  In this paper we present a quantum algorithm solving the triangle finding
problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where
$n$ denotes the number of vertices in the graph. This improves the previous
upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and
Santha. Our result shows, for the first time, that in the quantum query
complexity setting unweighted triangle finding is easier than its edge-weighted
version, since for finding an edge-weighted triangle Belovs and Rosmanis proved
that any quantum algorithm requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries.
Our result also illustrates some limitations of the non-adaptive learning graph
approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over
unweighted graphs, any quantum algorithm for triangle finding obtained using
this approach requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries as well. To
bypass the obstacles characterized by these lower bounds, our quantum algorithm
uses combinatorial ideas exploiting the graph-theoretic properties of triangle
finding, which cannot be used when considering edge-weighted graphs or the
non-adaptive learning graph approach.
