### Abstract

We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z_{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 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 language | English (US) |
---|---|

Pages (from-to) | 515-525 |

Number of pages | 11 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - 2000 |

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### All Science Journal Classification (ASJC) codes

- Hardware and Architecture

### Cite this

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**Improved quantum Fourier transform algorithm and applications.** / Hales, Lisa; Hallgren, Sean.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Improved quantum Fourier transform algorithm and applications

AU - Hales, Lisa

AU - Hallgren, Sean

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034505623&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034505623&partnerID=8YFLogxK

M3 - Article

SP - 515

EP - 525

JO - Annual Symposium on Foundations of Computer Science - Proceedings

JF - Annual Symposium on Foundations of Computer Science - Proceedings

SN - 0272-5428

ER -