Element Distinctness

Abstract

The element distinctness problem is the problem of determining whether the elements of a list are distinct. For example, suppose we have a list x with $N=9$ elements, which are in the range [20, 50], such as $x=(25, 27, 39, 43, 39, 35, 30, 42, 28)$. Note that the third and the fifth elements are equal. We say that the elements in positions 3 and 5 collide. As a decision problem, the goal is to answer “yes” if there is a collision or “no” if there is no collision. In order to simplify the description of the quantum algorithm that solves this problem, we assume that there is either one 2-collision or none. If there is a collision, then there are indices $i_1$ and $i_2$ such that $x_{i_1}=x_{i_2}$. With a small overhead, we can find explicitly the indices $i_1$ and $i_2$ when the algorithm returns “yes.”