放大量子傅里叶变换：解决局域周期问题
The Amplified Quantum Fourier Transform: solving the local period problem
摘要：
本文提出并分析了一种新的量子算法--放大量子傅里叶变换（Amplified-QFT），用于解决以下问题：局域周期问题：设L = {0，1. N-1}是N个标签的集合，并且令A是周期P的M个标签的子集，即形式A = {j：j = s + rP; r = 0，1. M-1}，其中P < sqrt（N）且M << N，并且其中M被假定为已知。给定一个oracle f：L->{0，1}，它在A上为1，在其他地方为0，找到局部周期P。本文的第一部分定义了放大QFT算法。论文的第二部分总结了本文的主要结果，并将放大QFT算法与量子傅里叶变换（QFT）和量子隐子群（QHS）算法在求解局部周期问题时进行了比较。结果表明，放大QFT是，平均而言，二次方快于QFT和QHS算法。本文第三部分给出了主要结果的详细证明，描述了从观测值y恢复P的方法，描述了恢复炮检距s的方法。
英文摘要：
This paper creates and analyses a new quantum algorithm called the Amplified Quantum Fourier Transform (Amplified-QFT) for solving the following problem: The Local Period Problem: Let L = {0,1...N-1} be a set of N labels and let A be a subset of M labels of period P, i.e. a subset of the form A = {j : j = s + rP; r = 0,1...M-1} where P < sqrt(N) and M << N, and where M is assumed known. Given an oracle f : L->{0,1} which is 1 on A and 0 elsewhere, find the local period P. A separate algorithm finds the offset s. The first part of the paper defines the Amplified-QFT algorithm. The second part of the paper summarizes the main results and compares the Amplified-QFT algorithm against the Quantum Fourier Transform (QFT) and Quantum Hidden Subgroup (QHS) algorithms when solving the local period problem. It is shown that the Amplified-QFT is, on average, quadratically faster than both the QFT and QHS algorithms. The third part of the paper provides the detailed proofs of the main results, describes the method of recovering P from an observation y and describes the method for recovering the offset s.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
