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

  • Markus Püschel
  • Martin Rötteler
  • Thomas Beth
Conference paper

DOI: 10.1007/3-540-46796-3_15

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

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(n2) in all cases.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Markus Püschel
    • 1
  • Martin Rötteler
    • 2
  • Thomas Beth
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceDrexel UniversityPhiladelphiaUSA
  2. 2.Institut für Algorithmen und Kognitive SystemeUniversität KarlsruheKarlsruheGermany

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