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arXiv:2609.31541

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

Guanyu Zhu, Shi Jie Samuel Tan, Ryohei Kobayashi, Po-Shen Hsin

quant-phcond-mat.str-elcs.IThep-thmath-phSubmitted Sep 25, 2026• Announced Sep 28, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Sep 25, 2026v2 · Oct 5, 2026
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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 gauge theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian $D_4$ topological order. The corresponding twisted Clifford-stabilizer qLDPC code can be obtained from gauging a new type of 0-form sub-complex symmetry from a qLDPC code defined on a general sheaf complex, and can be interpreted as a topological defect network of non-Abelian $D_4$ patches glued together with proper gapped interfaces. As an application, one can use this to realize a magic state fountain via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product (HGP) sheaf code. This includes a scheme of subdividing an arbitrary constant-rate 2D HGP code with parameters $[[n,Θ(n), Ω(n^{1/2})]]$ into a quantum sheaf code which allows preparation of $Θ(n^{1/2})$ logical CZ magic states 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 parameters $[[n,Θ(n^{1-ε}), Ω(n^{(1-ε)/2})]]$ for arbitrary small $ε$, one can prepare $Θ(n^{1-ε})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.

Comments
68 pages of main text and 17 figures + appendices

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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 gauge theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian D4D_4D4​ topological order. The corresponding twisted Clifford-stabilizer qLDPC code can be obtained from gauging a new type of 0-form sub-complex symmetry from a qLDPC code defined on a general sheaf complex, and can be interpreted as a topological defect network of non-Abelian D4D_4D4​ patches glued together with proper gapped interfaces. As an application, one can use this to realize a magic state fountain via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product (HGP) sheaf code. This includes a scheme of subdividing an arbitrary constant-rate 2D HGP code with parameters [[n,Θ(n),Ω(n1/2)]][[n,Θ(n), Ω(n^{1/2})]][[n,Θ(n),Ω(n1/2)]] into a quantum sheaf code which allows preparation of Θ(n1/2)Θ(n^{1/2})Θ(n1/2) logical CZ magic states 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 parameters [[n,Θ(n1−ε),Ω(n(1−ε)/2)]][[n,Θ(n^{1-ε}), Ω(n^{(1-ε)/2})]][[n,Θ(n1−ε),Ω(n(1−ε)/2)]] for arbitrary small εεε, one can prepare Θ(n1−ε)Θ(n^{1-ε})Θ(n1−ε) CZ magic states in parallel and hence achieve an almost-constant magic rate.
A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical non-Clifford gates, one needs to resort to 3D product-code constructions. In this work, we extend Kitaev's framework of non-Abelian topological codes on manifolds to non-Abelian qLDPC codes (realized as Clifford-stabilizer codes) and the corresponding combinatorial topological quantum field theories (TQFT) defined on Poincaré CW complexes and certain types of general chain complexes. We also construct the spacetime path integrals as topological invariants on these complexes. Remarkably, we show that native non-Clifford logical gates can be realized using constant-rate 2D hypergraph-product codes and their Clifford-stabilizer variants. This is achieved by a spacetime path integral effectively implementing the addressable gauging measurement of a new type of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford gates and become higher-form symmetries when lifted to higher-dimensional CW complexes or manifolds. Building on this structure, we apply the gauging protocol to the magic state fountain scheme for parallel preparation of O(n)O(\sqrt{n})O(n​) disjoint CZ magic states with code distance of O(n)O(\sqrt{n})O(n​), using a total number of nnn qubits.
Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to establish code families with constant encoding rate and linear distance, simultaneously achieving optimal asymptotic scaling in both parameters. In particular, we characterize explicit logical representatives of the full gauged code space, and provide a fundamental treatment of non-Abelian code distance in the sense of Knill--Laflamme based on combining expansion with cleaning to establish protection against arbitrary errors. We further construct an almost-good family with entire code spaces exhibiting long-range magic. Moreover, we use linearly addressable cup product to directly generate linearly many encoded magic states from gauging measurements of Clifford operators. These results extend good qLDPC codes beyond the Pauli stabilizer setting, lay the foundation for pursuing the no low-energy trivial magic conjecture and shed light on non-Abelian quantum phases and orders beyond geometric locality.
