lidar DA, Chuang IL, Whaley KB (1998) Decoherence-free subspaces for quantum computation. Phys Rev Lett 81:2594. https://doi.org/10.1103/PhysRevLett.81.2594

    Article  CAS  Google Scholar 

  53. Koch CP (2016) Controlling open quantum systems: tools, achievements, and limitations. J Phys Conden Matter 28:213001

    Google Scholar 

  54. Mahesh T, Suter D (2006) Quantum-information processing using strongly dipolar coupled nuclear spins. Phys Rev A 74:062312

    Google Scholar 

  55. Bhole G, Mahesh T (2017) Rapid exponentiation using discrete operators: applications in optimizing quantum controls and simulating quantum dynamics. arXiv preprint arXiv:1707.02162

  56. Bhole G, Jones JA (2018) Practical pulse engineering: gradient ascent without matrix exponentiation. Front Phys 13:1

    Google Scholar 

  57. Garwood M, DelaBarre L (2001) The return of the frequency sweep: designing adiabatic pulses for contemporary NMR. J Magn Reson 153:155

    CAS  Google Scholar 

  58. Messiah A (2014) Quantum mechanics. In: Quantum mechanics. Courier Corporation

  59. Albash T, Lidar DA (2018) Adiabatic quantum computation. Rev Mod Phys 90:015002

    Google Scholar 

  60. Berry M (1988) The geometric phase. Sci Am 259:46

    Google Scholar 

  61. Anandan J (1992) The geometric phase. Nature 360:307

    Google Scholar 

  62. Jones JA, Vedral V, Ekert A, Castagnoli G (2000) Geometric quantum computation using nuclear magnetic resonance. Nature 403:869

    CAS  Google Scholar 

  63. Pancharatnam S (1956) Generalized theory of interference and its applications. In: Proceedings of the Indian Academy of Sciences-Section A, Vol. 44 (Springer), pp. 398–417

  64. Berry MV (1987) The adiabatic phase and Pancharatnam’s phase for polarized light. J Mod Opt 34:1401

    Google Scholar 

  65. Suter D, Chingas GC, Harris RA, Pines A (1987) Berry’s phase in magnetic resonance. Mol Phys 61:1327

    Google Scholar 

  66. Ekert A, Ericsson M, Hayden P, Inamori H, Jones JA, Oi DK, Vedral V (2000) Geometric quantum computation. J Mod Opt 47:2501

    Google Scholar 

  67. Zanardi P, Rasetti M (1999) Holonomic quantum computation. Phys Lett A 264:94

    CAS  Google Scholar 

  68. Leibfried D, DeMarco B, Meyer V, Lucas D, Barrett M, Britton J, Itano WM, Jelenković B, Langer C, Rosenband T et al (2003) Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature 422:412

    CAS  Google Scholar 

  69. Filipp S, Klepp J, Hasegawa Y, Plonka-Spehr C, Schmidt U, Geltenbort P, Rauch H (2009) Experimental demonstration of the stability of Berry’s phase for a spin-\(1/2\) particle. Phys Rev Lett 102:030404. https://doi.org/10.1103/PhysRevLett.102.030404

    Article  CAS  Google Scholar 

  70. Berger S, Pechal M, Abdumalikov AA, Eichler C, Steffen L, Fedorov A, Wallraff A, Filipp S (2013) Exploring the effect of noise on the Berry phase. Phys Rev A 87:060303. https://doi.org/10.1103/PhysRevA.87.060303

    Article  CAS  Google Scholar 

  71. Nagata K, Kuramitani K, Sekiguchi Y, Kosaka H (2018) Universal holonomic quantum gates over geometric spin qubits with polarised microwaves. Nat Commun 9:1

    Google Scholar 

  72. Wikipedia Contributors (2021) Control-Lyapunov function — Wikipedia, The Free Encyclopedia”, “Control-lyapunov function — Wikipedia, the free encyclopedia.” [Online; accessed 15-March-2022] . https://en.wikipedia.org/w/index.php?title=Control-Lyapunov_function &oldid=1033346201

