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.

翻译：量子计数是一种关键的量子算法，旨在确定数据库中标记元素的数量。该算法基于量子相位估计算法，并利用Grover算法的演化算子，因其非平凡特征值依赖于标记元素的数量。由于Grover算法可视为完全图上的量子游走，扩展量子计数的一种自然方法是在非完全图上使用基于量子游走搜索的演化算子来替代Grover算子。本文通过分析具有任意数量标记顶点的完全二分图上的硬币量子游走来探索这一扩展。我们证明演化算子的某些特征值依赖于标记顶点数量，并据此表明量子相位估计算法可用于获取标记顶点的数量。使用我们的算法估计二分图中标记顶点数量的时间复杂度与原始量子计数算法基本一致。