Simpler quantum counting

A simpler quantum counting algorithm based on amplitude amplification is presented. This algorithm is bounded by O(sqrt(N/M)) calls to the controlled-Grover operator where M is the number of marked states and N is the total number of states in the search space. This algorithm terminates within log(sqrt(N/M)) consecutive measurement steps when the probability p1 of measuring the state |1> is at least 0.5, and the result from the final step is used in estimating M by a classical post processing. The purpose of controlled-Grover iteration is to increase the probability p1. This algorithm requires less quantum resources in terms of the width and depth of the quantum circuit, produces a more accurate estimate of M, and runs significantly faster than the phase estimation-based quantum counting algorithm when the ratio M/N is small. We compare the two quantum counting algorithms by simulating various cases with a different M/N ratio, such as M/N > 0.125 or M/N < 0.001.