Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
Quantum vs. classical communication and computation
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
A lower bound on the quantum query complexity of read-once functions
Journal of Computer and System Sciences
Quantum verification of matrix products
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Quantum Query Complexity of Some Graph Problems
SIAM Journal on Computing
Quantum Algorithms for the Triangle Problem
SIAM Journal on Computing
Quantum Query Complexity of Boolean Functions with Small On-Sets
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Quantum search on bounded-error inputs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Any AND-OR Formula of Size $N$ Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer
SIAM Journal on Computing
Subcubic Equivalences between Path, Matrix and Triangle Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Improved output-sensitive quantum algorithms for Boolean matrix multiplication
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Reflections for quantum query algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Boolean Function Complexity: Advances and Frontiers
Boolean Function Complexity: Advances and Frontiers
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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−ε) gates.