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行走，将量子计数扩展至非完全图上的量子行走搜索演化算子（而非Grover算子）是一种自然途径。本文通过分析完全二部图上带任意数量标记顶点的硬币量子行走来探索这一扩展。我们证明演化算子的部分特征值依赖于标记顶点数量，并据此表明量子相位估计可用于获取标记顶点数。对于二部图中标记顶点数量的估计，我们算法的时间复杂度与原量子计数算法基本一致。