Suppose we have two graphs, given by two different adjacency matrices $G_0$ and $G_1$. We wish to know whether the graphs are isomorphic.
It's been a folklore result for a long time that if we can find a way permute the adjacency matrices so as to prepare the state:
$$\sum_{b\in\{0,1\},i\in S_n}\vert b\rangle\vert \pi_i(G_b)\rangle$$
then a simple $\mathsf{SWAP}$ test on the first register will tell whether the graphs are isomorphic.