On the role of dealing with quantum coherence in amplitude amplification

Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique develops key ideas of the Grover search algorithm. The original formulation by Grover has been reformulated in order to to make building blocks of the algorithm as generally as possible. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. There are practical problems, in which we may have a priori information about the database. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. We try to understand how prior knowledge and other modifications of algorithm blocks should be exploited properly. As measures of coherence, the geometric coherence and the relative entropy of coherence are mainly considered. In terms of the relative entropy of coherence, complementarity relations with the success probability seem to be the most expository. The general relations presented are illustrated within several model scenarios of amplitude amplification process.