The Quantum Query Complexity of Read-Many Formulas [chapter]

Andrew M. Childs, Shelby Kimmel, Robin Kothari
2012 Lecture Notes in Computer Science  
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(minn, sqrtS, n^1/2
more » ... 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(minn, sqrtS, 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 fatcat:sj4wkylm7bb5jmdrou7wvhuf4i