Quantum Queries on Permutations with a Promise

  • Authors:

  • Rūsiņš Freivalds;Kazuo Iwama

  • Affiliations:

  • Department of Computer Science, University of Latvia, Riga, Latvia LV-1459;Department of Communications and Computer Engineering, School of Informatics, Kyoto University, Kyoto, Japan 606-8501

  • Venue:
  • CIAA '09 Proceedings of the 14th International Conference on Implementation and Application of Automata
  • Year:
  • 2009

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Abstract

This paper studies quantum query complexities for deciding (exactly or with probability 1.0) the parity of permutations of n numbers, 0 through n *** 1. Our results show quantum mechanism is quite strong for this non-Boolean problem as it is for several Boolean problems: (i) For n = 3, we need a single query in the quantum case whereas we obviously need two queries deterministically. (ii) For even n , n /2 quantum queries are sufficient whereas we need n *** 1 queries deterministically. (iii) Our third result is for the problem deciding whether the given permutation is the identical one. For this problem, we show that there is a nontrivial promise such that if we impose that promise to the input of size n = 4m , then we need only two quantum queries, while at least 2m + 2 ( = n /2 + 2) deterministic queries are necessary.