Quantum walk algorithm for element distinctness
Andris Ambainis
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(N2/3) query quantum algorithm. This improves the previous O(N3/4) query quantum algorithm of Buhrman et.al. (quant-ph/0007016) and matches the lower bound by Shi (quant-ph/0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(Nk/(k+1)) query quantum algorithm.
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