Theory of (Co)homological Invariants on Quantum LDPC Codes
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Quantum Physics arXiv:2603.25831 (quant-ph) [Submitted on 26 Mar 2026] Title:Theory of (Co)homological Invariants on Quantum LDPC Codes Authors:Zimu Li, Yuguo Shao, Fuchuan Wei, Yiming Li, Zi-Wen Liu View a PDF of the paper titled Theory of (Co)homological Invariants on Quantum LDPC Codes, by Zimu Li and 4 other authors View PDF Abstract:With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that $\tilde{\Theta}(N)$ independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics. Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph) Cite as: arXiv:2603.25831 [quant-ph] (or arXiv:2603.25831v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.25831 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Zimu Li [view email] [v1] Thu, 26 Mar 2026 18:50:36 UTC (101 KB) Full-text links: Access Paper: View a PDF of the paper titled Theory of (Co)homological Invariants on Quantum LDPC Codes, by Zimu Li and 4 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: cs cs.IT math math-ph math.IT math.MP 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)