The 27-qubit Counterexample to the LU-LC Conjecture is Minimal

Quantum Physics [Submitted on 26 Mar 2026] The 27-qubit Counterexample to the LU-LC Conjecture is Minimal Nathan Claudet It was once conjectured that two graph states are local unitary (LU) equivalent if and only if they are local Clifford (LC) equivalent. This so-called LU-LC conjecture was disproved in 2007, as a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered. We prove that this counterexample to the LU-LC conjecture is minimal. In other words, for graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide. This result is obtained by studying the structure of 2-local complementation, a special case of the recently introduced r-local complementation, and a generalization of the well-known local complementation. We make use of a connection with triorthogonal codes and Reed-Muller codes. Subjects: Quantum Physics (quant-ph); Discrete Mathematics (cs.DM) Cite as: arXiv:2603.25219 [quant-ph]   (or arXiv:2603.25219v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.25219 Focus to learn more Submission history From: Nathan Claudet [view email] [v1] Thu, 26 Mar 2026 09:18:10 UTC (27 KB) Access Paper: view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.DM References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)