Quantum Lower Bound for the Collision Problem
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2001
- DOI:
- 10.48550/arXiv.quant-ph/0111102
- arXiv:
- arXiv:quant-ph/0111102
- Bibcode:
- 2001quant.ph.11102A
- Keywords:
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- Quantum Physics;
- Computational Complexity
- E-Print:
- 10 pages plus 4 page appendix, no figures. Submitted to STOC'2002