Family of Grover’s quantum-searching algorithms

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 $O\left(\sqrt{N}\right),$ 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.