The Quantum Query Complexity of AC0

We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from $[2n-2]$ to $[n]$ is surjective. These results yield a nearly linear $\Omega(n/\log n)$ lower bound on the quantum query complexity of $\cl{AC}^0$. The best previous lower bound known for any $\cl{AC^0}$ function was the $\Omega ((n/\log n)^{2/3})$ bound given by Aaronson and Shi’s $\Omega(n^{2/3})$ lower bound for the element distinctness problem.

💡 Research Summary

The paper investigates the quantum query complexity of two fundamental decision problems—distinguishing a 2‑to‑1 function from an “almost” 2‑to‑1 function, and testing surjectivity of a function—and uses these results to derive a near‑linear lower bound for the quantum query complexity of any function computable by constant‑depth, polynomial‑size circuits (the class AC⁰).

A 2‑to‑1 function f: