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Algebraic synthesis of time-optimal unitaries in SU(2) with alternating controls

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An Erratum to this article was published on 18 April 2016

Abstract

We present an algebraic framework to study the time-optimal synthesis of arbitrary unitaries in SU(2), when the control set is restricted to rotations around two non-parallel axes in the Bloch sphere. Our method bypasses commonly used control-theoretical techniques and easily imposes necessary conditions on time-optimal sequences. In a straightforward fashion, we prove that time-optimal sequences are solely parametrized by three rotation angles and derive general bounds on those angles as a function of the relative rotation speed of each control and the angle between the axes. Results are substantially different whether both clockwise and counterclockwise rotations about the given axes are allowed, or only clockwise rotations. In the first case, we prove that any finite time-optimal sequence is composed at most of five control concatenations, while for the more restrictive case, we present scaling laws on the maximum length of any finite time-optimal sequence. The bounds we find for both cases are stricter than previously published ones and severely constrain the structure of time-optimal sequences, allowing for an efficient numerical search of the time-optimal solution. Our results can be used to find the time-optimal evolution of qubit systems under the action of the considered control set and thus potentially increase the number of realizable unitaries before decoherence.

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Notes

  1. The appearance of the region with \(\pi /2<\alpha <2\pi /3\) is not intuitive; since the marked zones only reflect necessary conditions for time-optimality, we independently confirm the existence of the two disjoint regions with numerical simulations.

  2. During the preparation of this work, it was proven in [19] by a different method that finite sequences can be at most 4-long.

References

  1. Khaneja, N., Brockett, R., Glaser, S.J.: Time optimal control in spin systems. Phys. Rev. A 63, 032308-1–032308-13 (2001)

    Article  ADS  Google Scholar 

  2. Boozer, A.D.: Time-optimal synthesis of SU(2) transformations for a spin-1/2 system. Phys. Rev. A 85, 012317-1–012317-8 (2012)

    Article  ADS  Google Scholar 

  3. Garon, A., Glaser, sj, Sugny, D.: Time-optimal control of SU(2) quantum operations. Phys. Rev. A 88, 043422-1–043422-12 (2013)

    Article  ADS  Google Scholar 

  4. Boscain, U., Chitour, Y.: Time-optimal synthesis for left-invariant control systems on SO(3). SIAM J. Control Optim. 44, 111–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boscain, U., Mason, P.: Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47, 062101-1–062101-29 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Boscain, U., Grönberg, F., Long, R., Rabitz, H.: Time minimal trajectories for two-level quantum systems with two bounded controls. arXiv:1211.0666v1, 42 pp. (2012)

  7. Jelezko, F., Gaebel, T., Popa, I., Gruber, A., Wrachtrup, J.: Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 92, 076401-1–076401-4 (2004)

    Article  ADS  Google Scholar 

  8. Childress, L., Gurudev Dutt, M.V., Taylor, J.M., Zibrov, A.S., Jelezko, F., Wrachtrup, J., Hemmer, P.R., Lukin, M.D.: Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281–285 (2006)

    Article  ADS  Google Scholar 

  9. Hodges, J.S., Yang, J.C., Ramanathan, C., Cory, D.G.: Universal control of nuclear spins via anisotropic hyperfine interactions. Phys. Rev. A 78, 010303-1–010303-4 (2008)

    Article  ADS  Google Scholar 

  10. Trelat, E.: Optimal control and applications to aerospace: some results and challenges. J. Optim. Theory Appl. 154, 713–758 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1961)

    Google Scholar 

  12. Carlini, A., Hosoya, A., Koike, T., Okudaira, Y.: Time-optimal quantum evolution. Phys. Rev. Lett 96, 060503-1–060503-4 (2006)

    Article  ADS  MATH  Google Scholar 

  13. Billig, Y.: Time-optimal decompositions in SU(2). Quantum Inf. Process 12, 955–971 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Jurdjevic, V., Sussmann, H.J.: Control systems on Lie groups. J. Differ. Equ. 12, 313–329 (1972). doi:10.1016/0022-0396(72)90035-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Lowenthal, F.: Uniform finite generation of the rotation group. Rocky Mt. J. Math. 1, 575–586 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  16. Piovan, G., Bullo, F.: On coordinate-free rotation decomposition: Euler angles about arbitrary axes. IEEE Trans. Robot. 28, 728–733 (2012)

