Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $n$-vertex graph contains a triangle, clique or star of some size. For a general subgraph $H$ with $k$ vertices, we show that $H$ containment can be solved with quantum query complexity $O(n^{2-\frac{2}{k}-g(H)})$, with $g(H)$ a strictly positive function of $H$. This is better than $\tilde{O}\s{n^{2-2/k}}$ by Magniez et al. These results are obtained in the learning graph model of Belovs.