Sipser-Spielman meets Dijkgraaf-Witten: non-Abelian qLDPC codes via twisted sheaf gauge theory and almost-constant-overhead magic state fountain

Abstract: Sipser-Spielman codes and Dijkgraaf-Witten twisted gauge theories are among the most profound ideas in the last few decades in the areas of computer science and mathematical physics respectively. This work unifies them in the same framework using the language of sheaf cohomology, and produces a new framework of twisted sheaf guage theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian $D_4$ topological order. The corresponding twisted qLDPC code can be obtained from gauging a new type of 0-form sub-complex $\mathbb{Z}_2<sup>3$ symmetry from a qLDPC code defined on a sheaf complex, and can be interpreted as a topological defect network of non-Abelain $D_4$ patches glued together with proper gapped interfaces. As an application, one can use this to realize a \textit{magic state fountain} via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product on a sheaf complex. This includes a scheme of subdividing an arbitrary constant-rate 2D hypergraph-product code with parameter $[[n,Θ(n), Ω(n<sup>{1/2})]]$ into a quantum sheaf code which allows preparation of $Θ(n<sup>{1/2})$ in parallel, equivalent to the recent geometric construction using the code-to-manifold mapping in (arXiv:2601.06736). Moreover, using the recent sheaf complex and algebraic code constructions by Golowich-Tamo-Zhu (arXiv:2609.27801) with parameter $[[n,Θ(n^{1-ε}), Ω(n<sup>{(1-ε)/2})]]$ for arbitrary small $ε$, one can prepare $Θ(n<sup>{1-ε})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.