2002/11/20 by Cristopher Moore, Daniel Rockmore, Moore, Cristopher +7 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Graph theory and applications #Machine Learning and Algorithms #Quantum Physics (quant-ph) #Topological and Geometric Data Analysis #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0211124
16 pages
openalex publication_date 2002/11/20 · arxiv created 2003/02/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a subgroup H of a group G must be determined from a quantum state y uniformly supported on a left coset of H. These hidden subgroup problems are then solved by Fourier sampling: the quantum Fourier transform of y is computed and measured. When the underlying group is non-Abelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups of semidirect products of Zp by Zq, where q divides (p-1) and q = p / polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the ``forgetful'' Abelian method are insufficient for these groups. We extend this to an information-theoretic solution for the hidden subgroup problem over semidirect products of Zp by \Zq where q divides (p-1) and, in particular, the Affine groups Ap. Finally, we prove a closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.