Quantum Algorithms for the Triangle Problem

TLDR: In this paper, Buhrman et al. presented two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph, or reject if the triangle is triangle free.

Abstract: We present two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject if $G$ is triangle free The first algorithm uses combinatorial ideas with Grover Search and makes $\tilde{O}(n^{10/7})$ queries The second algorithm uses $\tilde{O}(n^{13/10})$ queries and is based on a design concept of Ambainis [in Proceedings of the $45$th IEEE Symposium on Foundations of Computer Science, 2004, pp 22-31] that incorporates the benefits of quantum walks into Grover Search [L Grover, in Proceedings of the Twenty-Eighth ACM Symposium on Theory of Computing, 1996, pp 212-219] The first algorithm uses only $O(\log n)$ qubits in its quantum subroutines, whereas the second one uses $O(n)$ qubits The Triangle Problem was first treated in [H Buhrman et al, SIAM J Comput, 34 (2005), pp 1324-1330], where an algorithm with $O(n+\sqrt{nm})$ query complexity was presented, where $m$ is the number of edges of $G$