This paper focuses on Grover’s quantum search algorithm, which is of paramount importance as a masterpiece of Quantum Computing software. Given the inherent probabilistic nature of quantum computers, quantum programs based on Grover’s algorithm need to be run a number of times in order to generate a histogram of candidate values for solutions, which are then checked to identify the valid ones. In this paper, the distribution of the required number of shots to find all or a fraction of all the solutions to the Grover’s search problem is studied. Firstly, considering the similarity of the probability problem with the well-known coupon collector’s problem, two formulae are obtained from asymptotic results on the distribution of the required number of shots, as the number of problem solutions grows. These expressions allow to compute the number of shots required to ensure that, with probability p, all or a fraction of all the solutions are found. Secondly, the probability mass function of the required number of shots is derived, which serves as a benchmark to assess the validity of the asymptotic approximations derived previously. A comparison between the two approaches is presented and, as a result, a rule of thumb to decide under which circumstances employ one or the other is proposed.

中文翻译：

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Grover 搜索算法的镜头数量的确定

本文重点介绍了 Grover 的量子搜索算法，该算法作为量子计算软件的杰作，具有至关重要的意义。考虑到量子计算机固有的概率性质，基于 Grover 算法的量子程序需要运行多次，才能生成解决方案候选值的直方图，然后检查这些值以识别有效的值。在本文中，研究了找到 Grover 搜索问题的全部或部分解所需的镜头数量的分布。首先，考虑到概率问题与著名的优惠券收集问题的相似性，随着问题解数量的增加，从所需镜头数分布的渐近结果得到两个公式。这些表达式允许计算所需的射击次数，以确保以概率 p 找到所有或部分解决方案。其次，推导了所需射击次数的概率质量函数，作为评估先前推导的渐近近似有效性的基准。对这两种方法进行了比较，并由此提出了决定在什么情况下采用其中一种方法的经验法则。