Sharp error bounds on quantum Boolean summation in various settings

  • Authors:

  • Marek Kwas;Henryk Woźniakowski

  • Affiliations:

  • Department of Computer Science, Columbia University, 1214 Amsterdam Ave. MC0401, New York, NY and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsz ...;Department of Computer Science, Columbia University, 1214 Amsterdam Ave. MC0401, New York, NY and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsz ...

  • Venue:
  • Journal of Complexity
  • Year:
  • 2004

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Abstract

We study the quantum summation (QS) algorithm of Brassard et al. (see Brassard et al. (in: S.J. Lomonaco, H.E. Brandt (Eds.) Quantum Computation and Information, American Mathematical Society Providence, RI, 2002)) that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in Brassard et al. (2002) in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using M - 1 quantum queries is 3/4 πM-1 with probability 8/π2, which improves the error bound πM-1 + π2M-2 of Brassard et al. (2002). We also present error bounds with probabilities p ∈ (½,8/π2 and show that they are sharp for large M and NM-1. In the average-probabilistic setting, we prove that the QS algorithm has error of order min{M-1,N-1/2} iff M is divisible by 4. This bound is optimal, as recently shown in Papageorgiou (Average case quantum lower bounds for computing the Boolean means, this issue). For M not divisible by 4, the QS algorithm is far from being optimal if M ≪ N1/2 since its error is proportional to M-1.