The Quantum Query Complexity of AC0

Paul Beame; Widad Machmouchi

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arXiv · 1008.2422

The Quantum Query Complexity of AC0

Paul Beame·Widad Machmouchi

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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 $Θ(n)$ 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 $Ω(n/\log n)$ 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 $Ω((n/\log n)^{2/3})$ bound given by Aaronson and Shi's $Ω(n^{2/3})$ lower bound for the element distinctness problem.

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Paul Beame, Widad Machmouchi. 2012-01-31. The Quantum Query Complexity of AC0. https://arxiv.org/abs/1008.2422

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