### Abstract

We employ concepts and tools from the theory of finite permutation groups in order to analyse the Hidden Subgroup Problem via Quantum Fourier Sampling (QFS) for the symmetric group. We show that under very general conditions both the weak and the random-strong form (strong form with random choices of basis) of QFS fail to provide any advantage over classical exhaustive search. In particular we give a complete characterisation of polynomial size subgroups, and of primitive subgroups, that can be distinguished from the identity subgroup with the above methods. Furthermore, assuming a plausible group theoretic conjecture for which we give supporting evidence, we show that weak and random-strong QFS for the symmetric group have no advantage whatsoever over classical search.

Original language | English (US) |
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Pages | 1118-1125 |

Number of pages | 8 |

State | Published - 2005 |

Event | Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms - Vancouver, BC, United States Duration: Jan 23 2005 → Jan 25 2005 |

### Other

Other | Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | Vancouver, BC |

Period | 1/23/05 → 1/25/05 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

*The hidden subgroup problem and permutation group theory*. 1118-1125. Paper presented at Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Vancouver, BC, United States.