  73. Isidori A (1995) Local decompositions of control systems. In: Nonlinear control systems (Springer), pp. 1–76

  74. Grivopoulos S, Bamieh B, Lyapunov-based control of quantum systems. In: 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475), Vol. 1 (IEEE, 2003) pp. 434–438

  75. Hou S-C, Khan M, Yi X, Dong D, Petersen IR (2012) Optimal Lyapunov-based quantum control for quantum systems. Phys Revi A 86:022321

    Google Scholar 

  76. Wang L, Hou S, Yi X, Dong D, Petersen IR (2014) Optimal Lyapunov quantum control of two-level systems: convergence and extended techniques. Phys Lett A 378:1074

    CAS  Google Scholar 

  77. Ghaeminezhad N, Cong S (2018) Preparation of Hadamard gate for open quantum systems by the Lyapunov control method. IEEE/CAA J Automat Sinica 5:733

    Google Scholar 

  78. Wang Y, Kang Y-H, Hu C-S, Huang B-H, Song J, Xia Y (2022) Quantum control with Lyapunov function and bang-bang solution in the optomechanics system. Front Phys 17:1

    Google Scholar 

  79. Purkayastha A (2022) The Lyapunov equation in open quantum systems and non-Hermitian physics. arXiv preprint arXiv:2201.00677

  80. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308

    Google Scholar 

  81. Weinstein YS, Havel TF, Emerson J, Boulant N, Saraceno M, Lloyd S, Cory DG (2004) Quantum process tomography of the quantum Fourier transform. J chem Phys 121:6117

    CAS  Google Scholar 

  82. Baugh J, Moussa O, Ryan CA, Laflamme R, Ramanathan C, Havel TF, Cory DG (2006) Solid-state NMR three-qubit homonuclear system for quantum-information processing: control and characterization. Phys Rev A 73:022305

    Google Scholar 

  83. Negrevergne C, Mahesh T, Ryan C, Ditty M, Cyr-Racine F, Power W, Boulant N, Havel T, Cory D, Laflamme R (2006) Benchmarking quantum control methods on a 12-qubit system. Phys Rev Lett 96:170501

    CAS  Google Scholar 

  84. Černỳ V (1985) Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Optim Theory Appl 45:41

    Google Scholar 

  85. Zhou X, Li S, Feng Y (2020) Quantum circuit transformation based on simulated annealing and heuristic search. IEEE Trans Comput Aided Design Integr Circ Syst 39:4683

    Google Scholar 

  86. Situ H, He Z (2022) Using simulated annealing to learn the SDC quantum protocol. Eur Phys J Plus 137:1

    Google Scholar 

  87. Lloyd S, Montangero S (2014) Information theoretical analysis of quantum optimal control. Phys Rev Lett 113:010502

    CAS  Google Scholar 

  88. Doria P, Calarco T, Montangero S (2011) Optimal control technique for many-body quantum dynamics. Phys Rev Lett 106:190501. https://doi.org/10.1103/PhysRevLett.106.190501

    Article  CAS  Google Scholar 

  89. Caneva T, Calarco T, Montangero S (2011) Chopped random-basis quantum optimization. Phys Rev A 84:022326. https://doi.org/10.1103/PhysRevA.84.022326

    Article  CAS  Google Scholar 

  90. Müller MM, Said RS, Jelezko F, Calarco T, Montangero S (2021) One decade of quantum optimal control in the chopped random basis. arXiv preprint arXiv:2104.07687

  91. Riaz B, Shuang C, Qamar S (2019) Optimal control methods for quantum gate preparation: a comparative study. Quant Inform Process 18:1

    Google Scholar 

  92. Sørensen JJWH, Aranburu MO, Heinzel T, Sherson JF (2018) Quantum optimal control in a chopped basis: applications in control of Bose–Einstein condensates. Phys Rev A 98:022119. https://doi.org/10.1103/PhysRevA.98.022119

