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/ √log(1/ε)) queries. We also prove a lower bound of Ω(n/ log (1/ε)) in both cases, as well as lower bounds of Ω(log n) in the randomized case and Ω(√ 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. © 2014 Springer-Verlag.
CITATION STYLE
Aaronson, S., Ambainis, A., Balodis, K., & Bavarian, M. (2014). Weak parity. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8572 LNCS, pp. 26–38). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_3
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