The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.

中文翻译：

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具有多个标记顶点的基于量子游走的搜索算法

量子游走是开发量子算法的有力工具，其通常基于在具有多个标记顶点的图中搜索顶点，其中Ambainis解决元素独特性问题的量子算法是最闪亮的例子。在这项工作中，我们解决了使用基于量子游走的搜索算法在任意图上具有多个标记顶点的查找标记顶点的时间复杂度的解析表达式的计算问题，扩展了以前基于 Szegedy 量子游走的分析方法，可以是仅适用于二部图。基于在二维晶格和超立方体上创造的量子行走的两个例子展示了我们方法的细节。