computational complexity

, Volume 22, Issue 2, pp 429–462 | Cite as

A strong direct product theorem for quantum query complexity

Article

Abstract

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma—computing the parity of k copies of f with fewer than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.

Keywords

Quantum query complexity adversary method strong direct product theorem XOR lemma 

Subject classification

68Q12 68Q17 81P68 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  2. 2.NEC Laboratories AmericaPrincetonUSA
  3. 3.QuIC - ULB, Ecole Polytechnique de BruxellesUniversité Libre de BruxellesBrusselsBelgium

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