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 Θ( √ 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, √ S, n1/2G1/4}) 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 Ω(min{n, √ S, n1/2G1/4}) 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 Ω̃(n) queries. Applications of these results include a Ω̃(n) 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 circuit of linear size that can only be evaluated by a formula with Ω(n) gates.