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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/ n) lower bound on the quantum query complexity of AC^0. The best previous lower bound known for any AC^0 function was the Ω ((n/ n)^2/3) bound given by Aaronson and Shi's Ω(n^2/3) lower bound for the element distinctness problem.arXiv:1008.2422v2 fatcat:ef2zcpeqifa2nid4vitmtjecla