Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
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
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.
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
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