Unbounded-Error Quantum Query Complexity

  • Authors:

  • Ashley Montanaro;Harumichi Nishimura;Rudy Raymond

  • Affiliations:

  • Department of Computer Science, University of Bristol, UK;School of Science, Osaka Prefecture University, Japan;Tokyo Research Laboratory, IBM Japan, Japan

  • Venue:
  • ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
  • Year:
  • 2008

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Abstract

This work studies the quantum query complexity of Booleanfunctions in an unbounded-error scenario where it is onlyrequired that the query algorithm succeeds with a probabilitystrictly greater than 1/2. We first show that, just as in thecommunication complexity model, the unbounded-error quantum querycomplexity is exactly half of its classical counterpart for any(partial or total) Boolean function. Next, connecting the query andcommunication complexity results, we show that the "black-box"approach to convert quantum query algorithms into communicationprotocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even inthe unbounded-error setting. We also study a related setting,called the weakly unbounded-error setting. In contrast tothe case of communication complexity, we show a tightmultiplicative θ(logn) separation betweenquantum and classical query complexity in this setting for apartial Boolean function.