The quantum query complexity of AC0

Authors:
Paul Beame;Widad Machmouchi
Affiliations:
Computer Science and Engineering, University of Washington, Seattle, WA;Computer Science and Engineering, University of Washington, Seattle, WA
Venue:
Quantum Information & Computation
Year:
2012

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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 AC0. The best previous lower bound known for any AC0 function was the Ω((n/log n)2/3) bound given by Aaronson and Shi's Ω(n2/3) lower bound for the element distinctness problem [1].