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

  • Markus Püschel
  • Martin Rötteler
  • Thomas Beth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1719)


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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  1. 1.
    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Physical Review A, 52(5):3457–3467, November 1995.CrossRefGoogle Scholar
  2. 2.
    R. Beals. Quantum computation of Fourier transforms over the symmetric groups. In Proc. STOC 97, El Paso, Texas, 1997.Google Scholar
  3. 3.
    Th. Beth. Verfahren der Schnellen Fouriertransformation. Teubner, 1984.Google Scholar
  4. 4.
    Th. Beth. On the computational complexity of the general discrete Fourier transform. Theoretical Computer Science, 51:331–339, 1987.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Clausen. Fast generalized Fourier transforms. Theoretical Computer Science, 67:55–63, 1989.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Clausen and U. Baum. Fast Fourier Transforms. BI-Verlag, 1993.Google Scholar
  7. 7.
    James W. Cooley and John W. Tukey. An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation, 19:297–301, 1965.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D. Coppersmith. An Approximate Fourier Transform Useful for Quantum Factoring. Technical Report RC 19642, IBM Research Division, 1994.Google Scholar
  9. 9.
    W.C. Curtis and I. Reiner. Methods of Representation Theory, volume 1. Interscience, 1981.Google Scholar
  10. 10.
    P. Diaconis and D. Rockmore. Efficient computation of the Fourier transform on finite groups. Amer. Math. Soc., 3(2):297–332, 1990.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    S. Egner. Zur Algorithmischen Zerlegungstheorie Linearer Transformationen mit Symmetrie. PhD thesis, Universität Karlsruhe, Informatik, 1997.Google Scholar
  12. 12.
    S. Egner and M. Püschel. AREP-A Package for Constructive Representation Theory and Fast Signal Transforms. GAP share package, 1998.
  13. 13.
    A. Fijany and C. P. Williams. Quantum Wavelet Transfoms: Fast Algorithms and Complete Circuits. In Proc. NASA conference QCQC 98, LNCS vol. 1509, 1998.Google Scholar
  14. 14.
    The GAP Team, Lehrstuhl D für Mathematik, RWTH Aachen, Germany and School of Mathematical and Computational Sciences, U. St. Andrews, Scotland. GAP-Groups, Algorithms, and Programming, Version 4, 1997.Google Scholar
  15. 15.
    M. Grassl, W. Geiselmann, and Th. Beth. Quantum Reed-Solomon Codes. In Proc. of the AAECC-13 (this volume), 1999.Google Scholar
  16. 16.
    P. H∅yer. Efficient Quantum Transforms. LANL preprint quant-ph/9702028, February 1997.Google Scholar
  17. 17.
    B. Huppert. Endliche Gruppen, volume I. Springer, 1983.Google Scholar
  18. 18.
    A. Yu. Kitaev. QuantumMeasurements and the Abelian Stabilizer Problem. LANL preprint quant-ph/9511026, November 1995.Google Scholar
  19. 19.
    D. Maslen and D. Rockmore. Generalized FFTs-A survey of some recent results. In Proceedings of IMACS Workshop in Groups and Computation, volume 28, pages 182–238, 1995.Google Scholar
  20. 20.
    T. Minkwitz. Algorithmensynthese für lineare Systeme mit Symmetrie. PhD thesis, Universität Karlsruhe, Informatik, 1993.Google Scholar
  21. 21.
    T. Minkwitz. Extension of Irreducible Representations. AAECC, 7:391–399, 1996.MATHMathSciNetGoogle Scholar
  22. 22.
    C. Moore and M. Nilsson. Some notes on parallel quantum computation. LANL preprint quant-ph/9804034, April 1998.Google Scholar
  23. 23.
    M. Püschel. Konstruktive Darstellungstheorie und Algorithmengenerierung. PhD thesis, Universität Karlsruhe, Informatik, 1998. Translated in [24].Google Scholar
  24. 24.
    M. Püschel. Constructive representation theory and fast signal transforms. Technical Report Drexel-MCS-1999-1, Drexel University, Philadelphia, 1999. Translation of [23].Google Scholar
  25. 25.
    D. Rockmore. Some applications of generalized FFT’s. In Proceedings of DIMACS Workshop in Groups and Computation, volume 28, pages 329–370, 1995.MathSciNetGoogle Scholar
  26. 26.
    J. P. Serre. Linear Representations of Finite Groups. Springer, 1977.Google Scholar
  27. 27.
    P. W. Shor. Algorithms for Quantum Computation: Discrete Logarithm and Factoring. In Proc. FOCS 94, pages 124–134. IEEE Computer Society Press, 1994.Google Scholar

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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