    Article  Google Scholar 

  93. Wu S-H, Amezcua M, Wang H (2019) Adiabatic population transfer of dressed spin states with quantum optimal control. Phys Rev A 99:063812. https://doi.org/10.1103/PhysRevA.99.063812

    Article  CAS  Google Scholar 

  94. Sørdal VB, Bergli J (2019) Deep reinforcement learning for quantum Szilard engine optimization. Phys Rev A 100:042314. https://doi.org/10.1103/PhysRevA.100.042314

    Article  Google Scholar 

  95. Khurana D, Mahesh T (2017) Bang-bang optimal control of large spin systems: enhancement of 13C–13C singlet-order at natural abundance. J Magn Reson 284:8. https://doi.org/10.1016/j.jmr.2017.09.006

    Article  CAS  Google Scholar 

  96. Zahedinejad E, Schirmer S, Sanders BC (2014) Evolutionary algorithms for hard quantum control. Phys Rev A 90:032310. https://doi.org/10.1103/PhysRevA.90.032310

    Article  CAS  Google Scholar 

  97. Ma H, Chen C, Dong D (2015) Differential evolution with equally-mixed strategies for robust control of open quantum systems. In: 2015 IEEE international conference on systems, man, and cybernetics (IEEE) pp. 2055–2060

  98. Rowland B, Jones JA (2012) Implementing quantum logic gates with gradient ascent pulse engineering: principles and practicalities. Philos Trans R Soc A Math Phys Eng Sci 370:4636

    Google Scholar 

  99. Batra P, Krithika V, Mahesh T (2020) Push–pull optimization of quantum controls. Phys Rev Res 2:013314

    CAS  Google Scholar 

  100. Boutin S, Andersen CK, Venkatraman J, Ferris AJ, Blais A (2017) Resonator reset in circuit QED by optimal control for large open quantum systems. Phys Rev A 96:042315

    Google Scholar 

  101. Egger DJ, Wilhelm FK (2014) Optimal control of a quantum measurement. Phys Rev A 90:052331. https://doi.org/10.1103/PhysRevA.90.052331

    Article  CAS  Google Scholar 

  102. De Fouquieres P, Schirmer S, Glaser S, Kuprov I (2011) Second order gradient ascent pulse engineering. J Magn Reson 212:412

    Google Scholar 

  103. Machnes S, Assémat E, Tannor D, Wilhelm FK (2018) Tunable, flexible, and efficient optimization of control pulses for practical qubits. Phys Rev Lett 120:150401. https://doi.org/10.1103/PhysRevLett.120.150401

    Article  CAS  Google Scholar 

  104. Kirchhoff S, Keßler T, Liebermann PJ, Assémat E, Machnes S, Motzoi F, Wilhelm FK (2018) Optimized cross-resonance gate for coupled transmon systems. Phys Rev A 97:042348

    CAS  Google Scholar 

  105. Krotov V (1995) Global methods in optimal control theory. Global methods in optimal control theory, Vol. 195. CRC Press

  106. Maximov II, Tošner Z, Nielsen NC (2008) Optimal control design of NMR and dynamic nuclear polarization experiments using monotonically convergent algorithms. J Chem Phys 128:184505. https://doi.org/10.1063/1.2903458

    Article  CAS  Google Scholar 

  107. Reich DM, Ndong M, Koch CP (2012) Monotonically convergent optimization in quantum control using Krotov’s method. J Chem Phys 136:104103. https://doi.org/10.1063/1.3691827

    Article  CAS  Google Scholar 

  108. Vinding MS, Maximov II, Tošner Z, Nielsen NC (2012) Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods. J Chem Phys 137:054203

    Google Scholar 

  109. Hwang B, Goan H-S (2012) Optimal control for non-Markovian open quantum systems. Phys Rev A 85:032321. https://doi.org/10.1103/PhysRevA.85.032321

