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

2026-09-25

AI Summary

This study addresses the critical bottlenecks in fault-tolerant quantum computing: constructing quantum low-density parity-check (qLDPC) codes with non-Abelian topological order and enabling efficient magic state preparation. To this end, it unifies Sipser-Spielman codes with Dijkgraaf-Witten theory to propose a twisted gauge framework based on sheaf cohomology. By defining zero-form subcomplex symmetries and non-Abelian patch-gluing models combined with hypergraph product code subdivision techniques, this work transcends conventional Abelian topological constraints to construct qLDPC codes exhibiting D4 topological order. Consequently, it achieves parallel magic state fountain distillation scaling from Ω(n^{1/2}) to Ω(n^{1−ε}), attaining near-constant overhead while preserving high encoding rates, thereby substantially reducing the resource requirements for fault-tolerant quantum computation.

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 $\mathbb{Z}_2^3$ 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 $[[n,Θ(n), Ω(n^{1/2})]]$ into a quantum sheaf code which allows preparation of $Θ(n^{1/2})$ 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 $[[n,Θ(n^{1-ε}), Ω(n^{(1-ε)/2})]]$ for arbitrary small $ε$, one can prepare $Θ(n^{1-ε})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.

Problem

• quantum LDPC codes
• non-Abelian topological order
• magic state distillation
• sheaf cohomology
• Dijkgraaf-Witten theory

Innovation

• non-Abelian qLDPC codes
• twisted sheaf gauge theory
• magic state fountain
• Dijkgraaf-Witten theory
• topological defect network