Quantum Algorithm for Triangle Finding in Sparse Graphs

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent \(\tilde{O}(n^{5/4})\)-query algorithm given by Le Gall [FOCS 2014] for triangle finding over dense graphs (here n denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with \(O(n^{5/4-\epsilon })\) queries for some constant \(\epsilon >0\) whenever the graph has at most \(O(n^{2-c})\) edges for some constant \(c>0\).