# An Exact Quantum Hidden Subgroup Algorithm and Applications to Solvable Groups

Imran, M and Ivanyos, Gábor
(2022)
*An Exact Quantum Hidden Subgroup Algorithm and Applications to Solvable Groups.*
QUANTUM INFORMATION & COMPUTATION, 22 (9-10).
pp. 770-789.
ISSN 1533-7146
10.26421/QIC22.9-10-4

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## Abstract

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in Z(mk)(n). The algorithm uses the quantum Fourier transform modulo m and does not require factorization of m. For smooth m, i.e., when the prime factors of m are of size (log m)(O(1)), the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general m, the algorithm of Mosca and Zalka is available. Even for m = 3 and k = 1 our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as m. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.

Item Type: | Article |
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Uncontrolled Keywords: | Primitive roots; Abelian group; Computer Science, Theory & Methods; Query complexity; Physics, Particles & Fields; Quantum Science & Technology; Hidden Subgroup Problem; exact quantum algorithm; CIRCUIT FAMILIES; |

Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Divisions: | Informatics Laboratory |

SWORD Depositor: | MTMT Injector |

Depositing User: | MTMT Injector |

Date Deposited: | 29 Jun 2022 15:48 |

Last Modified: | 10 Aug 2022 08:09 |

URI: | https://eprints.sztaki.hu/id/eprint/10362 |

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