Quantum Algorithms for Finding Claws, Collisions and Triangles

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an N 3/4 log N quantum upper bound for the element distinctness problem (contrasting with N log N classical complexity). We also give an algorithm to finding a triangle in a graph more efficiently than classically.