Quantum counting is a key quantum algorithm that aims to determine the number
of marked elements in a database. This algorithm is based on the quantum phase
estimation algorithm and uses the evolution operator of Grover's algorithm
because its non-trivial eigenvalues are dependent on the number of marked
elements. Since Grover's algorithm can be viewed as a quantum walk on a
complete graph, a natural way to extend quantum counting is to use the
evolution operator of quantum-walk-based search on non-complete graphs instead
of Grover's operator. In this paper, we explore this extension by analyzing the
coined quantum walk on the complete bipartite graph with an arbitrary number of
marked vertices. We show that some eigenvalues of the evolution operator depend
on the number of marked vertices and using this fact we show that the quantum
phase estimation can be used to obtain the number of marked vertices. The time
complexity for estimating the number of marked vertices in the bipartite graph
with our algorithm aligns closely with that of the original quantum counting
algorithm.Comment: 12 pages, 3 figures, title changed, references adde