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An Efficient Quantum Algorithm for Finding Hidden Parabolic Subgroups in the General Linear Group

  • Thomas Decker
  • Gábor Ivanyos
  • Raghav Kulkarni
  • Youming Qiao
  • Miklos Santha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)

Abstract

In the theory of algebraic groups, parabolic subgroups form a crucial building block in the structural studies. In the case of general linear groups over a finite field \(\mathbb{F}_q\), given a sequence of positive integers n 1, …, n k , where n = n 1 + … + n k , a parabolic subgroup of parameter (n 1, …, n k ) in GL\(_n(\mathbb{F}_q)\) is a conjugate of the subgroup consisting of block lower triangular matrices where the ith block is of size n i . Our main result is a quantum algorithm of time polynomial in logq and n for solving the hidden subgroup problem in GL\(_n(\mathbb{F}_q)\), when the hidden subgroup is promised to be a parabolic subgroup. Our algorithm works with no prior knowledge of the parameter of the hidden parabolic subgroup. Prior to this work, such an efficient quantum algorithm was only known for minimal parabolic subgroups (Borel subgroups), for the case when q is not much smaller than n (G. Ivanyos: Quantum Inf. Comput., Vol. 12, pp. 661-669).

Keywords

Parabolic Subgroup Quantum Algorithm Discrete Logarithm Borel Subgroup General Linear Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Thomas Decker
    • 5
  • Gábor Ivanyos
    • 1
  • Raghav Kulkarni
    • 2
  • Youming Qiao
    • 2
    • 3
  • Miklos Santha
    • 2
    • 4
  1. 1.Institute for Computer Science and ControlHungarian Academy of SciencesBudapestHungary
  2. 2.Centre for Quantum TechnologiesNational University of SingaporeSingapore
  3. 3.Centre for Quantum Computation and Intelligent SystemsUniversity of TechnologySydneyAustralia
  4. 4.LIAFAUniv. Paris Diderot, CNRSParisFrance
  5. 5.EXASOLNurembergGermany

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