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

• URL: http://arxiv.org/abs/2601.22489v1
• Date: Fri, 30 Jan 2026 02:59:06 GMT
• Title: Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes
• Authors: Mohammad Rowshan,
• Abstract summary: Quantum low-density parity-check (qLDPC) codes promise constant-rate, linear-distance families with bounded-weight checks.<n>No explicit emphqubit qLDPC family is known that simultaneously has constant rate, linear distance, bounded stabilizer weight, and a native emphmagic-state fountain that prepares many non-Clifford resource states in constant depth.
• Score: 5.685589351789461
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $Ω(n^{1+γ})$ 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.

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