### Abstract

The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.

Original language | English (US) |
---|---|

Pages (from-to) | 916-934 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - Jun 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Cite this

*SIAM Journal on Computing*,

*32*(4), 916-934. https://doi.org/10.1137/S009753970139450X

}

*SIAM Journal on Computing*, vol. 32, no. 4, pp. 916-934. https://doi.org/10.1137/S009753970139450X

**The hidden subgroup problem and quantum computation using group representations.** / Hallgren, Sean; Russell, Alexander; Ta-Shma, Amnon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The hidden subgroup problem and quantum computation using group representations

AU - Hallgren, Sean

AU - Russell, Alexander

AU - Ta-Shma, Amnon

PY - 2003/6/1

Y1 - 2003/6/1

N2 - The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.

AB - The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.

UR - http://www.scopus.com/inward/record.url?scp=0141534114&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141534114&partnerID=8YFLogxK

U2 - 10.1137/S009753970139450X

DO - 10.1137/S009753970139450X

M3 - Article

AN - SCOPUS:0141534114

VL - 32

SP - 916

EP - 934

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -