Solutions to the hidden subgroup problem on some metacyclic groups

  • Authors:

  • Demerson N. Gonçalves;Renato Portugal;Carlos M. M. Cosme

  • Affiliations:

  • National Laboratory for Scientific Computation, Petrópolis, RJ, Brazil;National Laboratory for Scientific Computation, Petrópolis, RJ, Brazil;Department of Mathematics, Teófilo, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Otoni, MG, Brazil

  • Venue:
  • TQC'09 Proceedings of the 4th international conference on Theory of Quantum Computation, Communication, and Cryptography
  • Year:
  • 2009

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Abstract

We describe a new polynomial-time quantum algorithm that solves the hidden subgroup problem (HSP) for a special class of metacyclic groups, namely $\mathbb{Z}_{p} \rtimes \mathbb{Z}_{q^s}$, with q |(p−1) and p/q=poly(log p), where p, q are any odd prime numbers and s is any positive integer. This solution generalizes previous algorithms presented in the literature. In a more general setting, without imposing a relation between p and q, we obtain a quantum algorithm with time and query complexity $2^{O(\sqrt{\log p})}$. In any case, those results improve the classical algorithm, which needs ${\Omega}(\sqrt{p})$ queries.