Skip to main content

Advertisement

Springer Nature Link
Log in

Fourier 1-norm and quantum speed-up

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Understanding quantum speed-up over classical computing is fundamental for the development of efficient quantum algorithms. In this paper, we study such problem within the framework of the quantum query model, which represents the probability of output \(x \in \{0,1\}^n\) as a function \(\pi (x)\). We present a classical simulation for output probabilities \(\pi \), whose error depends on the Fourier 1-norm of \(\pi \). Such dependence implies upper-bounds for the quotient between the number of queries applied by an optimal classical algorithm and our quantum algorithm, respectively. These upper-bounds show a strong relation between Fourier 1-norm and quantum parallelism. We show applications to query complexity.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. For details and up-to-date examples, the reader may refer to S. Jordan, Quantum Algorithm Zoo, https://math.nist.gov/quantum/zoo/.

References

  1. Aaronson, S.: Limitations of quantum advice and one-way communication. In: Proceedings of 19th IEEE Annual Conference on Computational Complexity, pp. 320–332. IEEE (2004)

  2. Aaronson, S.: BQP and the polynomial hierarchy. In: Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pp. 141–150. ACM (2010)

  3. Aaronson, S., Ambainis, A.: The need for structure in quantum speedups. Theory Comput. 10(6), 133–166 (2014)

    Article  MathSciNet  Google Scholar 

  4. Aaronson, S., Ambainis, A., Iraids, J., Kokainis, M., Smotrovs, J.: Polynomials, quantum query complexity, and Grothendieck’s inequality. In: Proceedings of the 31st Conference on Computational Complexity, CCC’16, pp. 25:1–25:19. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Germany (2016). https://doi.org/10.4230/LIPIcs.CCC.2016.25

  5. Abbott, A.A., Calude, C.S.: Understanding the quantum computational speed-up via de-quantisation. In: Cooper, S.B., Panangaden, P., Kashefi, E. (eds.) Proceedings Sixth Workshop on Developments in Computational Models: Causality, Computation, and Physics, Edinburgh, Scotland, 9–10th July 2010, Electronic Proceedings in Theoretical Computer Science, vol. 26, pp. 1–12. Open Publishing Association (2010). https://doi.org/10.4204/EPTCS.26.1

  6. Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)

    Article  MathSciNet  Google Scholar 

  7. Ambainis, A., Gruska, J., Zheng, S.: Exact quantum algorithms have advantage for almost all Boolean functions. Quantum Inf. Comput. 15(5–6), 435–452 (2015)

    MathSciNet  Google Scholar 

  8. Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for hamiltonian circuits and matchings. J. Comput. Syst. Sci. 18(2), 155–193 (1979)

    Article  MathSciNet  Google Scholar 

  9. Barnum, H., Saks, M., Szegedy, M.: Quantum query complexity and semi-definite programming. In: Proceedings of the 18th IEEE Annual Conference on Computational Complexity, pp. 179–193 (2003). https://doi.org/10.1109/CCC.2003.1214419

  10. Beals, R., Buhrman, H., Cleve, R., Mosca, M., De Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)

    Article  MathSciNet  Google Scholar 

  11. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447 (1966)

    Article  ADS  MathSciNet  Google Scholar 

  12. Buhrman, H., De Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002)

    Article  MathSciNet  Google Scholar 

  13. Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100(5), 050502 (2008)

    Article  ADS  Google Scholar 

  14. Datta, A., Vidal, G.: Role of entanglement and correlations in mixed-state quantum computation. Phys. Rev. A 75(4), 042310 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  15. De Wolf, R.: A brief introduction to fourier analysis on the Boolean cube. Theory Comput. Grad. Surv. 1, 1–20 (2008)

    Article  Google Scholar 

  16. Deutsch, D.: Quantum theory, the Church–Turing principle and the universal quantum computer. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 400, pp. 97–117. The Royal Society (1985)

  17. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 439, pp. 553–558. The Royal Society (1992)

  18. Grillo, S., Marquezino, F.: Quantum query as a state decomposition. Theor. Comput. Sci. 736, 62–75 (2018). https://doi.org/10.1016/j.tcs.2018.03.017

    Article  MathSciNet  MATH  Google Scholar 

  19. Howard, M., Wallman, J., Veitch, V., Emerson, J.: Contextuality supplies the ’magic’ for quantum computation. Nature 510(7505), 351–355 (2014)

    Article  ADS  Google Scholar 

  20. Jozsa, R.: Entanglement and quantum computation. In: Huggett, S.A., Mason, L.J., Tod, K.P., Tsou, S.T., Woodhouse, N.M.J. (eds.) The Geometric Universe: Science, Geometry, and the Work of Roger Penrose, vol. 27, pp. 369–379. Oxford University Press, Oxford (1998)

    Google Scholar 

  21. Jozsa, R., Linden, N.: On the role of entanglement in quantum-computational speed-up. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 459, pp. 2011–2032. The Royal Society (2003)

  22. Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81(25), 5672 (1998)

    Article  ADS  Google Scholar 

  23. Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. In: Hooker, C.A. (eds.) The Logico-Algebraic Approach to Quantum Mechanics, vol. 5a, pp. 293–328. Springer, Dordrecht (1975). https://doi.org/10.1007/978-94-010-1795-4_17

  24. Montanaro, A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53(12), 122206 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  25. O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  26. Ozols, M., Roetteler, M., Roland, J.: Quantum rejection sampling. ACM Trans. Comput. Theory 5(3), 11 (2013)

    Article  MathSciNet  Google Scholar 

  27. Reichardt, B.W., Spalek, R.: Span-program-based quantum algorithm for evaluating formulas. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 103–112. ACM (2008)

  28. Rötteler, M.: Quantum algorithms for highly non-linear Boolean functions. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 448–457. SIAM (2010)

  29. Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91(14), 147902 (2003)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors thank R. Portugal, S. Collier, J. Szwarcfiter, and E. Galvão for useful discussions and suggestions. This work was initiated while SAG was at the Federal University of Rio de Janeiro, Brazil. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro - Brasil (FAPERJ), JCNE Grant E-26/203.235/2016.

Author information

Authors and Affiliations

  1. Universidad Autónoma de Asunción, Jejuí 667, Asunción, Brazil

    Sebastián Alberto Grillo

  2. Universidade Federal do Rio de Janeiro, Avenida Horacio Macedo, 2030, COPPE/H319, Rio de Janeiro, Brazil

    Franklin de Lima Marquezino

Authors
  1. Sebastián Alberto Grillo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Franklin de Lima Marquezino
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Franklin de Lima Marquezino.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grillo, S.A., de Lima Marquezino, F. Fourier 1-norm and quantum speed-up. Quantum Inf Process 18, 99 (2019). https://doi.org/10.1007/s11128-019-2208-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2208-7

Keywords