



@article {1222,
	title = {The quantum query complexity of read-many formulas},
	journal = {Lecture Notes in Computer Science},
	volume = {7501},
	year = {2012},
	month = {2012/09/10},
	pages = {337-348},
	abstract = {  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.
},
	doi = {10.1007/978-3-642-33090-2_30},
	url = {http://arxiv.org/abs/1112.0548v1},
	author = {Andrew M. Childs and Shelby Kimmel and Robin Kothari}
}
