The Quantum Query Complexity of Read-Many Formulas

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 can be repeated) 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 can be repeated many times. 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 $\tilde\Omega(n^{5/9})$ queries. Applications of these results include a $\tilde\Omega(n^{19/18})$ 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 AC0 circuit of linear size that can only be evaluated by a formula with Ω(n2−e) gates.