Skip to main content
SpringerLink
Log in

Multifractality in fidelity sequences of optimized Toffoli gates

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We analyze the multifractality in the fidelity sequences of several engineered Toffoli gates. Using quantum control methods, we consider several optimization problems whose global solutions realize the gate in a chain of three qubits with XY Heisenberg interaction. Applying a minimum number of control pulses assuring a fidelity above 99 % in the ideal case, we design stable gates that are less sensitive to variations in the interqubits couplings. The most stable gate has the fidelity above 91 % with variations about 0.1 %, for up to 10 % variation in the nominal couplings. We perturb the system by introducing a single source of 1 / f noise that affects all the couplings. In order to quantify the performance of the proposed optimized gates, we calculate the fidelity of a large set of optimized gates under prescribed levels of coupling perturbation. Then, we run multifractal analysis on the sequence of attained fidelities. This way, gate performance can be assessed beyond mere average results, since the chosen multifractality measure (the width of the multifractal spectrum) encapsulates into a single performance indicator the spread of fidelity values around the mean and the presence of outliers. The higher the value of the performance indicator the more concentrated around the mean the fidelity values are and rarer is the occurrence of outliers. The results of the multifractal analysis on the fidelity sequences demonstrate the effectiveness of the proposed optimized gate implementations, in the sense they are rendered less sensitive to variations in the interqubits coupling strengths.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. http://lps.lncc.br/index.php/demonstracoes/emd-damf.

References

  1. Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62(02), 331–358 (1974)

    Article  ADS  MATH  Google Scholar 

  2. Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I., Shraiman, B.: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33(2), 1141 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Meneveau, C., Sreenivasan, K.: The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484 (1991)

    Article  ADS  MATH  Google Scholar 

  4. Muzy, J., Bacry, E., Arnéodo, A.: Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys. Rev. Lett. 67(25), 3515–3518 (1991)

    Article  ADS  Google Scholar 

  5. Mandelbrot, B., Fisher, A., Calvet, L.: A multifractal model of asset returns. Cowles Foundation Discussion Papers (1997)

  6. Stanley, H., Meakin, P.: Multifractal phenomena in physics and chemistry. Nature 335(6189), 405–409 (1988)

    Article  ADS  Google Scholar 

  7. Abry, P., Baraniuk, R., Flandrin, P., Riedi, R., Veitch, D.: Multiscale nature of network traffic. IEEE Signal Proc. Mag. 19(3), 28–46 (2002)

    Article  ADS  Google Scholar 

  8. Jafari, G., Pedram, P., Hedayatifar, L.: Long-range correlation and multifractality in Bach’s inventions pitches. J. Stat. Mech: Theory Exp. 2007(04), P04012 (2007)

    Article  Google Scholar 

  9. Lovejoy, S., Schertzer, D.: Haar wavelets, fluctuations and structure functions: convenient choices for geophysics. Nonlinear Process. Geophys. 19, 513–527 (2012)

    Article  ADS  Google Scholar 

  10. Ausloos, M.: Generalized hurst exponent and multifractal function of original and translated texts mapped into frequency and length time series. Phys. Rev. E 86(3), 031108 (2012)

    Article  ADS  Google Scholar 

  11. Mirlin, A.D.: Statistics of energy levels and eigenfunctions in disordered systems. Phys. Rep. 326(5), 259–382 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Evers, F., Mirlin, A.D.: Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008)

    Article  ADS  Google Scholar 

  13. Rodriguez, A., Vasquez, L.J., Römer, R.A.: Multifractal analysis with the probability density function at the three-dimensional Anderson transition. Phys. Rev. Lett. 102, 106406 (2009)

    Article  ADS  Google Scholar 

  14. Rodriguez, A., Vasquez, L.J., Roemer, R.A.: Optimisation of multifractal analysis at the 3D Anderson transition using box-size scaling. Eur. Phys. J. B Condens. Matter Complex Syst. 67(1), 77–82 (2009)

    Article  Google Scholar 

  15. Rodriguez, A., Vasquez, L.J., Slevin, K., Römer, R.A.: Critical parameters from a generalized multifractal analysis at the Anderson transition. Phys. Rev. Lett. 105, 046403 (2010)

    Article  ADS  Google Scholar 

  16. Rodriguez, A., Vasquez, L.J., Slevin, K., Römer, R.A.: Multifractal finite-size scaling and universality at the Anderson transition. Phys. Rev. B 84, 134209 (2011)

    Article  ADS  Google Scholar 

  17. Burmistrov, I.S., Gornyi, I.V., Mirlin, A.D.: Multifractality at Anderson transitions with Coulomb interaction. Phys. Rev. Lett. 111, 066601 (2013)

    Article  ADS  Google Scholar 

  18. Huckestein, B.: Scaling theory of the integer quantum hall effect. Rev. Mod. Phys. 67, 357–396 (1995)

    Article  ADS  Google Scholar 

  19. Evers, F., Mildenberger, A., Mirlin, A.D.: Multifractality of wave functions at the quantum hall transition revisited. Phys. Rev. B 64, 241303 (2001)

