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Exact solutions for time-optimal control of spin \(I=1\) by NMR

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Abstract

We consider the problem of time-optimal control of quadrupole nucleus with the spin \(I=1\) by NMR. In contrast to the conventional methods based on selective pulses, the control is implemented using nonselective pulses separated by free-evolution intervals. Using the Cartan decomposition, the system of equations is obtained for finding parameters of a control field. Partial time-optimal solutions for the important single-qutrit gates (selective rotations and quantum Fourier transform) are found. The strong dependence of minimum gate implementation times on global phase of the gate is observed. The analytical values of minimum times are consistent with the numerical data.

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References

  1. Brif, C., Chakrabarti, R., Rabitz, H.: Control of quantum phenomena: past, present and future. New J. Phys. 12, 075008 (2010)

    Article  ADS  Google Scholar 

  2. Jones, J.A.: Quantum computing with NMR. Prog. NMR. Spectrosc. 59, 91 (2011)

    Article  Google Scholar 

  3. Wu, R., Zhang, J., Li, Ch., Long, G., Tarn, T.: Control problems in quantum systems. Chin. Sci. Bull. 57, 18 (2012)

    Google Scholar 

  4. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  5. Valiev, K.A., Kokin, A.A.: Quantum Computers: Hopes and Reality. Regul. Khaot. Din, Izhevsk (2001)

    Google Scholar 

  6. Schulte-Herbrüggen, T., Spörl, A., Khaneja, N., Glaser, S.J.: Optimal control-based efficient synthesis of building blocks of quantum algorithms: a perspective from network complexity towards time complexity. Phys. Rev. A 72, 042331 (2005)

    Article  ADS  Google Scholar 

  7. Koike, T., Okudaira, Y.: Time complexity and gate complexity. Phys. Rev. A 82, 042305 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Carlini, A., Hosoya, A., Koike, T., Okudaira, Y.: Time-optimal unitary operations. Phys. Rev. A 75, 042308 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bin, L., ZuHuan, Y., ShaoMing, F., XianQing, L.-J.: Time optimal quantum control of two-qubit systems. Sci. China G 56, 2116 (2013)

    Google Scholar 

  11. Yuan, H., Wei, D., Zhang, Y., Glaser, S., Khaneja, N.: Efficient synthesis of quantum gates on indirectly coupled spins. Phys. Rev. A 89, 042315 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Moore Tibbetts, K.W., Brif, C., Grace, M.D., Donovan, A., Hocker, D.L., Ho, T.-S., Wu, R.-B., Rabitz, H.: Exploring the trade-off between fidelity- and time-optimal control of quantum unitary transformations. Phys. Rev. A 86, 062309 (2012)

    Article  ADS  Google Scholar 

  15. Gottesman, D.: Fault-tolerant quantum computation with higher-dimensional systems. Lect. Notes. Comput. Sci. 1509, 302 (1999)

    MathSciNet  Google Scholar 

  16. Muthukrishnan, A., Stroud, C.R.: Multivalued logic gates for quantum computation. J. Mod. Optics 49, 2115 (2002)

    Article  MATH  ADS  Google Scholar 

  17. Daboul, J., Wang, X., Sanders, B.C.: Quantum gates on hybrid qudits. J. Phys. A 36, 2525 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Greentree, A.D., Schirmer, S.G., Green, F., Hollenberg, L.C.L., Hamilton, A.R., Clark, R.G.: Maximizing the Hilbert space for a finite number of distinguishable quantum states. Phys. Rev. Lett. 92, 097901 (2004)

    Article  ADS  Google Scholar 

  19. Bechmann-Pasquinucci, H., Pers, A.: Quantum cryptography with 3-state systems. Phys. Rev. Lett. 85, 3313 (2002)

    Article  ADS  Google Scholar 

  20. Zobov, V.E., Pekhterev, D.I.: Adder on ternary base elements for a quantum computer. Pis’ma Zh. Eksp. Teor. Fiz. 89, 303 (2009) [JETP Lett. 89, 260 (2009)]

  21. Di, Y.-M., Wei, H.-R.: Elementary gates of ternary quantum logic circuit. arXiv:1105.5485 (2013)

  22. Klimov, A.B., Guzman, R., Retamal, J.C., Saavedra, C.: Qutrit quantum computer with trapped ions. Phys. Rev. A 67, 062313 (2003)

    Article  ADS  Google Scholar 

  23. Das, R., Mitra, A., Kumar, V., Kumar, A.: Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system. Int. J. Quantum Inf. 1, 387 (2003)

    Article  Google Scholar 

  24. Shauro, V. P., Pekhterev, D.I., Zobov, V.E.: A comparative analysis of two methods of realizing elementary logic operators for a quantum computer on qutrits. Izv. Vyssh. Uchebn. Zaved. Fiz. 6, 41 (2007) [Russ. Phys. J. 50, 566 (2007)]

  25. Vitanov, N.V.: Synthesis of arbitrary SU(3) transformations of atomic qutrits. Phys. Rev. A 85, 032331 (2012)

    Article  ADS  Google Scholar 

  26. Dogra, S., Dorai, K.: Determining the parity of a permutation using an experimental NMR qutrit. arXiv:1402.5026 (2014)

  27. Ernst, R.R., Bodenhausen, D., Wokaun, A.: Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Clarendon, Oxford (1987)

    Google Scholar 

  28. Schirmer, S.G., Fu, H., Solomon, A.I.: Complete controllability of quantum systems. Phys. Rev. A 63, 063410 (2001)

    Article  ADS  Google Scholar 

  29. Li, B., Yu, Z.-H., Fei, S.-M.: Geometry of quantum computation with qutrits. Sci. Rep. 3, 2594 (2013)

    ADS  Google Scholar 

  30. Zobov, V.E., Shauro, V.P.: Selective control of the states of a three-level quadrupole nucleus by means of nonselective rf pulses. Pis’ma Zh. Eksp. Teor. Fiz. 86, 260 (2007) [JETP Lett. 86, 230 (2007)]

  31. Zobov, V.E., Shauro, V.P.: Selective control of the states of multilevel quantum systems using nonselective rotation operators. Zh. Eksp. Teor. Fiz. 135, 10 (2009) [JETP 108, 5 (2009)]

  32. Shauro, V.P., Zobov, V.E.: Global phase and minimum time of quantum Fourier transform for qudits represented by quadrupole nuclei. Phys. Rev. A 88, 042320 (2013)

    Article  ADS  Google Scholar 

  33. Zobov, V.E., Shauro, V.P.: Effect of a phase factor on the minimum time of a quantum gate. Zh. Eksp. Teor. Fiz. 145, 25 (2014) [JETP 118, 18 (2014)]

  34. Zobov, V.E., Shauro, V.P.: On time-optimal NMR control of states of qutrits represented by quadrupole nuclei with the spin I=1. Zh. Eksp. Teor. Fiz. 140, 211 (2011) [JETP 113, 181 (2011)]

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Acknowledgments

The author thanks V.E. Zobov for fruitful discussions. This study was supported by the Russian Foundation for Basic Research, Project No. 14-07-31086.

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  1. Kirensky Institute of Physics, Russian Academy of Sciences, Siberian Branch, Akademgorodok 50, Bld. 38, Krasnoyarsk, Russia

    Vitaly Shauro

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  1. Vitaly Shauro
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Correspondence to Vitaly Shauro.

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Shauro, V. Exact solutions for time-optimal control of spin \(I=1\) by NMR. Quantum Inf Process 14, 2345–2355 (2015). https://doi.org/10.1007/s11128-015-0999-8

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