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Redondo Beach, California

Nov. 12, 2000 to Nov. 14, 2000

ISBN: 0-7695-0850-2

pp: 515

L. Hales , Group in Logic & the Methodology of Sci., California Univ., Berkeley, CA, USA

S. Hallgren , Group in Logic & the Methodology of Sci., California Univ., Berkeley, CA, USA

ABSTRACT

We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary inverse polynomial in n. This improves the method of A.Y. Kitaev (1995) which requires time quadratic in n. This algorithm also leads to a general and efficient Fourier sampling technique which improves upon the quantum Fourier sampling lemma of L. Hales and S. Hallgren (1997). As an application of this technique, we give a quantum algorithm which finds the period of an arbitrary periodic function, i.e. a function which may be many-to-one within each period. We show that this algorithm is efficient (polylogarithmic in the period of the function) for a large class of periodic functions. Moreover, using standard quantum lower-bound techniques, we show that this characterization is right. That is, this is the maximal class of periodic functions with an efficient quantum period-finding algorithm.

INDEX TERMS

Fourier transforms; quantum computing; computational complexity; polynomials; improved quantum Fourier transform algorithm; arbitrary inverse polynomial; Fourier sampling technique; quantum Fourier sampling lemma; arbitrary periodic function; periodic functions; standard quantum lower-bound techniques; maximal class; quantum period-finding algorithm

CITATION

L. Hales,
S. Hallgren,
"An improved quantum Fourier transform algorithm and applications",

*FOCS*, 2000, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2000, pp. 515, doi:10.1109/SFCS.2000.892139