The Phase Matrix

  • Peter Høyer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)

Abstract

Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical algorithm. Amplitude amplification is controlled by choosing two complex numbers φs and φt of unit norm, called phase factors. If the phases are well-chosen, amplitude amplification reduces the error of quantum algorithms, if not, it may increase the error. We give an analysis of amplitude amplification with a emphasis on the influence of the phase factors on the error of quantum algorithms. We introduce a so-called phase matrix and use it to give a straightforward and novel analysis of amplitude amplification processes. We show that we may always pick identical phase factors φs = φt with argument in the range \({{\pi}\over{3}}{\leq} {\rm arg}(\phi_{s}){\leq} {\pi}\). We also show that identical phase factors φs = φt with \(-{{\pi}\over{2}}< {\rm arg}(\phi_{s})< {{\pi}\over{2}}\) never leads to an increase in the error, generalizing a recent result of Lov Grover who shows that amplitude amplification becomes a quantum analogue of classical repetition if we pick phase factors φs = φt with \({\rm arg}(\phi_{s}) = {{\pi}\over{3}}\).

Keywords

Quantum Computing Algorithms Amplitude Amplification Randomized Algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46(4–5), 493–505 (1998)CrossRefGoogle Scholar
  2. 2.
    Brassard, G., Høyer, P.: An exact quantum polynomial-time algorithm for Simon’s problem. In: Proc. 5th Israeli Symp. Theory of Comput. and Systems, pp. 12–23 (1997)Google Scholar
  3. 3.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information: A Millennium Volume. AMS Contemp. Math. 305, 53–74 (2002)Google Scholar
  4. 4.
    Buhrman, H., Cleve, R., de Wolf, R., Zalka, C.: Bounds for small-error and zero-error quantum algorithms. In: Proc. 40th IEEE Symp. Found. Comput. Sci., pp. 358–368 (1999)Google Scholar
  5. 5.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th ACM Symp. Theory Comput., pp. 212–219 (1996)Google Scholar
  6. 6.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325–328 (1997)CrossRefGoogle Scholar
  7. 7.
    Grover, L.K.: A different kind of quantum search. quant-ph/0503205 (May 2005)Google Scholar
  8. 8.
    Long, G.-L., Li, X., Sun, Y.: Phase matching condition for quantum search with a generalized initial state. Phys. Lett. A 294(3-4), 143–152 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Long, G.-L., Li, Y.S., Zhang, W.L., Niu, L.: Phase matching in quantum searching. Phys. Lett. A 262(1), 27–34 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Tulsi, T., Grover, L.K., Patel, A.: A new algorithm for directed quantum search. quant-ph/0505007 (May 2005)Google Scholar
  11. 11.
    Xiao, L., Jones, J.: An NMR implementation of Grover’s fixed-point quantum search algorithm. quant-ph/0504054 (April 2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Peter Høyer
    • 1
  1. 1.Department of Computer ScienceUniversity of CalgaryCanada

Personalised recommendations