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Quantum Killer Apps: Quantum Fourier Transform and Search Algorithms

  • Bernard Zygelman
Chapter

Abstract

We present a brief overview of the Fourier series, the Fourier and discrete Fourier transforms and their applications. We discuss a quantum algorithm that encodes the Fourier transform of the mapping f : {0, 1}n →{0, 1} in an n-qubit register. It’s shown how the quantum Fourier transform (QFT) gate is constructed from single-qubit phase and two-qubit control gates. Due to the collapse postulate, the quantum Fourier transform for f is not available in a register query, but it does allow efficient period estimation. We illustrate how the QFT is exploited in the Shor algorithm for factoring large numbers. On the average, search for an item in an unordered list of size N requires N∕2 queries. We show how the Grover quantum algorithm improves on this figure of merit as it requires resources that scale as \(\sqrt {N}\).

References

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    Michael E. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge U. Press, 2011zbMATHGoogle Scholar
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    William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, 2007)Google Scholar
  3. 3.
    Wikipedia entry for Killer application. https://en.wikipedia.org/wiki/Killer_application
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    Paul, Zimmermann, Factorisation of RSA-220 with CADO-NFS, Cado-nfs-discuss, https://lists.gforge.inria.fr/pipermail/cado-nfs-discuss/2016-May/000626.html

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Bernard Zygelman
    • 1
  1. 1.Department of Physics and AstronomyUniversity of NevadaLas VegasUSA

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