Improved quantum Fourier transform algorithm and applications

Lisa Hales, Sean Hallgren

Research output: Contribution to journalConference article

68 Scopus citations

Abstract

We give an algorithm for approximating the quantum Fourier transform over an arbitrary Zp 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 Kitaev 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 [8]. 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 tight. That is, this is the maximal class of periodic functions with an efficient quantum period-finding algorithm.

Original languageEnglish (US)
Pages (from-to)515-525
Number of pages11
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
StatePublished - Dec 1 2000
Event41st Annual Symposium on Foundations of Computer Science (FOCS 2000) - Redondo Beach, CA, USA
Duration: Nov 12 2000Nov 14 2000

All Science Journal Classification (ASJC) codes

  • Hardware and Architecture

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