Weak Parity

Abstract

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2 + ε fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/ε)) queries, as well as a quantum algorithm that makes $O(n/\sqrt{\log(1/\varepsilon)})$ queries. We also prove a lower bound of $\Omega\left( n/\log\left( 1/\varepsilon\right) \right)$ in both cases, as well as lower bounds of Ω(logn) in the randomized case and $\Omega(\sqrt{\log n})$ in the quantum case for any ε > 0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.