The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups

Quantum Physics arXiv:2608.05321 (quant-ph) [Submitted on 5 Aug 2026] Title:The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups Authors:Mauro E.S. Morales View a PDF of the paper titled The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups, by Mauro E.S. Morales View PDF Abstract:Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively. In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form $G=A\rtimes_{\varphi} \mathbb{Z}_{p^k}$, with $A$ finite Abelian, $p$ prime, $k\in \mathbb{N}$ and the action of $\mathbb Z_{p^k}$ is generated by the scalar automorphism $a\mapsto\mu a$, for some $\mu\in\mathbb Z_{\operatorname{Exp}(A)}^\times$, where $\mathrm{Exp}(A)$ is the exponent of $A$. Our algorithm is efficient when $A$ has bounded generator rank and $\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)$. This includes the case $A=\mathbb{Z}_N$ for $N\in\mathbb{N}$ and $k=1$, studied by Bacon, Childs and van Dam (FOCS 2005), and $A=\mathbb{Z}_{q^r}$ with $q$ prime and $r\in \mathbb{N}$ studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003). Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2608.05321 [quant-ph] (or arXiv:2608.05321v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.05321 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Mauro Morales [view email] [v1] Wed, 5 Aug 2026 18:25:25 UTC (60 KB) Full-text links: Access Paper: View a PDF of the paper titled The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups, by Mauro E.S. MoralesView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-08 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)