Representation-theoretical properties of the approximate quantum Fourier transform (original) (raw)

References

  1. Aharonov, D., Landau, Z., Makowsky, J.: The quantum FFT can be classically simulated. ArXiv quant-ph/0611156 (2006) (preprint)
  2. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995)
    Article Google Scholar
  3. Barenco, A., Ekert, A., Suominen, K.-A., Törmä, P.: Approximate qantum Fourier transform and decoherence. Phys. Rev. A 54(1), 139–146 (1996)
    Article MathSciNet Google Scholar
  4. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1997)
    Article MATH MathSciNet Google Scholar
  5. Beth, Th.: Verfahren der Schnellen Fourier transformation. Teubner (1984)
  6. Beth, Th.: On the computational complexity of the general discrete Fourier transform. Theor. Comput. Sci. 51, 331–339 (1987)
    Article MATH MathSciNet Google Scholar
  7. Beth, Th., Fumy, W., Mühlfeld, R.: Zur algebraischen diskreten Fourier-Transformation. Arch. Math. 40, 238–244 (1983)
    Article MATH Google Scholar
  8. Beth, Th., Rötteler, M.: Quantum algorithms: applicable algebra and quantum physics. In: Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, vol.173, Springer Texts in Modern Physics, pp. 96–150. Springer, Heidelberg (2001)
  9. Brassard, G., Høyer, P.: An exact polynomial-time algorithm for Simon’s problem. In: Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems (ISTCS), pp. 12–23. IEEE Computer Society Press (1997)
  10. Cheung, D.: Using generalized quantum Fourier transforms in quantum phase estimation algorithms. PhD thesis, University of Waterloo (2003)
  11. Cheung, D.: Improved bounds for the approximate QFT. In: Proceedings of the Winter International Symposium on Information and Communication Technologies (WISICT’04), vol.58. ACM International Conference Proceeding Series, pp. 1–6, Cancun, Mexico (2004)
  12. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. Lond. Ser. A 454(1969), 339–354 (1998)
    MATH MathSciNet Google Scholar
  13. Cleve, R., Watrous, J.: Fast parallel circuits for the quantum Fourier transform. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS’00), pp. 526–536. IEEE Computer Society (2000)
  14. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)
    Article MATH MathSciNet Google Scholar
  15. Coppersmith, D.: An approximate Fourier transform useful in quantum factoring. Technical Report RC 19642, IBM Research Division, see also ArXiv preprint quant-ph/0201067 (1994)
  16. Curtis, W.C., Reiner, I.: Representation Theory of Finite Groups and Algebras. Wiley, London (1962)
    MATH Google Scholar
  17. Golub, G., Van Loan, Ch.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)
    MATH Google Scholar
  18. Griffiths, R., Niu, C.: Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett. 76(17), 3228–3231 (1996)
    Article Google Scholar
  19. Grohmann, B.: Slim normal bases and basefield transforms. Appl. Algebra Eng. Commun. Comput. 18, 397–406 (2007)
    Article MATH MathSciNet Google Scholar
  20. Hales, L., Hallgren, S.: Quantum Fourier sampling simplified. In: Proceedings of the Symposium on Theory of Computing (STOC’99), pp. 330–338 (1999)
  21. Hales, L., Hallgren, S.: An improved quantum Fourier transform algorithm and applications. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS’00), pp. 515–525. IEEE Computer Society (2000)
  22. Hallgren, S.: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. J. ACM 54(1), 1–19 (2007)
    Article MathSciNet Google Scholar
  23. Hong, J., Vetterli, M., Duhamel, P.: Basefield transforms with the convolution property. Proc. IEEE 82(3), 400–412 (1994)
    Article Google Scholar
  24. Jozsa, R.: Quantum algorithms and the Fourier transform. Proc. R. Soc. Lond. A 454, 323–337 (1998)
    Article MATH MathSciNet Google Scholar
  25. Kitaev, A.Yu.: Quantum measurements and the abelian stabilizer problem. ArXiv quant–ph/9511026 (1995) (preprint)
  26. Kitaev, A.Yu.: Quantum computations: algorithms and error correction. Russ. Math. Surv. 52(6), 1191–1249 (1997)
    Article MATH MathSciNet Google Scholar
  27. Klappenecker, A.: Basefield transforms derived from character tables. In: Proceedings of the 1997 International Conference on Acoustics, Speech, and Signal Processing (ICASSP’97), pp. 1997–2000 (1997)
  28. Lang, S.: Algebra. Addison-Wesley, Reading (1993)
    MATH Google Scholar
  29. Mosca, M., Ekert, A.: The hidden subgroup problem and eigenvalue estimation on a quantum computer. In: Quantum Computing and Quantum Communications, QCQC’98, Palm Springs, vol. 1509, Lecture Notes in Computer Science, pp. 174–188. Springer, Heidelberg (1998)
  30. Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
    MATH Google Scholar
  31. Shor, P.W.: Algorithms for quantum computation: discrete logarithm and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134, Los Alamitos. Institute of Electrical and Electronic Engineers Computer Society Press (1994)
  32. Steidl, G.: Generalization of the algebraic discrete Fourier transform with application to fast convolutions. Linear Algebra Appl. 139, 181–206 (1990)
    Article MATH MathSciNet Google Scholar
  33. Yoran, N., Short, A.J.: Efficient classical simulation of the approximate quantum Fourier transform. ArXiv quant–ph/0611241 (2006) (preprint)
  34. Zilic, Z., Radecka, K.: Scaling and better approximating quantum Fourier transforms by higher radices. IEEE Trans. Comput. 56(2), 202–207 (2007)
    Article MathSciNet Google Scholar

Download references