    Article  ADS  Google Scholar 

  20. Evers, F., Mildenberger, A., Mirlin, A.D.: Multifractality at the spin quantum hall transition. Phys. Rev. B 67, 041303 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Meenakshisundaram, N., Lakshminarayan, A.: Multifractal eigenstates of quantum chaos and the Thue–Morse sequence. Phys. Rev. E 71, 065303 (2005)

    Article  ADS  Google Scholar 

  22. Martin, J., Giraud, O., Georgeot, B.: Multifractality and intermediate statistics in quantum maps. Phys. Rev. E 77, 035201 (2008)

    Article  ADS  Google Scholar 

  23. Martin, J., García-Mata, I., Giraud, O., Georgeot, B.: Multifractal wave functions of simple quantum maps. Phys. Rev. E 82, 046206 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  24. Bandyopadhyay, J.N., Wang, J., Gong, J.: Generating a fractal butterfly Floquet spectrum in a class of driven SU(2) systems: eigenstate statistics. Phys. Rev. E 81, 066212 (2010)

    Article  ADS  Google Scholar 

  25. Wołoszyn, M., Spisak, B.J.: Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields. Eur. Phys. J. B Condens. Matter Complex Syst. 85(1), 1–7 (2012)

    Article  Google Scholar 

  26. García-Mata, I., Martin, J., Giraud, O., Georgeot, B.: Multifractality of quantum wave packets. Phys. Rev. E 86, 056215 (2012)

    Article  ADS  Google Scholar 

  27. Mirlin, A.D., Fyodorov, Y.V., Dittes, F.M., Quezada, J., Seligman, T.H.: Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices. Phys. Rev. E 54, 3221–3230 (1996)

    Article  ADS  Google Scholar 

  28. Kravtsov, V.E., Muttalib, K.A.: New class of random matrix ensembles with multifractal eigenvectors. Phys. Rev. Lett. 79, 1913–1916 (1997)

    Article  ADS  Google Scholar 

  29. Bogomolny, E., Giraud, O.: Multifractal dimensions for all moments for certain critical random-matrix ensembles in the strong multifractality regime. Phys. Rev. E 85, 046208 (2012)

    Article  ADS  Google Scholar 

  30. Fyodorov, Y.V., Ossipov, A., Rodriguez, A.: The anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices. J. Stat. Mech: Theory Exp. 2009(12), L12001 (2009)

    Article  Google Scholar 

  31. Atas, Y.Y., Bogomolny, E.: Multifractality of eigenfunctions in spin chains. Phys. Rev. E 86, 021104 (2012)

    Article  ADS  Google Scholar 

  32. Luitz, D.J., Alet, F., Laflorencie, N.: Universal behavior beyond multifractality in quantum many-body systems. Phys. Rev. Lett. 112, 057203 (2014)

    Article  ADS  Google Scholar 

  33. Jia, X., Subramaniam, A.R., Gruzberg, I.A., Chakravarty, S.: Entanglement entropy and multifractality at localization transitions. Phys. Rev. B 77, 014208 (2008)

    Article  ADS  Google Scholar 

  34. Giraud, O., Martin, J., Georgeot, B.: Entropy of entanglement and multifractal exponents for random states. Phys. Rev. A 79, 032308 (2009)

    Article  ADS  Google Scholar 

  35. Pellegrini, F., Montangero, S.: Fractal fidelity as a signature of quantum chaos. Phys. Rev. A 76(5), 052327 (2007)

    Article  ADS  Google Scholar 

  36. Bin, Y., Gang, D., Xiao-Ping, M.: Fractals in the open quantum kicked top model. Commun. Nonlinear Sci. Numer. Simul. 15(10), 2967–2973 (2010)

    Article  ADS  Google Scholar 

  37. Stojanović, V.M., Fedorov, A., Wallraff, A., Bruder, C.: Quantum-control approach to realizing a Toffoli gate in circuit QED. Phys. Rev. B 85(5), 054504 (2012)

    Article  ADS  Google Scholar 

  38. Moqadam, J.K., Portugal, R., Svaiter, N.F., de Oliveira Corrêa, G.: Analyzing the Toffoli gate in disordered circuit QED. Phys. Rev. A 87, 042324 (2013)

    Article  ADS  Google Scholar 

  39. Welter, G.S., Esquef, P.A.A.: Multifractal analysis based on amplitude extrema of intrinsic mode functions. Phys. Rev. E 87, 032916 (2013)

    Article  ADS  Google Scholar 

  40. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  41. Monz, T., Kim, K., Hänsel, W., Riebe, M., Villar, A.S., Schindler, P., Chwalla, M., Hennrich, M., Blatt, R.: Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009)

