The quantum query complexity of evaluating any read-once formula with n
black-box input bits is Theta(sqrt(n)). However, the corresponding problem for
read-many formulas (i.e., formulas in which the inputs have fanout) is not well
understood. Although the optimal read-once formula evaluation algorithm can be
applied to any formula, it can be suboptimal if the inputs have large fanout.
We give an algorithm for evaluating any formula with n inputs, size S, and G
gates using O(min{n, sqrt{S}, n^{1/2} G^{1/4}}) quantum queries. Furthermore,
we show that this algorithm is optimal, since for any n,S,G there exists a
formula with n inputs, size at most S, and at most G gates that requires
Omega(min{n, sqrt{S}, n^{1/2} G^{1/4}}) queries. We also show that the
algorithm remains nearly optimal for circuits of any particular depth k >= 3,
and we give a linear-size circuit of depth 2 that requires Omega (n^{5/9})
queries. Applications of these results include a Omega (n^{19/18}) lower bound
for Boolean matrix product verification, a nearly tight characterization of the
quantum query complexity of evaluating constant-depth circuits with bounded
fanout, new formula gate count lower bounds for several functions including
PARITY, and a construction of an AC^0 circuit of linear size that can only be
evaluated by a formula with Omega(n^{2-epsilon}) gates.
