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

TL;DR

This paper establishes structural conditions under which quantum LDPC codes can support a native, constant-depth $ ext{CCZ}$ magic-state fountain, linking code properties to the ability to generate non-Clifford resource states efficiently.

Contribution

It introduces a coding-theoretic framework using magic-friendly triples and hypergraph models to identify when qLDPC codes support parallel $ ext{CCZ}$ gates and magic-state fountains.

Findings

Identifies algebraic conditions for $ ext{CCZ}$ magic-state support in qLDPC codes.

Shows that large collections of magic-friendly triples enable parallel $ ext{CCZ}$ gates.

Reduces the problem to a combinatorial counting problem in asymptotically good codes.

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.