Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω(N) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.

中文翻译：

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量子行走很难找到相邻的顶点

量子遍历对于设计在各种搜索问题上优于经典算法的量子算法非常有用。但是，大多数论文都考虑包含单个标记元素的搜索空间。我们表明，如果搜索空间包含多个标记元素，则它们的位置可能会严重影响搜索的性能。更具体地说，我们研究了在一般图形上通过量子游走进行的搜索，并显示了标记顶点的一大类配置，为此，通过量子游走进行的搜索需要Ω（N）步骤，也就是说，它无法比经典的穷举搜索更快。对于两个或多个相邻标记顶点的某些放置，会发生所展示的配置。分析是针对二维网格和超立方体进行的，然后将其推广到任何图。此外，我们考虑发现效果的算法应用。我们研究了二部图中完美匹配的检测问题。我们将发现的效果用作算法的构建块，并构造量子算法，对于特定的图类，量子算法的性能要优于其经典类似物。