We establish a systematic mathematical framework for constructing and computing (co)homological invariants that induce logical gates on quantum low-density parity-check (qLDPC) codes. By synthesizing tools from graph theory, algebraic topology, and number theory, we study constructions ranging from hypergraph product (HGP) codes to sheaf codes. Using commutative algebra, we generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a longstanding challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, {we establish a nearly linear family of independent cup product cochains on almost-good qLDPC codes. Under a separate nonvanishing pairing condition, they are expected to yield parallel logical multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes, we construct interlaced families through odd-degree lifts that preserve compatible invariant pairings, allowing nontrivial, constant-depth logical gates throughout an infinite family to be certified on a constant-size HGP seed. Treating local code constructions and expansion bounds as external inputs, our framework identifies the compatibility conditions for combining large dimension and distance, local testability, and constant-depth non-Clifford gates, thereby provides a mathematical foundation for separate constructions that realize these properties simultaneously.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for nnn-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of nnn-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)(n+1)(n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23\mathbb{Z}_2^3Z23​ gauge theory (equivalent to D4\mathbb{D}_4D4​ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d)O(d)O(d) rounds via code switching. In 3D, we construct a transversal logical T\sqrt{\text{T}}T​ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24\mathbb{Z}_2^4Z24​ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)(n+1)(n+1)-th level of Clifford hierarchy in nnn spatial dimension.
A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the N1/3N^{1/3}N1/3 distance barrier
We develop a topological theory for fault-tolerant quantum computation in quantum low-density parity-check (qLDPC) codes. We show that there exist hidden simplicial or CW complex structures encoding the topological data for all qLDPC and CSS codes obtained from product construction by generalizing the Freedman-Hastings code-to-manifold mapping. This is achieved by building manifolds from the Tanner graphs of the skeleton classical or quantum codes, which further form a product manifold and an associated thickened product code defined on its triangulation. One can further deformation retract the manifold back to a CW complex which supports a non-topological code with minimal overhead suitable for near-term implementation. Both types of codes admit cohomology operations including cup product which can induce non-Clifford gates. When applying this mapping to a 3D hypergraph product code obtained from the product of 3 copies of good classical expander codes, we obtain non-Clifford logical CCZ gates via constant depth circuits on a code with constant stabilizer weight w=O(1)w=O(1)w=O(1), constant rate K=Θ(N)K=Θ(N)K=Θ(N), and polynomial distance D=Ω(N1/3)D=Ω(N^{1/3})D=Ω(N1/3). When applied to logical CCZ on 3D homological product codes consisting of the product of a pair of good quantum and classical LDPC codes, we can further improve the distance to D=Ω(N)D=Ω(\sqrt{N})D=Ω(N​) exceeding the N1/3N^{1/3}N1/3 distance barrier implied by the Bravyi-König bound for conventional topological codes with the aid of non-Euclidean geometries. Our work suggests that it is feasible to apply native logical non-Clifford gates on qLDPC codes or directly inject high-fidelity magic states as resources ('magic state fountain') without the distillation process. For the homological product construction, the fountain can inject Θ(N)Θ(\sqrt{N})Θ(N​) magic states in parallel in a single round.
Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good qLDPC codes with profound underlying mathematics. In this work, we generalize Poincaré duality from manifolds to both classical and quantum codes defined via sheaf theory on ttt-dimensional cell complexes. Viewing important code properties including the encoding rate, code distance, local testability soundness, and efficient decoders as parameters of the underlying (co)chain complexes, we rigorously prove a duality relationship between the iii-th chain and the (t−i)(t-i)(t−i)-th cochain of sheaf codes. We further build multiplicative structures such as cup and cap products on sheaved chain complexes, inspired by the standard notions of multiplicative structures and Poincaré duality on manifolds. This immediately leads to an explicit isomorphism between (co)homology groups of sheaf codes via a cap product. As an application, we obtain transversal disjoint logical CZ\mathrm{C}ZCZ gates with kCZ=Θ(n)k_{\mathrm{C}Z}=Θ(n)kCZ​=Θ(n) on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of CCZ\mathrm{C}\mathrm{C}ZCCZ gates as well as for higher order controlled-ZZZ that are provably logical operations on the code space. We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes. Mathematically, our results are built on establishing the equivalence between sheaf cohomology in the derived-functor sense, Čech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.