Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Childs, Andrew M.; Kothari, Robin; Ozols, Maris; Roetteler, Martin http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-43203
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Easy and Hard Functions for the Boolean Hidden Shift Problem

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Abstract

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice. Our algorithm for random functions is based on performing the pretty good measurement on several copies of a certain state; its analysis relies on the Fourier transform. We also use this approach to improve the quantum rejection sampling approach to the Boolean hidden shift problem.

BibTeX - Entry

@InProceedings{childs_et_al:LIPIcs:2013:4320,
  author =	{Andrew M. Childs and Robin Kothari and Maris Ozols and Martin Roetteler},
  title =	{{Easy and Hard Functions for the Boolean Hidden Shift Problem}},
  booktitle =	{8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)},
  pages =	{50--79},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-55-2},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{22},
  editor =	{Simone Severini and Fernando Brandao},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2013/4320},
  URN =		{urn:nbn:de:0030-drops-43203},
  doi =		{10.4230/LIPIcs.TQC.2013.50},
  annote =	{Keywords: Boolean hidden shift problem, quantum algorithms, query complexity, Fourier transform, bent functions}
}

Keywords: Boolean hidden shift problem, quantum algorithms, query complexity, Fourier transform, bent functions
Seminar: 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)
Issue date: 2013
Date of publication: 2013


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