We introduce the concepts of Grover operators and Grover kernels to systematically analyze Grover's searching algorithms. Then we investigate a one-parameter family of quantum searching algorithms of Grover type and we show that the standard Grover algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order O(N), with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.
This paper illustrated the optimality of Grover quantum search algorithm, and simulated the number of iterations and the specific implementation steps of quantum search algorithm with QCL in Linux operating systems, then validated the time complexity of Grover's quantum searching algorithm is O(√(N)) while the algorithm's time complexity on classical computers is O(N). The Grover quantum search algorithm were shown to improve the performance significantly in the random database search problem. This conclusion was drawn from the simulation in the Linux operating systems adopted to quantum computation language. The present study highlighted the importance of high efficiency of the Grover quantum search algorithm to achieve the best possible performance in such cases. The existence of such efficiency is an outstanding advantage of quantum search algorithm over the conventional approach.