Quantum walk algorithm for element distinctness

We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness ( the problem of finding two equal items among N given items), we get an O(N-2/3) query quantum algorithm. This improves the previous O(N-3/4) quantum algorithm of Buhrman et al. [SIAM J. Comput., 34 ( 2005), pp. 1324-1330] and matches the lower bound of Aaronson and Shi [J. ACM, 51 ( 2004), pp. 595-605]. We also give an O(Nk/(k+1)) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.