Quantum Algorithms for Element Distinctness
  We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp, and imply an O(N^{3/4} log N) quantum upper bound for the element distinctness problem in the comparison complexity model (contrasting with Theta(N log N) classical complexity). We also prove a lower bound of Omega(N^{1/2}) comparisons for this problem and derive bounds for a number of related problems. 
Harry Buhrman, Christoph Durr, Mark Heiligman, Peter Hoyer, Frederic   Magniez, Miklos Santha, Ronald de Wolf