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Control landscapes for high-fidelity generation of C-NOT and C-PHASE gates with coherent and environmental driving (2405.14069v1)

Published 23 May 2024 in quant-ph

Abstract: High fidelity generation of two-qubit gates is important for quantum computation, since such gates are components of popular universal sets of gates. Here we consider the problem of high fidelity generation of two-qubit C-NOT and C-PHASE (with a detailed study of C-Z) gates in presence of the environment. We consider the general situation when qubits are manipulated by coherent and incoherent controls; the latter is used to induce generally time-dependent decoherence rates. For estimating efficiency of optimization methods for high fidelity generation of these gates, we study quantum control landscapes which describe the behaviour of the fidelity as a function of the controls. For this, we generate and analyze the statistical distributions of best objective values obtained by incoherent GRadient Ascent Pulse Engineering (inGRAPE) approach. We also apply inGRAPE and stochastic zero-order method to numerically estimate minimal infidelity values. The results are different from the case of single-qubit gates and indicate a smooth trap-free behaviour of the fidelity.

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References (121)
  1. The quantum technologies roadmap: a European community view. New J. Phys., 20(8):080201, 2018. doi:10.1088/1367-2630/aad1ea.
  2. Quantum technology: from research to application. Appl. Phys. B, 122(5):130, 2016. doi:10.1007/s00340-016-6353-8.
  3. M. Nielsen and I. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, Cambridge, 2010. doi:10.1017/CBO9780511976667.
  4. Classical and Quantum Computation / Translated from the Russian by L.J. Senechal. American Mathematical Soc., Providence, Rhode Island, 2002. doi:10.1090/gsm/047.
  5. K.A. Valiev. Quantum computers and quantum computations. Physics-Uspekhi, 48(1):1–36, 2005. doi:10.1070/PU2005v048n01ABEH002024.
  6. D. Gottesman. The Heisenberg representation of quantum computers, 1998. arXiv:quant-ph/9807006, doi:10.48550/arXiv.quant-ph/9807006.
  7. Quantum control landscapes for generation of H𝐻{H}italic_H and T𝑇{T}italic_T gates in an open qubit with both coherent and environmental drive. Photonics, 10(11):1200, 2023. doi:10.3390/photonics10111200.
  8. Quantum circuits with mixed states. In Proc. of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 20–30, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276708.
  9. V.E. Tarasov. Quantum computer with mixed states and four-valued logic. J. Phys. A, 35(25):5207–5235, 2002. URL: https://dx.doi.org/10.1088/0305-4470/35/25/305, doi:10.1088/0305-4470/35/25/305.
  10. J.I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett., 74:4091–4094, 1995. URL: https://link.aps.org/doi/10.1103/PhysRevLett.74.4091, doi:10.1103/PhysRevLett.74.4091.
  11. Quantum Zeno dynamics. Phys. Lett. A, 275(1–2):12–19, 2000. doi:10.1016/S0375-9601(00)00566-1.
  12. From the quantum Zeno to the inverse quantum Zeno effect. Phys. Rev. Lett., 86(13):2699–2703, 2001. doi:10.1103/PhysRevLett.86.2699.
  13. S. Mancini and R. Bonifacio. Quantum Zeno-like effect due to competing decoherence mechanisms. Phys. Rev. A, 64(4):042111, 2001. doi:10.1103/PhysRevA.64.042111.
  14. Control of decoherence: Dynamical decoupling versus quantum Zeno effect: A case study for trapped ions. Int. J. Quantum Chem., 98(2):160–172, 2004. doi:10.1002/qua.10870.
  15. Quantum control by von Neumann measurements. Phys. Rev. A, 74(5):052102, 2006. doi:10.1103/PhysRevA.74.052102.
  16. Quantum state and entanglement protection in finite temperature environment by quantum feed-forward control. Eur. Phys. J. Plus, 136(8):851, 2021. doi:10.1140/epjp/s13360-021-01861-7.
  17. The quantum speed limit time of a qubit in amplitude-damping channel with weak measurement controls. Eur. Phys. J. Plus, 138(5):440, 2023. doi:10.1140/epjp/s13360-023-04028-8.
  18. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe. EPJ Quantum Technol., 9:19, 2022. doi:10.1140/epjqt/s40507-022-00138-x.
  19. D.J. Tannor. Introduction to Quantum Mechanics: A Time Dependent Perspective. Univ. Science Books, Sausilito, CA, 2007. URL: https://uscibooks.aip.org/books/introduction-to-quantum-mechanics-a-time-dependent- perspective/.
