  This paper concerns the Grover algorithm that permits to make amplification
of quantum states previously tagged by an Oracle. Grover's algorithm allows
searches in an unstructure database of n entries finding a marked element with
a quadratic speedup. The algorithm requires a predefined number of runs to
succeed with probability close to one.This article provides a description of
the amplitude amplification quantum algorithm mechanism in a very short
computational way, based on tensor products and provides a geometric
presentation of the successive system states. All the basis changes are fully
described to provide an alternative to the wide spread Grover description based
only on matrices and complex tensor computation. Our experiments encompass
numerical evaluations of circuit using the Qiskit library of IBM that meet the
theoretical considerations
