  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.