    Article  Google Scholar 

  17. Murray, R.M., Sastry, S.S., Zexiang, L.: A Mathematical Introduction to Robotic Manipulation, 1st edn. CRC Press Inc, Boca Raton (1994)

    MATH  Google Scholar 

  18. Agrachev, A.A., Gamkrelidze, R.V.: Symplectic geometry for optimal control. In: Sussman, H.J. (ed.) Nonlinear Controllability and Optimal Control, vol. 133, pp. 263–277. M. Dekker, New York (1990)

    Google Scholar 

  19. Billig, Y.: Optimal attitude control with two rotation axes. arXiv:1409.3102, 22 pp. (2014)

  20. Khaneja, N.: Switched control of electron nuclear spin systems. Phys. Rev. A 76, 032326-1–032326-8 (2007)

    Article  ADS  Google Scholar 

  21. Mitrikas, G., Sanakis, Y., Papavassiliou, G.: Ultrafast control of nuclear spins using only microwave pulses: towards switchable solid-state quantum gates. Phys. Rev. A 81, 020305-1–020305-4 (2010)

    Article  ADS  Google Scholar 

  22. Morton, J.J.L., Tyryshkin, A.M., Brown, R.M., Shankar, S., Lovett, B.W., Ardavan, A., Schenkel, T., Haller, E.E., Ager, J.W., Lyon, S.A.: Solid-state quantum memory using the 31P nuclear spin. Nature 455, 1085–1088 (2008)

    Article  ADS  Google Scholar 

  23. Morton, J.J.L., Tyryshkin, A.M., Ardavan, A., Benjamin, S.C., Porfyrakis, K., Lyon, S.A., Briggs, G.A.D.: Bang–bang control of fullerene qubits using ultrafast phase gates. Nat. Phys. 2, 40–43 (2006)

  24. Assémat, E., Lapert, M., Zhang, Y., Braun, M., Glaser, S.J., Sugny, D.: Simultaneous time-optimal control of the inversion of two spin-1/2 particles. Phys. Rev. A 82, 013415-1–013415-6 (2010)

  25. Aiello, C.D., Cappellaro, P.: Time-optimal control by a quantum actuator. Phys. Rev. A 91, 042340-1–042340-8 (2015)

  26. Burgarth, D., Maruyama, K., Murphy, M., Montangero, S., Calarco, T., Nori, F., Plenio, M.B.: Scalable quantum computation via local control of only two qubits. Phys. Rev. A 81, 040303-1–040303-4 (2010)

  27. Zhang, Y., Ryan, C.A., Laflamme, R., Baugh, J.: Coherent control of two nuclear spins using the anisotropic hyperfine interaction. Phys. Rev. Lett. 107, 170503-1–170503-5 (2011)

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Acknowledgments

This work was supported in part by the US Air Force Office of Scientific Research through the Young Investigator Program. C.D.A acknowledges support from Schlumberger. The authors would like to thank Seth Lloyd for discussions; exchanges with Ugo Boscain and Domenico D’Alessandro are also gratefully acknowledged.

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    Authors and Affiliations

    1. Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

      Clarice D. Aiello

    2. Dipartimento di Fisica, Università Degli Studi di Torino and INFN, Sezione di Torino, 10125, Turin, Italy

      Michele Allegra

    3. Department of Physics, Harvard University, Cambridge, MA, 02138, USA

      Börge Hemmerling

    4. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

      Xiaoting Wan

    5. Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

      Paola Cappellaro

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    1. Clarice D. Aiello
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    Correspondence to Clarice D. Aiello.

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    Michele Allegra and Börge Hemmerling have contributed equally to this work.

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    Aiello, C.D., Allegra, M., Hemmerling, B. et al. Algebraic synthesis of time-optimal unitaries in SU(2) with alternating controls. Quantum Inf Process 14, 3233–3256 (2015). https://doi.org/10.1007/s11128-015-1045-6

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    • DOI: https://doi.org/10.1007/s11128-015-1045-6

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