  20. V. Letokhov. Laser Control of Atoms and Molecules. Oxford Univ. Press, 2007. URL: https://global.oup.com/academic/product/laser-control-of-atoms-and- molecules-9780199697137
  21. A.L. Fradkov. Cybernetical Physics: From Control of Chaos to Quantum Control. Springer, Berlin, Heidelberg, 2007. doi:10.1007/978-3-540-46277-4.
  22. Control of quantum phenomena: Past, present and future. New J. Phys., 12:075008, 2010. doi:10.1088/1367-2630/12/7/075008.
  23. Special issue on quantum control. J. Opt. B Quantum Semiclass. Opt., 7(10):S177–S177, 2005. doi:10.1088/1464-4266/7/10/E01.
  24. Quantum control theory and applications: a survey. IET Control Theory Appl., 4:2651–2671, 2010. doi:10.1049/iet-cta.2009.0508.
  25. M. Shapiro and P. Brumer. Quantum Control of Molecular Processes. Second, Revised and Enlarged Edition. Wiley–VCH Verlag, Weinheim, 2012. doi:10.1002/9783527639700.
  26. J. Gough. Principles and applications of quantum control engineering. Phil. Trans. R. Soc. A, 370:5241–5258, 2012. doi:10.1098/rsta.2012.0370.
  27. S. Cong. Control of Quantum Systems: Theory and Methods. John Wiley & Sons, Hoboken, 2014.
  28. The modelling of quantum control systems. Sci. Bull., 60:1493–1508, 2015. doi:10.1007/s11434-015-0863-3.
  29. C.P. Koch. Controlling open quantum systems: Tools, achievements, and limitations. J. Phys.: Condens. Matter, 28(21):213001, 2016. doi:10.1088/0953-8984/28/21/213001.
  30. D. D’Alessandro. Introduction to Quantum Control and Dynamics. Chapman & Hall, Boca Raton, 2nd edition, 2021. doi:10.1201/9781003051268.
  31. I. Kuprov. Spin: From Basic Symmetries to Quantum Optimal Control. Springer, Cham, 2023. doi:10.1007/978-3-031-05607-9.
  32. A tutorial on optimal control and reinforcement learning methods for quantum technologies. Phys. Lett. A, 434:128054, 2022. doi:10.1016/j.physleta.2022.128054.
  33. Uncomputability and complexity of quantum control. Sci. Rep., 10(11):1–10, 2020. doi:10.1038/s41598-019-56804-1.
  34. Introduction to the Pontryagin maximum principle for quantum optimal control. PRX Quantum, 2:030203, 2021. doi:10.1103/PRXQuantum.2.030203.
  35. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson., 172(2):296–305, 2005. doi:10.1016/j.jmr.2004.11.004.
  36. Optimal control for generating quantum gates in open dissipative systems. J. Phys. B, 44:154013, 2011. doi:10.1088/0953-4075/44/15/154013.
  37. D. Lucarelli. Quantum optimal control via gradient ascent in function space and the time-bandwidth quantum speed limit. Phys. Rev. A, 97(6):062346, 2018. doi:10.1103/PhysRevA.97.062346.
  38. GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls. J. Phys. A: Math. Theor., 56(30):305303, 2023. doi:10.1088/1751-8121/ace13f.
  39. Accelerated Newton-Raphson GRAPE methods for optimal control. Phys. Rev. Res., 5(1):L012042, 2023. doi:10.1103/PhysRevResearch.5.L012042.
  40. Gradient flows for optimization in quantum information and quantum dynamics: foundations and applications. Rev. Math. Phys., 22:597–667, 2010. doi:10.1142/S0129055X10004053.
  41. Control of photochemical branching: novel procedures for finding optimal pulses and global upper bounds, pages 347–360. Springer US, Boston, MA, 1992. doi:10.1007/978-1-4899-2326-4_24.
  42. Optimal control of the spin system on a basis of the global improvement method. Autom. Remote Control, 72(6):1213–1220, 2011. doi:10.1134/S0005117911060075.
  43. Optimal quantum control of Bose-Einstein condensates in magnetic microtraps: Comparison of GRAPE and Krotov optimization schemes. Phys. Rev. A, 90:033628, 2014. doi:10.1103/PhysRevA.90.033628.
  44. Optimal control theory for a unitary operation under dissipative evolution. New J. Phys., 16(5), 2014. doi:10.1088/1367-2630/16/5/055012.
  45. Corrigendum: Optimal control theory for a unitary operation under dissipative evolution (2014 new j. phys.16 055012). New J. Phys., 23(3):039501, 2021. doi:10.1088/1367-2630/abe970.