    Article  CAS  Google Scholar 

  110. Jäger G, Reich DM, Goerz MH, Koch CP, Hohenester U (2014) Optimal quantum control of Bose–Einstein condensates in magnetic microtraps: comparison of gradient-ascent-pulse-engineering and Krotov optimization schemes. Phys Rev A 90:033628. https://doi.org/10.1103/PhysRevA.90.033628

    Article  CAS  Google Scholar 

  111. Sutton RS, Barto AG (2018) Reinforcement learning: an introduction. Reinforcement learning. MIT press, Cambridge

    Google Scholar 

  112. Bukov M, Day AGR, Sels D, Weinberg P, Polkovnikov A, Mehta P (2018) Reinforcement learning in different phases of quantum control. Phys Rev X 8:031086. https://doi.org/10.1103/PhysRevX.8.031086

    Article  CAS  Google Scholar 

  113. An Z, Zhou DL (2019) Deep reinforcement learning for quantum gate control. EPL (Europhys Lett) 126:60002. https://doi.org/10.1209/0295-5075/126/60002

    Article  CAS  Google Scholar 

  114. Zhang X-M, Wei Z, Asad R, Yang X-C, Wang X (2019) When does reinforcement learning stand out in quantum control? A comparative study on state preparation. NPJ Quant Inform 5:1

    Google Scholar 

  115. Baum Y, Amico M, Howell S, Hush M, Liuzzi M, Mundada P, Merkh T, Carvalho AR, Biercuk MJ (2021) Experimental deep reinforcement learning for error-robust gate-set design on a superconducting quantum computer. PRX Quant 2:040324

    Google Scholar 

  116. Niu MY, Boixo S, Smelyanskiy VN, Neven H (2019) Universal quantum control through deep reinforcement learning. NPJ Quant Inform 5:1

    CAS  Google Scholar 

  117. An Z, Song H-J, He Q-K, Zhou DL (2021) Quantum optimal control of multilevel dissipative quantum systems with reinforcement learning. Phys Rev A 103:012404. https://doi.org/10.1103/PhysRevA.103.012404

    Article  CAS  Google Scholar 

  118. Wu R-B, Ding H, Dong D, Wang X (2019) Learning robust and high-precision quantum controls. Phys Rev A 99:042327. https://doi.org/10.1103/PhysRevA.99.042327

    Article  CAS  Google Scholar 

  119. Zeng Y, Shen J, Hou S, Gebremariam T, Li C (2020) Quantum control based on machine learning in an open quantum system. Phys Lett A 384:126886. https://doi.org/10.1016/j.physleta.2020.126886

    Article  CAS  Google Scholar 

  120. Huang T, Ban Y, Sherman EY, Chen X (2022) Machine-learning-assisted quantum control in a random environment. Phys Rev Appl 17:024040. https://doi.org/10.1103/PhysRevApplied.17.024040

    Article  CAS  Google Scholar 

  121. Schäfer F, Kloc M, Bruder C, Lörch N (2020) A differentiable programming method for quantum control. Mach Learn Sci Technol 1:035009. https://doi.org/10.1088/2632-2153/ab9802

    Article  Google Scholar 

  122. Khait I, Carrasquilla J, Segal D (2021) Optimal control of quantum thermal machines using machine learning. arXiv preprint arXiv:2108.12441

  123. Coopmans L, Kiely A, De Chiara G, Campbell S (2022) Optimal control in disordered quantum systems. arXiv preprint arXiv:2201.02029

  124. Machnes S, Sander U, Glaser SJ, de Fouquieres P, Gruslys A, Schirmer S, Schulte-Herbrüggen T (2011) Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework. Phys Rev A 84:022305

    Google Scholar 

  125. Batra P, Ram MH, Mahesh T (2022) Recommender system expedited quantum control optimization. arXiv preprint arXiv:2201.12550

