Quantum Algorithms for the Triangle Problem

We present two new quantum algorithms that either find a triangle (a copy K-3) in an undirected graph C on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes (O) over tilde (n(10/7)) queries. The second algorithm uses (O) over tilde (n(13/10)) queries, and it is based on a new design concept of Ambainis [6] that incorporates the benefits of quantum walks int,o Grover search [18]. The first algorithm uses only O(logn) (whits in its quantum subroutines, whereas the second One uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + root n vertical bar E vertical bar) query complexity was presented (here vertical bar E vertical bar is the number of edges of G).