Computing with qLDPC Codes by Climbing the Chain Map Hierarchy

Quantum Physics arXiv:2609.02999 (quant-ph) [Submitted on 2 Sep 2026] Title:Computing with qLDPC Codes by Climbing the Chain Map Hierarchy Authors:Rahul Sahay, David M. Long, Vedika Khemani View a PDF of the paper titled Computing with qLDPC Codes by Climbing the Chain Map Hierarchy, by Rahul Sahay and 2 other authors View PDF HTML (experimental) Abstract:We develop a framework for logical computation with qLDPC codes that places logical Pauli, Clifford, and non-Clifford operations on the same footing. This brings the simple homological description of Pauli logicals to the patchwork landscape of logical Clifford and non-Clifford operations, providing a tool for the discovery of new logical gates. In particular, we define the chain map hierarchy: a family of chain complexes whose homology classes encode logical unitary and code surgery operations at any level of the Clifford hierarchy, precisely analogous to the chain complex description of Pauli logicals. Consequently, intuition for Pauli logicals can be leveraged to discover new logical operations on qLDPC codes. For instance, the familiar ability to deform Pauli logicals with stabilizers---i.e. boundaries of the chain complex---becomes a way to search for constant-depth unitary implementations of (non-)Clifford logical gates. Using this strategy, we discover constant-depth unitary implementations of the full logical Clifford group on any number of blocks of the 2D toric code, including within a single block, and addressable logical CCZ gates on any triple of logical qubits on any number of blocks of the 3D toric code. Beyond manifold codes, we find addressable non-Clifford gates on codes with many encoded qubits. The chain map hierarchy naturally encompasses and extends other constructions of logical gadgets, for instance providing a universal parameterization of cup products. As such, our work provides a unified, useful, and intuitive language for computing with qLDPC codes. Comments: Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el) Cite as: arXiv:2609.02999 [quant-ph] (or arXiv:2609.02999v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.02999 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Rahul Sahay [view email] [v1] Wed, 2 Sep 2026 18:00:00 UTC (789 KB) Full-text links: Access Paper: View a PDF of the paper titled Computing with qLDPC Codes by Climbing the Chain Map Hierarchy, by Rahul Sahay and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 Change to browse by: cond-mat cond-mat.str-el 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?)