The quantum query complexity of read-many formulas

TL;DR

This paper establishes optimal quantum algorithms and lower bounds for evaluating read-many formulas, extending understanding beyond read-once formulas and impacting complexity bounds for various circuit evaluation problems.

Contribution

The paper introduces a nearly optimal quantum algorithm for evaluating formulas with large fanout and proves matching lower bounds, advancing quantum query complexity theory for read-many formulas.

Findings

Quantum query complexity for read-many formulas is Theta(min{n, sqrt{S}, n^{1/2} G^{1/4}}).

The algorithm is nearly optimal for circuits of any depth k >= 3.

Applications include lower bounds for Boolean matrix product verification and circuit evaluation complexity.

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.