用格基约简的高斯算法恢复Shor算法的周期
Recovering the Period in Shor's Algorithm with Gauss' Algorithm for Lattice Basis Reduction
摘要：
Shor的算法包含一个经典的后处理部分，我们的目标是创建一个有效的，可理解的方法，除了连分数。设r为未知正整数。假设我们以某个恒定的概率获得形式为x=[ N k/r ]的随机正整数，其中[.]是有理数的下限或上限，k是从{0，1，.，r-1}，并且N是可以选择的参数。从这样的样本中恢复r的问题恰好发生在Shor算法的经典后处理部分。量子部分（量子相位估计）使得获得这样的样本成为可能，其中r是Z_n的单位群中某个元素a的阶，n是要分解的数。肖尔证明连分数算法可以有效地恢复r，因为如果N>2r^2，那么k/r在最低项中出现为x/N的收敛之一，这是由于连分数的标准结果。我们在这里提出了一种替代方法，用于恢复r的基础上的高斯算法格基减少，使我们能够有效地找到最短的非零向量的两个向量生成的格。我们的方法与基于连分数的方法一样有效，但更容易理解它为什么工作的所有细节。
英文摘要：
Shor's algorithm contains a classical post-processing part for which we aim to create an efficient, understandable method aside from continued fractions. Let r be an unknown positive integer. Assume that with some constant probability we obtain random positive integers of the form x=[ N k/r ] where [.] is either the floor or ceiling of the rational number, k is selected uniformly at random from {0,1,...,r-1}, and N is a parameter that can be chosen. The problem of recovering r from such samples occurs precisely in the classical post-processing part of Shor's algorithm. The quantum part (quantum phase estimation) makes it possible to obtain such samples where r is the order of some element a of the unit group of Z_n and n is the number to be factored. Shor showed that the continued fraction algorithm can be used to efficiently recover r, since if N>2r^2 then k/r appears in lowest terms as one of the convergents of x/N due to a standard result on continued fractions. We present here an alternative method for recovering r based on the Gauss algorithm for lattice basis reduction, allowing us to efficiently find the shortest nonzero vector of a lattice generated by two vectors. Our method is about as efficient as the method based on continued fractions, yet it is much easier to understand all the details of why it works.
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