 
Summary: Quantum Lower Bound for the Collision Problem
Scott Aaronson #
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
# # n 1/5
# on the number of queries needed by a quantum
computer to solve this problem with bounded error prob
ability. The best known upper bound is O # n 1/3
# , 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# # 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.
