Quantum Query Complexity of Boolean Functions with Small On-Sets

  • Andris Ambainis
  • Kazuo Iwama
  • Masaki Nakanishi
  • Harumichi Nishimura
  • Rudy Raymond
  • Seiichiro Tani
  • Shigeru Yamashita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)


The main objective of this paper is to show that the quantum query complexity Q(f) of an N-bit Boolean function f is bounded by a function of a simple and natural parameter, i.e., M = |{x|f(x) = 1}| or the size of f’s on-set. We prove that: (i) For \(poly(N)\le M\le 2^{N^d}\) for some constant 0 < d < 1, the upper bound of Q(f) is \(O(\sqrt{N\log M / \log N})\). This bound is tight, namely there is a Boolean function f such that \(Q(f) = \Omega(\sqrt{N\log M / \log N})\). (ii) For the same range of M, the (also tight) lower bound of Q(f) is \(\Omega(\sqrt{N})\). (iii) The average value of Q(f) is bounded from above and below by \(Q(f) = O(\log M +\sqrt{N})\) and \(Q(f) = \Omega (\log M/\log N+ \sqrt{N})\), respectively. The first bound gives a simple way of bounding the quantum query complexity of testing some graph properties. In particular, it is proved that the quantum query complexity of planarity testing for a graph with n vertices is Θ(N 3/4) where \(N = \frac{n(n-1)}{2}\).


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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andris Ambainis
    • 1
  • Kazuo Iwama
    • 2
  • Masaki Nakanishi
    • 3
  • Harumichi Nishimura
    • 4
  • Rudy Raymond
    • 5
  • Seiichiro Tani
    • 6
    • 7
  • Shigeru Yamashita
    • 3
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia
  2. 2.School of InformaticsKyoto UniversityKyotoJapan
  3. 3.Graduate School of Information ScienceNAISTNaraJapan
  4. 4.School of ScienceOsaka Prefecture UniversityOsakaJapan
  5. 5.Tokyo Research LaboratoryIBM JapanKanagawaJapan
  6. 6.NTT Communication Science LaboratoriesNTT CorporationKyotoJapan
  7. 7.JST ERATO-SORST QCI ProjectTokyoJapan

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