Quantum walk algorithm for element distinctness

TL;DR

This paper introduces a quantum walk-based algorithm that significantly improves the efficiency of solving the element distinctness problem and its generalizations, achieving optimal query complexity.

Contribution

The authors develop a new quantum walk algorithm that reduces query complexity for element distinctness and its generalizations, matching known lower bounds.

Findings

Achieves O(N^{2/3}) query complexity for element distinctness.

Provides an O(N^{k/(k+1)}) query algorithm for finding k equal items.

Improves upon previous algorithms and matches theoretical lower bounds.

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.