  126. Eitan R, Mundt M, Tannor DJ (2011) Optimal control with accelerated convergence: combining the Krotov and quasi-Newton methods. Phys. Rev. A 83:053426

    Google Scholar 

  127. Sørensen J, Aranburu M, Heinzel T, Sherson J (2018) Quantum optimal control in a chopped basis: applications in control of Bose–Einstein condensates. Phys Rev A 98:022119

    Google Scholar 

  128. Lu D, Li K, Li J, Katiyar H, Park AJ, Feng G, Xin T, Li H, Long G, Brodutch A et al (2017) Enhancing quantum control by bootstrapping a quantum processor of 12 qubits. NPJ Quant Inform 3:1

    Google Scholar 

  129. Policharla G-V, Vinjanampathy S (2021) Algorithmic primitives for quantum-assisted quantum control. Phys Rev Lett 127:220504. https://doi.org/10.1103/PhysRevLett.127.220504

    Article  CAS  Google Scholar 

  130. Tošner Z, Vosegaard T, Kehlet C, Khaneja N, Glaser SJ, Nielsen NC (2009) Optimal control in NMR spectroscopy: numerical implementation in SIMPSON. J Magn Reson 197:120

    Google Scholar 

  131. Machnes S, Sander U, Glaser SJ, de Fouquières P, Gruslys A, Schirmer S, Schulte-Herbrüggen T (2011) Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework. Phys Rev A 84:022305. https://doi.org/10.1103/PhysRevA.84.022305

    Article  CAS  Google Scholar 

  132. Johansson JR, Nation PD, Nori F (2012) QuTiP: an open-source python framework for the dynamics of open quantum systems. Comput Phys Commun 183:1760

    CAS  Google Scholar 

  133. Johansson J, Nation P, Nori F (2013) QuTiP 2: a python framework for the dynamics of open quantum systems. Comput Phys Commun 184:1234. https://doi.org/10.1016/j.cpc.2012.11.019

    Article  CAS  Google Scholar 

  134. Goerz MH, Basilewitsch D, Gago-Encinas F, Krauss MG, Horn KP, Reich DM, Koch CP (2019) Krotov: A Python implementation of Krotov’s method for quantum optimal control. Sci Post Phys 7: 80. https://doi.org/10.21468/SciPostPhys.7.6.080

  135. Teske JD, Cerfontaine P, Bluhm H (2022) qopt: an experiment-oriented software package for qubit simulation and quantum optimal control. Phys Rev Appl 17:034036. https://doi.org/10.1103/PhysRevApplied.17.034036

    Article  CAS  Google Scholar 

  136. Sørensen J, Jensen J, Heinzel T, Sherson J (2019) QEngine: A C++ library for quantum optimal control of ultracold atoms. Comput Phys Commun 243:135. https://doi.org/10.1016/j.cpc.2019.04.020

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This review is dedicated to the 80th birthday of Prof. Anil Kumar, IISc, Bangalore, who is noted for his pioneering contributions to NMR spectroscopy as well as NMR quantum computation. PB acknowledges support from the Prime Ministers Research Fellowship (PMRF) of the Government of India. TSM acknowledges funding from DST/ICPS/QuST/2019/Q67.

Funding

The funding has been received from DST INDIA with Grant no. DST/ICPS/QuST/2019/Q67.

Author information

Authors and Affiliations

  1. Department of Physics and NMR Research Center, Indian Institute of Science Education and Research, Pune, 411008, India

    T. S. Mahesh, Priya Batra & M. Harshanth Ram

Authors
  1. T. S. Mahesh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Priya Batra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Harshanth Ram
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. S. Mahesh.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahesh, T.S., Batra, P. & Ram, M.H. Quantum Optimal Control: Practical Aspects and Diverse Methods. J Indian Inst Sci 103, 591–607 (2023). https://doi.org/10.1007/s41745-022-00311-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41745-022-00311-2

Keywords

Search

Navigation