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：本文提出了三种基于高阶模式（即模式聚类）的图聚类量子算法。第一种算法直接应用Grover搜索，另外两种利用量子近似计数，且在所有算法中，相较于现有最快经典算法，均能在多种场景下实现平方根量级的加速。为在聚类中应用近似计数，我们证明对于一般加权图，边权重上存在的常数（相对）误差对谱聚类的性能影响甚微。最后，我们拓展了模式聚类的原始分析，以更深入理解模式中多个"锚节点"的作用，以及该聚类方法所能及所不能捕捉的关系类型。