We present an efficient exact version of Shor's order finding algorithm when a multiple $m$ of the order $r$ is known. The algorithm consists of two main ingredients. The first ingredient is the exact quantum Fourier transform proposed by Mosca and Zalka in [MZ03]. The second ingredient is an amplitude amplification version of Brassard and Hoyer in [BH97] combined with some ideas from the exact discrete logarithm procedure by Mosca and Zalka in [MZ03]. As applications, we show how the algorithm derandomizes the quantum algorithm for primality testing proposed by Donis-Vela and Garcia-Escartin in [DVGE18], and serves as a subroutine of an efficient exact quantum algorithm for finding primitive elements in arbitrary finite fields in a slightly less general computational model proposed by Mosca in [Mos02].
翻译:我们提出了一个高效的Shor订单查找算法的精确版本,当知道该订单的多百万美元美元时,该算法由两个主要成分组成。第一个成分是Mosca和Zalka在[MZ03]中提议的精确量子Fourier变异。第二个成分是[BH97]中的Brassard和Hoyer振幅放大版,加上Mosca和Zalka在[MZ03]中提出的精确离散对数程序的一些想法。作为应用,我们展示了该算法如何将Donis-Vela和Garcia-Escartin在[DVGE18]中提议的初量度测试量算算法作例外处理,并作为一种高效精确量算法的子,用于在Mosca在[Mos02]中提议的略为不那么一般的计算模型中,在任意的有限地区找到原始元素。