The Quantum Black-Box Complexity of Majority

Abstract. We describe a quantum black-box network computing the majority of N bits with zero-sided error ɛ using only

queries: the algorithm returns the correct answer with probability at least 1 - ɛ , and ``I don't know'' otherwise. Our algorithm is given as a randomized ``XOR decision tree'' for which the number of queries on any input is strongly concentrated around a value of at most 2/3N . We provide a nearly matching lower bound of

on the expected number of queries on a worst-case input in the randomized XOR decision tree model with zero-sided error o(1) . Any classical randomized decision tree computing the majority on N bits with zero-sided error 1/2 has cost N .

Authors and Affiliations

1. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA. hayest@math.uchicago.edu., , , , , , US

   Hayes

2. Department of Computer Science, University of Chicago, 1100 E. 58th Street, Chicago, IL 60637, USA. kutin@cs.uchicago.edu., , , , , , US

   Kutin

3. Computer Sciences Department, University of Wisconsin, 1210 W. Dayton Street, Madison, WI 53706, USA. dieter@cs.wisc.edu., , , , , , US

   van Melkebeek

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