Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.25831

Theory of (Co)homological Invariants on Quantum LDPC Codes

Zimu Li, Yuguo Shao, Fuchuan Wei, Yiming Li, Zi-Wen Liu

quant-phcs.ITmath-phSubmitted Mar 26, 2026• Announced Mar 30, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Mar 26, 2026v2 · Oct 6, 2026
arXivHTMLPDF

Abstract

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.

Related papers

Poincaré Duality and Multiplicative Structures on Quantum Codes

Yiming Li, Zimu Li, Zi-Wen Liu +1 more

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 $t$-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 $i$-th chain and the $(t-i)$-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 $\mathrm{C}Z$ gates with $k_{\mathrm{C}Z}=Θ(n)$ on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of $\mathrm{C}\mathrm{C}Z$ gates as well as for higher order controlled-$Z$ 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.

quant-pharXiv:2512.21922•Dec 29, 2025•0 citations

Non-Abelian sheaf quantum LDPC codes: good and magical

Zimu Li, Fuchuan Wei, Zhengyi Han +1 more

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.

quant-pharXiv:2609.30159•Sep 25, 2026•0 citations

Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

Yiming Li, Zimu Li, Zi-Wen Liu

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check (qLDPC) codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-$Z$. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.

quant-pharXiv:2604.01874•Apr 3, 2026•0 citations

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

Guanyu Zhu

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 $w=O(1)$, constant rate $K=Θ(N)$, and polynomial distance $D=Ω(N^{1/3})$. 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 $D=Ω(\sqrt{N})$ exceeding the $N^{1/3}$ 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 $Θ(\sqrt{N})$ magic states in parallel in a single round.

arXiv:2501.19375•Jan 31, 2025•0 citations

No-go theorems for logical gates on product quantum codes

Xiaojun Fu, Han Zheng, Zimu Li +1 more

Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.

arXiv:2507.16797•Jul 22, 2025•0 citations

Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes

Yiming Li, Zimu Li, Zhengyi Han +1 more

We achieve linearly many parallelizable and addressable transversal logical multi-controlled-Z gates with asymptotically optimal parameters simultaneously on good quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to good qLTC constructions. To establish the nontriviality of the logical operation, we express the cup product pairing as a coefficient in a product of Moore determinants and prove that this coefficient is nonzero. Moreover, we use the symmetries of the covering spaces to construct large spaces of cocycles. By multiplying the initial cocycles by suitable elements of these spaces, we obtain linearly many independent pairings, each selected by a corresponding cycle constructed using Poincaré duality. This yields linearly many parallelizable and addressable multi-controlled-Z gates and enables asymptotically constant-overhead magic state distillation with good qLDPC codes.

quant-pharXiv:2609.26691•Sep 23, 2026•0 citations
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.
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.
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 exhibit nontrivial transversal logical multi-controlled-ZZZ gates on [ ⁣[N,Θ(N),Θ~(N)] ⁣][\![N,Θ(N),\tildeΘ(N)]\!][[N,Θ(N),Θ~(N)]] quantum low-density parity-check (qLDPC) codes with soundness Θ~(1)\tildeΘ(1)Θ~(1), combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-ZZZ. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.
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 error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.
We achieve linearly many parallelizable and addressable transversal logical multi-controlled-Z gates with asymptotically optimal parameters simultaneously on good quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to good qLTC constructions. To establish the nontriviality of the logical operation, we express the cup product pairing as a coefficient in a product of Moore determinants and prove that this coefficient is nonzero. Moreover, we use the symmetries of the covering spaces to construct large spaces of cocycles. By multiplying the initial cocycles by suitable elements of these spaces, we obtain linearly many independent pairings, each selected by a corresponding cycle constructed using Poincaré duality. This yields linearly many parallelizable and addressable multi-controlled-Z gates and enables asymptotically constant-overhead magic state distillation with good qLDPC codes.