Quantum walk algorithm for element distinctness
<|reference_start|>Quantum walk algorithm for element distinctness: 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.<|reference_end|>
@article{ambainis2003quantum,
    title={Quantum walk algorithm for element distinctness},
    author={Andris Ambainis},
    journal={SIAM Journal on Computing, 37(1):210-239, 2007},
    year={2003},
    archivePrefix={arXiv},
    eprint={quant-ph/0311001},
    primaryClass={quant-ph cs.DS}
}