Quantum Algorithm for the Triangle Problem

We present a new quantum algorithm that either finds a triangle (a copy of K3) in an undirected graph G on n nodes, or it outputs “reject” if G is triangle free. The algorithm uses O(n) queries, and it is based on a new design concept of Ambainis [Amb03] that incorporates the benefits of quantum walks into Grover search [Gro96]. The algorithm both improves on, and is simpler than a recent algorithm of Szegedy [Sze03] which has Õ(n) query complexity. The Triangle Problem was first treated in [BDH01], where an algorithm with O( √ n|E|) query complexity was presented (here |E| is the number of edges of G).

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