  46. Reservoir engineering using quantum optimal control for qubit reset. New J. Phys., 21, 2019. 093054. URL: https://doi.org/10.1088/1367-2630/ab41ad.
  47. Krotov method for optimal control of closed quantum systems. Russian Math. Surveys, 74(5):851–908, 2019. doi:10.1070/RM9835.
  48. Effectiveness of the Krotov method in controlling open quantum systems, 2023. arXiv:2208.03114, doi:10.48550/arXiv.2208.03114.
  49. Hamilton–Jacobi–Bellman equations for quantum optimal feedback control. J. Opt. B Quantum Semiclass. Opt., 7(10):S237–S244, 2005. doi:10.1088/1464-4266/7/10/006.
  50. Chopped random-basis quantum optimization. Phys. Rev. A, 84:022326, 2011. doi:10.1103/PhysRevA.84.022326.
  51. One decade of quantum optimal control in the chopped random basis. Rep. Prog. Phys., 85:076001, 2022. doi:10.1088/1361-6633/ac723c.
  52. Optimal control with accelerated convergence: Combining the Krotov and quasi-Newton methods. Phys. Rev. A, 83:053426, 2011. doi:10.1103/PhysRevA.83.053426.
  53. Second order gradient ascent pulse engineering. J. Magn. Reson., 212(2):412–417, 2011. doi:10.1016/j.jmr.2011.07.023.
  54. R.S. Judson and H. Rabitz. Teaching lasers to control molecules. Phys. Rev. Lett., 68:1500–1503, 1992. doi:10.1103/PhysRevLett.68.1500.
  55. A. Pechen and H. Rabitz. Teaching the environment to control quantum systems. Phys. Rev. A, 73(6):062102, 2006. doi:10.1103/PhysRevA.73.062102.
  56. Y. Maday and G. Turinici. New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys., 118(18):8191–8196, 2003. doi:10.1063/1.1564043.
  57. Quantum theory of optical feedback via homodyne detection. Phys. Rev. Lett., 70(5):548–551, 1993. doi:10.1103/PhysRevLett.70.548.
  58. Quantum feedback control and classical control theory. Phys. Rev. A, 62(1):012105, 2000. doi:10.1103/PhysRevA.62.012105.
  59. S. Lloyd and L. Viola. Engineering quantum dynamics. Phys. Rev. A, 65(1):010101, 2001. doi:10.1103/PhysRevA.65.010101.
  60. Feedback control of quantum state reduction. IEEE Trans. Autom. Control, 50(6):768–780, 2005. doi:10.1109/TAC.2005.849193.
  61. S. Mancini and H.M. Wiseman. Optimal control of entanglement via quantum feedback. Phys. Rev. A, 75(1):012330, 2007. doi:10.1103/PhysRevA.75.012330.
  62. J.E. Gough. Principles and applications of quantum control engineering. Philos. Trans. Royal Soc. A, 370(1979):5241–5258, 2012. doi:10.1098/rsta.2012.0370.
  63. Design of feedback control laws for information transfer in spintronics networks. IEEE Trans. Autom. Control, 63(8):2523–2536, 2018. doi:10.1109/TAC.2017.2777187.
  64. Incoherent control of quantum systems with wavefunction-controllable subspaces via quantum reinforcement learning. IEEE Trans. Syst. Man. Cybern. B Cybern., 38(4):957–962, 2008. URL: http://dx.doi.org/10.1109/TSMCB.2008.926603, doi:10.1109/TSMCB.2008.926603.
  65. Universal quantum control through deep reinforcement learning. npj Quantum Information, 5:33, 2019. doi:10.1038/s41534-019-0141-3.
  66. Predicting quantum dynamical cost landscapes with deep learning. Phys. Rev. A, 105:012402, 2022. doi:10.1103/PhysRevA.105.012402.
  67. A modified deep q-learning algorithm for control of two-qubit systems. Proc. of 2021 IEEE Int. Conf. on Systems, Man, and Cybernetics (SMC), pages 3454–3459, 2021. doi:10.1109/SMC52423.2021.9658732.
  68. Control of the observables in the finite-level quantum systems. Autom. Remote Control, 66:734–745, 2005. doi:10.1007/s10513-005-0117-y.
  69. A. Pechen and S. Borisenok. Energy transfer in two-level quantum systems via speed gradient-based algorithm. IFAC-PapersOnLine, 48(11):446–450, 2015. doi:10.1016/j.ifacol.2015.09.226.
  70. Quantum machine learning. Nature, 549(7671):195–202, 2017. doi:10.1038/nature23474.
  71. Generation of density matrices for two qubits using coherent and incoherent controls. Lobachevskii J. Math., 42(10):2401–2412, 2021. doi:10.1134/S1995080221100176.
