A Review of Galois Qudits

Quantum Physics arXiv:2605.18981 (quant-ph) [Submitted on 18 May 2026] Title:A Review of Galois Qudits Authors:Adam Wills View a PDF of the paper titled A Review of Galois Qudits, by Adam Wills View PDF Abstract:Galois qudits are $q$-dimensional quantum systems whose choice of Pauli group encodes the arithmetic of some finite field $\mathbb{F}_q$. They differ from the more familiar modular qudit, which are the same quantum system but whose choice of Pauli group are the clock and shift operators, which encode the arithmetic of integer addition and multiplication modulo $q$. Galois qudits are a useful mathematical construct that allow us to leverage the mathematical tools that are native to the larger qudit while only physically building smaller qudits. In particular, a Galois qudit of dimension $q = 2^s$ is exactly the same thing as a collection of $s$ qubits, not only in its Hilbert space, but also in its Pauli group, and Clifford hierarchy. This formalism has found a lot of utility recently in constructing quantum error-correcting codes over qubits with useful properties. In this review, we build on existing literature to collect and formalise facts and proofs about Galois qudits over binary extension fields. We define them and their Clifford hierarchies, describe what it means to measure their Pauli operators, describe their stabiliser tableaux, formally define qudit-to-qubit mappings, and finally describe quantum Reed-Solomon codes. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2605.18981 [quant-ph] (or arXiv:2605.18981v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.18981 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Adam Wills [view email] [v1] Mon, 18 May 2026 18:03:27 UTC (30 KB) Full-text links: Access Paper: View a PDF of the paper titled A Review of Galois Qudits, by Adam WillsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-05 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?)