We present three quantum algorithms for clustering graphs based on higher-order patterns, known as motif clustering. One uses a straightforward application of Grover search, the other two make use of quantum approximate counting, and all of them obtain square-root like speedups over the fastest classical algorithms in various settings. In order to use approximate counting in the context of clustering, we show that for general weighted graphs the performance of spectral clustering is mostly left unchanged by the presence of constant (relative) errors on the edge weights. Finally, we extend the original analysis of motif clustering in order to better understand the role of multiple `anchor nodes' in motifs and the types of relationships that this method of clustering can and cannot capture.

翻译：我们为基于更高顺序模式的组合图提出了三种量子算法,称为“motif 群集 ” 。 一种是直接应用 Grover 搜索,其他两种是使用量子近似计算,它们都获得了平方根,就像在各种情况下最快速的经典算法上的加速。 为了在组合中使用近似计算,我们显示,对于一般加权图来说,光谱集的性能大部分由于边缘重量上存在恒定(相对)错误而没有变化。 最后,我们扩展了对motif 群集的原始分析,以便更好地了解多“锚节点”在运动中的作用以及这种组合方法能够并且无法捕捉到的关系类型。