Quantum Walk Algorithm for Element Distinctness

Abstract

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 $O(N^{k/(k+1)})$ query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.