Quantum walk algorithm for element distinctness

2003/11/30 by Andris Ambainis · 3 citations
Computer Science · Physics and Astronomy · #cs.DS #quant-ph

paper · pdf

published as SIAM Journal on Computing, 37(1):210-239, 2007 · 33 pages, 1 figure, v9 typos with signs corrected on pages 11-12

arxiv created 2014/04/30 · arxiv updated 2014/05/01

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) 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(N^k/(k+1)) query quantum algorithm.

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