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 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 (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 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.

• Type: Conference
• Year: 2002
• Where: STOC
• Authors: Scott Aaronson
• Keywords: Algorithms, Error Probability, Lower Bound, Quantum Complexity Theory, STOC 2002