Quantum Lower Bound for the Collision Problem

Scott Aaronson
(Professor Umesh Vazirani)
DARPA grant and NSF Graduate Fellowship

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 Ω(n1/5) 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(n1/3), but obtaining any lower bound better than Ω(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 Ω(n1/7) 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.

S. Aaronson, "Quantum Lower Bound for the Collision Problem," Proc. ACM Proc. ACM Symp. Theory of Computing, Montreal, Canada, May 2002.

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