Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
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 and S. Hallgren, "An improved quantum Fourier transform algorithm and applications," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 515.