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

arXiv:2609.31541 · quant-ph, cond-mat.str-el, cs.IT, hep-th, math-ph, math.IT, math.MP · Submitted 2026-09-25 · Read on arXiv

quant-ph, cond-mat.str-el, cs.IT, hep-th, math-ph, math.IT, math.MP

Submitted: 2026-09-25

Updated: 2026-10-05

Comments: 68 pages of main text and 17 figures + appendices

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Hypergraph — 56× in this paper · explained in 2 episodes
• Sheaf Codes — 27× in this paper · explained in 2 episodes
• Generalization — 15× in this paper · explained in 13 episodes
• Trace — 12× in this paper · explained in 2 episodes
• Topological Order — 9× in this paper · explained in 2 episodes
• Anyons — 7× in this paper · explained in 2 episodes
• Addressability — 5× in this paper · explained in 1 episode
• Degeneracy — 4× in this paper · explained in 3 episodes
• Hilbert Space — 4× in this paper · explained in 2 episodes
• Distillation — 3× in this paper · explained in 2 episodes
• Post-Selection — 3× in this paper · explained in 2 episodes
• Redundancy — 3× in this paper · explained in 2 episodes
• ACM Symposium on Theory of Computing — 2× in this paper
• Anyon Condensation — 2× in this paper · explained in 2 episodes
• Abundance — 1× in this paper · explained in 1 episode
• Adam — 1× in this paper · explained in 2 episodes
• Admissibility — 1× in this paper · explained in 2 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Domain Walls — 1× in this paper · explained in 2 episodes
• Electronic Colloquium on Computational Complexity — 1× in this paper
• Frustration — 1× in this paper · explained in 2 episodes
• Quantum Physics — 1× in this paper · explained in 11 episodes
• Quantum fluctuations — 1× in this paper · explained in 2 episodes
• Sparsity — 1× in this paper · explained in 4 episodes
• Topological Defects — 1× in this paper · explained in 2 episodes

Sources

• Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes
• Maximally Extendable Sheaf Codes
• Transversal non-Clifford gates for quantum LDPC codes on sheaves
• On Quantum Weight Reduction
• Good quantum LDPC codes with linear time decoder from lossless expanders
• An efficient decoder for a linear distance quantum LDPC code
• Single-shot decoding of good quantum LDPC codes
• Improved QLDPC Surgery: Logical Measurements and Bridging Codes
• Low-overhead fault-tolerant quantum computation by gauging logical operators
• Universal adapters between quantum LDPC codes
• Fast and Parallelizable Logical Computation with Homological Product Codes
• Fold-Transversal Clifford Gates for Quantum Codes
• Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
• Quantum Rainbow Codes: Achieving Linear Rate, Growing Distance and Transversal Non-Clifford Gates with Generalised Colour Codes
• Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes
• Cups and Gates I: Cohomology invariants and logical quantum operations
• Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry
• A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the $N^{1/3}$ distance barrier
• Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom
• Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes

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