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}) 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.

33 pages, 1 figure, v9 typos with signs corrected on pages 11-12

References in corpus (6)

• A Quantum Random Walk Search Algorithm
• Exponential algorithmic speedup by quantum walk
• Spatial search by quantum walk
• Spatial search and the Dirac equation
• Quantum Random Walks Hit Exponentially Faster
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• Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
• Quantum Algorithms and Covering Spaces