  72. State transfer and maintenance for non-markovian open quantum systems in a hybrid environment via lyapunov control method. Eur. Phys. J. Plus, 137(5):533, 2022. doi:10.1140/epjp/s13360-022-02713-8.
  73. QuOCS: The quantum optimal control suite. Comput. Phys. Commun., 291:108782, 2023. doi:10.1016/j.cpc.2023.108782.
  74. Simultaneous time-optimal control of the inversion of two spin-1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG particles. Phys. Rev. A, 82(1):013415, 2010. doi:10.1103/PhysRevA.82.013415.
  75. Optimal feedback control of two-qubit entanglement in dissipative environments. Phys. Rev. A, 94(1):012310, 2016. doi:10.1103/PhysRevA.94.012310.
  76. Universal feedback control of two-qubit entanglement. Phys. Rev. A, 96(1):012340, 2017. doi:10.1103/PhysRevA.96.012340.
  77. Optimal two-qubit gates in recurrence protocols of entanglement purification. Phys. Rev. A, 106(2):022422, 2022. doi:10.1103/PhysRevA.106.022422.
  78. Control of entanglement, single excited-state population and memory-assisted entropic uncertainty of two qubits moving in a cavity by using a classical driving field. Eur. Phys. J. Plus, 137(9):1065, 2022. doi:10.1140/epjp/s13360-022-03230-4.
  79. Time optimal realization of two-qubit entangler. Eur. Phys. J. Plus, 137(6):720, 2022. doi:10.1140/epjp/s13360-022-02904-3.
  80. Optimal state manipulation for a two-qubit system driven by coherent and incoherent controls. Quantum Inf. Process., 22:241, 2023. URL: https://doi.org/10.1007/s11128-023-03946-x.
  81. Speed limits for quantum gates in multiqubit systems. Physical Review A, 85(5):052327, 2012. doi:10.1103/PhysRevA.85.052327.
  82. Quantum speed limit for physical processes. Physical Review Letters, 110(5):050402, 2013. doi:10.1103/PhysRevLett.110.050402.
  83. Quantum optimally controlled transition landscapes. Science, 303(5666):1998–2001, 2004. doi:10.1126/science.1093649.
  84. T.-S. Ho and H. Rabitz. Why do effective quantum controls appear easy to find? Journal of Photochemistry and Photobiology A: Chemistry, 180(3):226–240, 2006. doi:10.1016/j.jphotochem.2006.03.038.
  85. Search complexity and resource scaling for the quantum optimal control of unitary transformations. Phys. Rev. A, 83:012326, 2011. doi:10.1103/PhysRevA.83.012326.
  86. Are there traps in quantum control landscapes? Phys. Rev. Lett., 106(12):120402, 2011. doi:10.1103/PhysRevLett.106.120402.
  87. A. Pechen and N. Il’in. Trap-free manipulation in the Landau-Zener system. Phys. Rev. A, 86(5):052117, 2012. doi:10.1103/PhysRevA.86.052117.
  88. Quantum control landscape for a Lambda-atom in the vicinity of second-order traps. Israel J. Chem., 52(5):467–472, 2012. doi:10.1002/ijch.201100165.
  89. P. De Fouquieres and S.G. Schirmer. A closer look at quantum control landscapes and their implication for control optimization. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 16(3):1350021, 2013. doi:10.1142/S0219025713500215.
  90. Quantum control landscape for a two-level system near the quantum speed limit. J. Phys. A: Math. Theor., 51(38), 2018. 385305. doi:10.1088/1751-8121/aad657.
  91. D.V. Zhdanov. Comment on ’Control landscapes are almost always trap free: a geometric assessment’. J. Phys. A: Math. Theor., 51:508001, 2018. doi:10.1088/1751-8121/aaecf6.
  92. Reply to comment on ’Control landscapes are almost always trap free: a geometric assessment’. J. Phys. A: Math. Theor., 51:508002, 2018. doi:10.1088/1751-8121/aaecf2.
  93. Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates. J. Phys. A: Math. Theor., 54:215303, 2021. doi:10.1088/1751-8121/abf45d.
  94. Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution. J. Phys. A, 42:205305, 2009. doi:10.1088/1751-8113/42/20/205305.
  95. Higher-order traps for some strongly degenerate quantum control systems. Russian Math. Surveys, 78(2):390–392, 2023. doi:10.4213/rm10069e.
