Majorization in General Grover's Algorithms: Efficient vs Non-efficient cases

We study the characterisation of efficient and non-efficient families of Grover's algorithms according to the majorization principle. We develop a geometrical interpretation based on the parameters that appears on these algorithms. Using this interpretation we observe a step-by-step majorization in all efficient Grover's algorithms, whereas the non-efficient Grover's algorithms fail to abide by this majorization principle. These Majorization results are first obtained from numerical calculations. Motivated by these numerical results, we also obtained an analytical demonstration. Finally, the geometrical interpretation found for these parameters is used to obtain additional results of Grover's algorithms that improves our understanding of how they work.