Weak Parity

  • Scott Aaronson
  • Andris Ambainis
  • Kaspars Balodis
  • Mohammad Bavarian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


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/log0.246(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.


Boolean Function Quantum Algorithm Query Complexity Full Version Quantum Walk 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Scott Aaronson
    • 1
  • Andris Ambainis
    • 2
  • Kaspars Balodis
    • 2
  • Mohammad Bavarian
    • 1
  1. 1.MITUSA
  2. 2.University of LatviaLatvia

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