  96. M. Elovenkova and A. Pechen. Control landscape of measurement-assisted transition probability for a three-level quantum system with dynamical symmetry. Quantum Rep., 5(3):526–545, 2023. doi:10.3390/quantum5030035.
  97. SciPy: the function scipy.optimize.dual_annealing. URL: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.dual_annealing.html (access: Dec. 29, 2023).
  98. Optimization by simulated annealing. Science, 220(4598):671–680, 1983. doi:10.1126/science.220.4598.671.
  99. C. Tsallis. Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys., 52:479–487, 1988. doi:10.1007/BF01016429.
  100. C. Tsallis and D.A. Stariolo. Generalized simulated annealing. Phys. A: Stat. Mech., 233(1–2):395–406, 1996. doi:10.1016/S0378-4371(96)00271-3.
  101. I. Andricioaei and J.E. Straub. Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide. Phys. Rev. E, 53(4):R3055(R), 1996. doi:10.1103/PhysRevE.53.R3055.
  102. Generalized simulated annealing algorithm and its application to the thomson model. Phys. Lett. A, 233(3):216–220, 1997. doi:10.1016/S0375-9601(97)00474-X.
  103. Y. Xiang and X.G. Gong. Efficiency of generalized simulated annealing. Phys. Rev. E, 62(3):4473, 2000. doi:10.1103/PhysRevE.62.4473.
  104. Mathematical structures related to the description of quantum states. Dokl. Math., 104(3):365–368, 2021. doi:10.1134/S1064562421060119.
  105. P. Maity and M. Purkait. Implementation of a holonomic 3-qubit gate using rydberg superatoms in a microwave cavity. Eur. Phys. J. Plus, 137:1299, 2022. doi:10.1140/epjp/s13360-022-03460-6.
  106. Fast route to equilibration. Phys. Rev. A, 101:052102, 2020. doi:10.1103/PhysRevA.101.052102.
  107. H. Spohn and J. Lebowitz. Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys., 38:109–142, 1978. doi:10.1002/9780470142578.ch2.
  108. A. Trushechkin. Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation. Phys. Rev. A, 103(6):062226, 2021. doi:10.1103/PhysRevA.103.062226.
  109. A. Trushechkin. Quantum master equations and steady states for the ultrastrong-coupling limit and the strong-decoherence limit. Phys. Rev. A, 106(4):042209, 2022. doi:10.1103/PhysRevA.106.042209.
  110. A. Pechen. Engineering arbitrary pure and mixed quantum states. Phys. Rev. A, 84(4):042106, 2011. doi:10.1103/PhysRevA.84.042106.
  111. Controllability of open quantum systems with Kraus-map dynamics. J. Phys. A, 40:5681–5693, 2007. doi:10.1088/1751-8113/40/21/015.
  112. All-optical input-agnostic polarization transformer via experimental Kraus-map control. Eur. Phys. J. Plus, 137:930, 2022. doi:10.1140/epjp/s13360-022-03104-9.
  113. Shaped incoherent light for control of kinetics: Optimization of up-conversion hues in phosphors. J. Chem. Phys., 149:054201, 2018. doi:10.1063/1.5035077.
  114. Open system perspective on incoherent excitation of light-harvesting systems. J. Phys. B At. Mol. Opt. Phys., 50(18):184003, 2017. doi:10.1088/1361-6455/aa8696.
  115. P. Brumer. Shedding (incoherent) light on quantum effects in light-induced biological processes. J. Phys. Chem. Lett., 9(18):184003, 2017. doi:10.1021/acs.jpclett.8b00874.
  116. Quantum feedback control in quantum photosynthesis. Phys. Rev. A, 106(3):032218, 2022. doi:10.1103/PhysRevA.106.032218.
  117. States, Effects, and Operations: Fundamental Notions of Quantum Theory. Springer, 1983. doi:10.1007/3-540-12732-1.
  118. A.S. Holevo. Quantum Systems, Channels, Information: A mathematical Introduction. 2nd Rev. and Expanded Ed. De Gruyter, Berlin, Boston, 2019. doi:10.1515/9783110642490.
  119. G.G. Amosov. On inner geometry of noncommutative operator graphs. Eur. Phys. J. Plus, 135(10):865, 2020. doi:10.1140/epjp/s13360-020-00871-1.
  120. A.F. Filippov. Differential Equations with Discontinuous Righthand Sides / Transl. from the edition published in Russian in 1985. Math. Appl. (Soviet Ser.), Vol. 18. Kluwer Acad. Publ., Dordrecht, 1988. doi:10.1007/978-94-015-7793-9.
  121. Control Theory from the Geometric Viewpoint. Springer, 2004. doi:10.1007/978-3-662-06404-7.

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