  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.
