{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-isomorphism-of-3-qubit-hadamards-and-e-8","title":"The Isomorphism of 3-Qubit Hadamards and $E_8$","arxiv_id":"2311.11918","date":"2023-11-20","proceeding":null,"authors":["J. G. Moxness"],"abstract":"This paper presents several notable properties of the matrix $\\mathbb{U}$ shown to be related to the isomorphism between $H_4$ and $E_8$. The most significant of these properties is that $\\mathbb{U}$.$\\mathbb{U}$ is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, $\\mathbb{U}$.$\\mathbb{U}$-$ (\\mathbb{U}$.$\\mathbb{U})^{-1}$ is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.","url_abs":"https://arxiv.org/abs/2311.11918v2","url_pdf":"https://arxiv.org/pdf/2311.11918v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-isomorphism-of-3-qubit-hadamards-and-e-8","repo_url":"https://github.com/jgmoxness/The_Isomorphism_of_3-Qubit_Hadamards_and_E8","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}