Determination of the number of shots for Grover’s search algorithm

Abstract

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.