Weak Parity
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Aaronson_Weak parity.pdf
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149.73 KB
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Author(s)
Aaronson, Scott
Ambainis, Andris
Balodis, Kaspars
Bavarian, Mohammad
Date Issued
2014
Journal
Automata, Languages, and Programming
Publisher
Springer-Verlag
Citation
Aaronson, Scott, Andris Ambainis, Kaspars Balodis, and Mohammad Bavarian. “Weak Parity.” Lecture Notes in Computer Science (2014): 26–38.
Version
Author's final manuscript
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[superscript 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(logn) in the randomized case and Ω(√logn) 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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/978-3-662-43948-7_3
