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 $Ω(n^{1+γ})$ magic-friendly triples whose supports have bounded per-qubit participation, then there exists a constant-depth circuit of physical $\CCZ$ gates implementing $Ω(n^γ)$ 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.

翻译：量子低密度奇偶校验（qLDPC）码有望实现具有有界权重校验的恒定码率、线性距离码族，近期研究已在多种（通常为非LDPC）码上实现了横向或恒定深度的非克利福德门。然而，目前尚未发现任何显式的量子比特qLDPC码族能同时具备恒定码率、线性距离、有界稳定子权重以及可在恒定深度制备大量非克利福德资源态的本征魔术态喷泉。本文采用结构分析方法，确定了CSS qLDPC码族必然支持恒定深度$\CCZ$魔术态喷泉的编码理论条件。核心要素包括：（i）$X$型逻辑算子的魔术友好三元组的代数概念，其定义为两两正交性及控制对角$\CCZ$相位的三重交叠形式；（ii）物理$\CCZ$电路的三均匀超图模型，结合将具有有界交叠的大量此类三元组转换为有界度超图的打包引理。我们的主要定理证明：若$n$量子比特上的CSS码族允许$Ω(n^{1+γ})$个支撑集具有每量子比特有界参与度的魔术友好三元组，则存在物理$\CCZ$门的恒定深度电路，可在保持距离至常数因子的前提下并行实现$Ω(n^γ)$个逻辑$\CCZ$门。对于渐进优的qLDPC码族（如量子坦纳码），这将本征$\CCZ$魔术态喷泉的存在性约化为逻辑$X$空间中计数与分布魔术友好三元组的具体组合问题。