Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments [article]

François Le Gall
2014 arXiv   pre-print
In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(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
more » ... and Rosmanis proved that any quantum algorithm requires Ω(n^9/7/√( 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 Ω(n^9/7/√( 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.
arXiv:1407.0085v2 fatcat:it5egcbufngx3oabx56u4ipk3e