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International Colloquium on Automata, Languages, and Programming

ICALP 2008: Automata, Languages and Programming pp 782–795Cite as

Superpolynomial Speedups Based on Almost Any Quantum Circuit

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5125))

Abstract

The first separation between quantum polynomial time and classical bounded-error polynomial time was due to Bernstein and Vazirani in 1993. They first showed a O(1) vs. Ω(n) quantum-classical oracle separation based on the quantum Hadamard transform, and then showed how to amplify this into a n O(1) time quantum algorithm and a n Ω(logn) classical query lower bound.

We generalize both aspects of this speedup. We show that a wide class of unitary circuits (which we call dispersing circuits) can be used in place of Hadamards to obtain a O(1) vs. Ω(n) separation. The class of dispersing circuits includes all quantum Fourier transforms (including over nonabelian groups) as well as nearly all sufficiently long random circuits. Second, we give a general method for amplifying quantum-classical separations that allows us to achieve a n O(1) vs. n Ω(logn) separation from any dispersing circuit.

An Erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-540-70575-8_73

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA

    Sean Hallgren

  2. Department of Mathematics, University of Bristol, Bristol, U.K.

    Aram W. Harrow

Authors
  1. Sean Hallgren
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  2. Aram W. Harrow
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Editor information

Editors and Affiliations

  1. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Luca Aceto

  2. Department of Computer Science, IT-Parken, University of Aarhus, Åbogade 34, 8200, ÅrhusN, Denmark

    Ivan Damgård

  3. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, UK

    Leslie Ann Goldberg

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavik, Iceland

    Magnús M. Halldórsson

  5. Department of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Anna Ingólfsdóttir

  6. Université de Bordeaux-1, LaBRI, 351, Cours de la Libération, 33405, Talence cedex, France

    Igor Walukiewicz

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Hallgren, S., Harrow, A.W. (2008). Superpolynomial Speedups Based on Almost Any Quantum Circuit. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_64

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  • DOI: https://doi.org/10.1007/978-3-540-70575-8_64

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70574-1

  • Online ISBN: 978-3-540-70575-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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