The Quantum Query Complexity of AC0

TL;DR

This paper establishes nearly linear lower bounds on the quantum query complexity for deciding properties of functions, including whether a function is 2-to-1 or surjective, implying high complexity for AC0 functions.

Contribution

It proves new nearly linear quantum lower bounds for specific function properties and extends these results to the complexity of AC0 functions, improving upon previous bounds.

Findings

Quantum algorithms require rac{n}{\u221a{ ext{log} n}} queries for certain function properties.

Lower bounds apply to deciding 2-to-1 and surjectivity of functions.

Results imply high quantum complexity for AC0 functions.

Abstract

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.