The quantum query complexity of certification

We study the quantum query complexity of finding a certificate for a
d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show
that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and
Theta(d^{k/2}) for 1-certificates. In particular, this shows that the
zero-error quantum query complexity of evaluating such formulas is
O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies
on the fact that the quantum adversary method obeys a direct sum theorem.