A learning graph based quantum query algorithm for finding constant-size subgraphs

We use the learning graph framework of Belovs to show that the quantum query complexity of determining if an $n$-vertex graph contains a fixed $k$-vertex subgraph $H$ is $O(n^{2-2/k-t})$ where $t=(2k-d-3)/(k(d+1)(m+2))$ and $d$ and $m$ are such that $H$ has a vertex of degree $d$ and $m+d$ edges. The previous best algorithm of Magniez et al. had complexity $\widetilde O(n^{2-2/k})$.