Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes

Quantum Physics arXiv:2601.22489 (quant-ph) [Submitted on 30 Jan 2026] Title:Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes Authors:Mohammad Rowshan View a PDF of the paper titled Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes, by Mohammad Rowshan View PDF HTML (experimental) Abstract:Quantum low-density parity-check (qLDPC) codes promise constant-rate, linear-distance families with bounded-weight checks, and recent work has realized transversal or constant-depth non-Clifford gates on various (often non-LDPC) codes. However, no explicit \emph{qubit} qLDPC family is known that simultaneously has constant rate, linear distance, bounded stabilizer weight, and a native \emph{magic-state fountain} that prepares many non-Clifford resource states in constant depth. We take a structural approach and identify coding-theoretic conditions under which a CSS qLDPC family necessarily supports a constant-depth $\CCZ$ magic-state fountain. The key ingredients are: (i) an algebraic notion of \emph{magic-friendly triples} of $X$-type logical operators, defined by pairwise orthogonality and a triple-overlap form controlling diagonal $\CCZ$ phases, and (ii) a 3-uniform hypergraph model of physical $\CCZ$ circuits combined with a packing lemma that turns large collections of such triples with bounded overlaps into bounded-degree hypergraphs. Our main theorem shows that if a CSS code family on $n$ qubits admits $\Omega(n^{1+\gamma})$ magic-friendly triples whose supports have bounded per-qubit participation, then there exists a constant-depth circuit of physical $\CCZ$ gates implementing $\Omega(n^{\gamma})$ logical $\CCZ$ gates in parallel while preserving distance up to a constant factor. For asymptotically good qLDPC families such as quantum Tanner codes, this reduces the existence of a native $\CCZ$ magic-state fountain to a concrete combinatorial problem about counting and distributing magic-friendly triples in the logical $X$ space. Comments: Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT) Cite as: arXiv:2601.22489 [quant-ph] (or arXiv:2601.22489v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.22489 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Mohammad Rowshan [view email] [v1] Fri, 30 Jan 2026 02:59:06 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes, by Mohammad RowshanView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 Change to browse by: cs cs.IT math math.IT 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?)