 
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
n1/5
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 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.
1. INTRODUCTION
