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Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1719))

Abstract

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of nonabelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2n is O( n 2) in all cases.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA

    Markus Püschel

  2. Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, D-76128, Karlsruhe, Germany

    Martin Rötteler & Thomas Beth

Authors
  1. Markus Püschel
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  2. Martin Rötteler
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  3. Thomas Beth
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, 2540 Dole St., # 483, Honolulu, HI, 96822, USA

    Marc Fossorier  & Shu Lin  & 

  2. Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo, 106, Japan

    Hideki Imai

  3. AAECC/IRIT, Université Paul Sabatier, 118, Route de Narbonne, F31067, Toulouse, Cedex, France

    Alain Poli

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© 1999 Springer-Verlag Berlin Heidelberg

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Püschel, M., Rötteler, M., Beth, T. (1999). Fast Quantum Fourier Transforms for a Class of Non-abelian Groups. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_15

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  • DOI: https://doi.org/10.1007/3-540-46796-3_15

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66723-0

  • Online ISBN: 978-3-540-46796-0

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