Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer
Andrew M. Childs, Ben W. Reichardt, Robert Špalek, Shengyu Zhang · arXiv (Cornell University) · 2007
For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-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.