Adversary Lower Bound for Element Distinctness with Small Range

Abstract: The Element Distinctness problem is to decide whether each character of an input string is unique. The quantum query complexity of Element Distinctness is known to be $\Theta(N<sup>{2/3})$; the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size $\Omega(N<sup>2)$. We construct a tight $\Omega(N<sup>{2/3})$ adversary lower bound for Element Distinctness with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.