Every NAND formula on N variables can be evaluated in time O(N^{1/2+eps})

For every NAND formula on N variables, there is a bounded-error O(N 1 2)-time quantum algorithm that evaluates this formula on a black-box input, for ε > 0 an arbitrarily small constant. It follows that the (2 − ε)-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.

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