Every NAND formula of size N can be evaluated in time O ( N 1 2 + ε ) on a quantum computer

For every NAND formula of size N , 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. Institute for Quantum Information, California Institute of Technology. Supported by NSF Grant PHY-0456720 and ARO Grant W911NF-05-1-0294. University of California, Berkeley. Supported by NSF Grant CCF-0524837 and ARO Grant DAAD 19-03-1-0082. Work conducted in part while visiting Caltech.