Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(n5/4), where n denotes the number of vertices in the graph. This improves the previous upper bound O(n9/7) = O(n1.285…) 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 O(n9/7 vlog n) queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous O(n9/7) upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires v(n9/7/ vlog 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.