In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

arXiv:quant-ph/0102014Feb 2, 20012 citations

ON THE COMPLEXITY OF THE HIDDEN SUBGROUP PROBLEM

Stephen Fenner, Yong Zhang

International Journal of Foundations of Computer ScienceDec 1, 20132 citations

ACM Symposium on Theory of Computing (STOC)

Hidden translation and orbit coset in quantum computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklós Sántha, Pranab Sen

Published Jan 1, 2003• 8 citations

Publisher

Abstract

DOI: 10.1145/780543.780544

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Stephen Fenner, Yong Zhang

We study the computational complexity of the HIDDEN SUBGROUP problem, a well-studied problem in quantum computing. First we show that several proposed generalizations or variants of this problem, including HIDDEN COSET, HIDDEN SHIFT, and ORBIT COSET, are all equivalent or reducible to HIDDEN SUBGROUP. Then we study the relationship between the decision version and search version of HIDDEN SUBGROUP over various group classes. We show that the two versions are polynomial-time equivalent over permutation groups, and over dihedral groups given the order of the group is smooth. Finally, we give nonadaptive program checkers for HIDDEN SUBGROUP and its decision version.We study the computational complexity of the HIDDEN SUBGROUP problem, a well-studied problem in quantum computing. First we show that several proposed generalizations or variants of this problem, including HIDDEN COSET, HIDDEN SHIFT, and ORBIT COSET, are all equivalent or reducible to HIDDEN SUBGROUP. Then we study the relationship between the decision version and search version of HIDDEN SUBGROUP over various group classes. We show that the two versions are polynomial-time equivalent over permutation groups, and over dihedral groups given the order of the group is smooth. Finally, we give nonadaptive program checkers for HIDDEN SUBGROUP and its decision version.

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\mathbb{Z}_p^n

, whenever

p

is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup and prove a powerful self-reducibility result: Translating Coset in a finite solvable group

G

is reducible to instances of Translating Coset in

G/N

and

N

, for appropriate normal subgroups

N

G

. Our self-reducibility framework, combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.