Quantum Lower Bound for the Collision Problem
<|reference_start|>Quantum Lower Bound for the Collision Problem: The collision problem is to decide whether a function X:{1,..,n}->{1,..,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^{1/5}) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O(n^{1/3}), but obtaining any lower bound better than Theta(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Theta(n^{1/7}) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.<|reference_end|>
@article{aaronson2001quantum,
    title={Quantum Lower Bound for the Collision Problem},
    author={Scott Aaronson},
    journal={arXiv preprint arXiv:quant-ph/0111102},
    year={2001},
    archivePrefix={arXiv},
    eprint={quant-ph/0111102},
    primaryClass={quant-ph cs.CC}
}