Quantum Lower Bound for the Collision Problem
Abstract
The collision problem is to decide whether a function X:{1,..,n}>{1,..,n} is onetoone or twotoone, 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:

arXiv eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:quantph/0111102
 Bibcode:
 2001quant.ph.11102A
 Keywords:

 Quantum Physics;
 Computational Complexity
 EPrint:
 10 pages plus 4 page appendix, no figures. Submitted to STOC'2002