    Article  ADS  Google Scholar 

  42. Lanyon, B.P., Barbieri, M., Almeida, M.P., Jennewein, T., Ralph, T.C., Resch, K.J., Pryde, G.J., Obrien, J.L., Gilchrist, A., White, A.G.: Simplifying quantum logic using higher-dimensional Hilbert spaces. Nat. Phys. 5(2), 134–140 (2009)

    Article  Google Scholar 

  43. Zahedinejad, E., Ghosh, J., Sanders, B.C.: High-fidelity single-shot Toffoli gate via quantum control. Phys. Rev. Lett. 114, 200502 (2015)

    Article  ADS  Google Scholar 

  44. Heule, R., Bruder, C., Burgarth, D., Stojanović, V.M.: Local quantum control of Heisenberg spin chains. Phys. Rev. A 82, 052333 (2010)

    Article  ADS  Google Scholar 

  45. Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman, New York (1982)

    MATH  Google Scholar 

  46. Mandelbrot, B., Van Ness, J.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Voss, R.F.: Random fractals: self-affinity in noise, music, mountains, and clouds. Phys. D 38(1), 362–371 (1989)

    Article  MathSciNet  Google Scholar 

  48. Arnéodo, A., Bacry, E., Muzy, J.: The thermodynamics of fractals revisited with wavelets. Phys. A 213(1), 232–275 (1995)

    Article  MATH  Google Scholar 

  49. Parisi, G., Frisch, U.: Fully developed turbulence and intermittency. In: Ghil, M., Benzi, R., Parisi, G. (eds.) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, pp. 84–88. North Holland, Amsterdam (1985)

    Google Scholar 

  50. Barral, J., Seuret, S.: From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11(5), 589–614 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kantelhardt, J., Zschiegner, S., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.: Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A 316(1), 87–114 (2002)

    Article  MATH  Google Scholar 

  52. Turiel, A., Pérez-Vicente, C., Grazzini, J.: Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study. J. Comput. Phys. 216(1), 362–390 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Oświȩcimka, P., Kwapień, J., Drożdż, S.: Wavelet versus detrended fluctuation analysis of multifractal structures. Phys. Rev. E 74(1), 016103 (2006)

    Article  Google Scholar 

  54. Huang, Y., Schmitt, F., Hermand, J., Gagne, Y., Lu, Z., Liu, Y.: Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: comparison study with detrended fluctuation analysis and wavelet leaders. Phys. Rev. E 84(1), 016208 (2011)

    Article  ADS  Google Scholar 

  55. Huang, N., Shen, Z., Long, S., Wu, M., Shih, H., Zheng, Q., Yen, N., Tung, C., Liu, H.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Philos. Trans. R. Soc. A 454(1971), 903–995 (1998)

    MathSciNet  MATH  Google Scholar 

  56. Mantica, G.: The global statistics of return times: return time dimensions versus generalized measure dimensions. J. Stat. Phys. 138(4–5), 701–727 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Shimizu, Y., Thurner, S., Ehrenberger, K.: Multifractal spectra as a measure of complexity in human posture. Fractals 10(01), 103–116 (2002)

    Article  Google Scholar 

  58. Paladino, E., Galperin, Y.M., Falci, G., Altshuler, B.L.: \(1{/}f\) noise: implications for solid-state quantum information. Rev. Mod. Phys. 86, 361–418 (2014)

    Article  ADS  Google Scholar 

  59. Dubertrand, R., García-Mata, I., Georgeot, B., Giraud, O., Lemarié, G., Martin, J.: Two scenarios for quantum multifractality breakdown. Phys. Rev. Lett. 112, 234101 (2014)

    Article  ADS  Google Scholar 

  60. Tsomokos, D., Hartmann, M., Huelga, S., Plenio, M.: Entanglement dynamics in chains of qubits with noise and disorder. New J. Phys. 9(3), 79 (2007)

    Article  ADS  Google Scholar 

  61. Tsomokos, D.I., Ashhab, S., Nori, F.: Fully connected network of superconducting qubits in a cavity. New J. Phys. 10(11), 113020 (2008)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

JKM acknowledges Grants PCI-DB 302866/2014-0 and PDJ 165941/2014-6, and GSW and PAAE acknowledge Grant 475566/2012-2, from Brazilian National Council for Scientific and Technological Development (CNPq).

Author information

Authors and Affiliations

  1. Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP, Brazil

    Jalil Khatibi Moqadam

  2. Laboratório Nacional de Computação Científica (LNCC), Petrópolis, RJ, Brazil

    Guilherme S. Welter & Paulo A. A. Esquef

Authors
  1. Jalil Khatibi Moqadam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guilherme S. Welter
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paulo A. A. Esquef
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Khatibi Moqadam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moqadam, J.K., Welter, G.S. & Esquef, P.A.A. Multifractality in fidelity sequences of optimized Toffoli gates. Quantum Inf Process 15, 4501–4520 (2016). https://doi.org/10.1007/s11128-016-1409-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-016-1409-6

Keywords

Search

Navigation