Quantum lower bounds for the collision and the element distinctness problems

Given a function f as an oracle, the collision problem is to find two distinct inputs i and j such that f(i)=f(j), under the promise that such inputs exist. In this paper, we prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function Ω ((n/r)¹3/) times, where n is the size of the domain and r|n. This lower bound matches, up to a constant factor, the upper bound of Brassard, Hoyer and Tapp (1997), which uses the quantum algorithm of Grover (1996) in a novel way. The previously best quantum lower bound is Ω ((n/r)¹5/) evaluations, due to Aaronson (2002). Our result implies a quantum lower bound of Ω (n²3/) queries to the inputs for another well studied problem, the element distinctness problem, which is to determine whether or not the given n real numbers are distinct. The previous best lower bound is Ω (√n) queries in the black-box model; and Ω (√n log n) comparisons in the comparisons-only model, due to Hoyer Neerbek, and Shi (2001).