在完全图中寻找子图的一条边的量子搜索模型
A quantum searching model finding one of the edges of a subgraph in a complete graph
摘要：
一些量子搜索模型是由微扰量子行走给出的。驱动一些受扰的量子行走，我们可能会很快找到一个高概率的目标。在本文中，我们建立了一个量子搜索模型，在一个完全图中找到一个给定的子图的一条边。构造模型的方法是用$+1$或$-1$标记弧，并在弧集合上用符号函数定义一个扰动量子行走。然后，我们尽可能快地检测标记为$-1$的边缘之一。这一想法由Segawa等人于2021年首次提出。他们只解决了子图形成匹配的情况，并通过组合论证得到，找到子图的一条边的时间比经典搜索模型快二次方。在本文中，我们证明了该模型是有效的任何子图，即，通过谱分析，我们得到了在完全图中寻找子图的一条边的二次加速。
英文摘要：
Some of the quantum searching models have been given by perturbed quantum walks. Driving some perturbed quantum walks, we may quickly find one of the targets with high probability. In this paper, we construct a quantum searching model finding one of the edges of a given subgraph in a complete graph. How to construct our model is that we label the arcs by $+1$ or $-1$, and define a perturbed quantum walk by the sign function on the set of arcs. After that, we detect one of the edges labeled $-1$ by the induced sign function as fast as possible. This idea was firstly proposed by Segawa et al. in 2021. They only addressed the case where the subgraph forms a matching, and obtained by a combinatorial argument that the time of finding one of the edges of the subgraph is quadratically faster than a classical searching model. In this paper, we show that the model is valid for any subgraph, i.e., we obtain by spectral analysis a quadratic speed-up for finding one of the edges of the subgraph in a complete graph.
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