Quantum measurements for hidden subgroup problems with optimal sample complexity

  • Authors:

  • Masahito Hayashi;Akinori Kawachi;Hirotada Kobayashi

  • Affiliations:

  • ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency, Bunkyo-ku, Tokyo, Japan and Graduate School of Information Sciences, Tohoku University, Aoba-ku, Senda ...;Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro-ku, Tokyo, Japan;Principles of Informatics Research Division, National Institute of Informatics, Chiyoda-ku, Tokyo, Japan

  • Venue:
  • Quantum Information & Computation
  • Year:
  • 2008

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Abstract

One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is Θ(log |H|/ log p) for a candidate set H of hidden subgroups in the case where the candidate nontrivial subgroups have the same prime order p, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important cases such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained by a variant of the pretty good measurement. This implies that the concept of the pretty good measurement is quite useful for identification of hidden subgroups over an arbitrary group with optimal sample complexity.