Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

Quantum Physics [Submitted on 2 Apr 2026] Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes Yiming Li, Zimu Li, Zi-Wen Liu We exhibit nontrivial transversal logical multi-controlled-Z gates on [\![N,\Theta(N),\tilde\Theta(N)]\!] quantum low-density parity-check codes and [\![N,\Theta(N),\tilde\Theta(N)]\!] quantum locally testable codes with soundness \tilde\Theta(1), combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-Z and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples. Comments: 30 pages Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph) Cite as: arXiv:2604.01874 [quant-ph]   (or arXiv:2604.01874v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.01874 Focus to learn more Submission history From: Yiming Li [view email] [v1] Thu, 2 Apr 2026 10:30:00 UTC (59 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.IT math math-ph math.IT math.MP 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?)