Chapter

Algorithms and Computation

Volume 5369 of the series Lecture Notes in Computer Science pp 907-918

# Quantum Query Complexity of Boolean Functions with Small On-Sets

• Andris AmbainisAffiliated withInstitute of Mathematics and Computer Science, University of Latvia
• , Kazuo IwamaAffiliated withSchool of Informatics, Kyoto University
• , Masaki NakanishiAffiliated withGraduate School of Information Science, NAIST
• , Harumichi NishimuraAffiliated withSchool of Science, Osaka Prefecture University
• , Rudy RaymondAffiliated withTokyo Research Laboratory, IBM Japan
• , Seiichiro TaniAffiliated withNTT Communication Science Laboratories, NTT CorporationJST ERATO-SORST QCI Project
• , Shigeru YamashitaAffiliated withGraduate School of Information Science, NAIST

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## Abstract

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 $$N = \frac{n(n-1)}{2}$$.