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

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 qubit qLDPC family is known that simultaneously has constant rate, linear distance, bounded stabilizer weight, and a native 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 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.