Generalized quantum counting algorithm for non-uniform amplitude distribution

We give generalized quantum counting algorithm to increase universality of quantum counting algorithm. Non-uniform initial amplitude distribution is possible due to the diversity of situations on counting problems or external noise in the amplitude initialization procedure. We give the reason why quantum counting algorithm is invalid on this situation. By modeling in three-dimensional space spanned by unmarked state, marked state and free state to the entire Hilbert space of n qubits, we find Grover iteration can be regarded as improper rotation in the space. This allows us to give formula to solve counting problem. Furthermore, we express initial amplitude distribution in the eigenvector basis of improper rotation matrix. This is necessary to obtain mathematical analysis of counting problem on various situations. Finally, we design four simulation experiments, the results of which show that compared with original quantum counting algorithm, generalized quantum counting algorithm wins great satisfaction from three aspects: (1) Whether initial amplitude distribution is uniform; (2) the diversity of situations on counting problems; and (3) whether phase estimation technique can get phase exactly.