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arXiv:2610.05214

Cups and Gates II: Operads, twists, and addressable intra-code gates

Nikolas P. Breuckmann, Margarita Davydova, Jens Niklas Eberhardt, Lucas Ferdinand Hallwirth, Nathanan Tantivasadakarn

quant-phcond-mat.str-elmath-phmath.ATSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$, enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2π/2^k)$. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.

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63 pages, 2 figures

Related papers

Cups and Gates I: Cohomology Invariants and Logical Quantum Operations

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Abstract We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $$\Lambda $$ Λ -fold cup product that can produce a logical operator in the $$\Lambda $$ Λ -th level of the Clifford hierarchy on $$\Lambda $$ Λ copies of the same quantum code, which we call the copy-cup gate . For any desired $$\Lambda $$ Λ , we can construct several families of quantum codes that support gates in the $$\Lambda $$ Λ -th level with various asymptotic code parameters.

arXiv:2410.16250•Communications in Mathematical Physics•Apr 4, 2026•6 citations

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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.

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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.

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Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of $\lceil d/2\rceil$ for distance-$d$ codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed $T$ gate on the $[[16,4,4]]$ BB code, $T$-type rotations on the $[[64,8,8]]$ BB code, and CCZ-, CS-, and $T$-type operations on the $[[36,6,6]]$ Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields C${}^r$Z gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.

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In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of Ck−1ZC^{k-1}ZCk−1Z logical gates that we called copy-cup on kkk copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from Z/2\mathbb{Z}/2Z/2 to Z/2k\mathbb{Z}/2^kZ/2k, enabling all diagonal logical gates such as single-qubit rotations R⁡(2π/2k)\operatorname{R}(2π/2^k)R(2π/2k). Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable TTT gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from E∞E_\inftyE∞​-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.
Abstract We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a Λ\Lambda Λ Λ -fold cup product that can produce a logical operator in the Λ\Lambda Λ Λ -th level of the Clifford hierarchy on Λ\Lambda Λ Λ copies of the same quantum code, which we call the copy-cup gate . For any desired Λ\Lambda Λ Λ , we can construct several families of quantum codes that support gates in the Λ\Lambda Λ Λ -th level with various asymptotic code parameters.
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.
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.
We develop a framework for logical computation with qLDPC codes that places logical Pauli, Clifford, and non-Clifford operations on the same footing. This brings the simple homological description of Pauli logicals to the patchwork landscape of logical Clifford and non-Clifford operations, providing a tool for the discovery of new logical gates. In particular, we define the chain map hierarchy: a family of chain complexes whose homology classes encode logical unitary and code surgery operations at any level of the Clifford hierarchy, precisely analogous to the chain complex description of Pauli logicals. Consequently, intuition for Pauli logicals can be leveraged to discover new logical operations on qLDPC codes. For instance, the familiar ability to deform Pauli logicals with stabilizers---i.e. boundaries of the chain complex---becomes a way to search for constant-depth unitary implementations of (non-)Clifford logical gates. Using this strategy, we discover constant-depth unitary implementations of the full logical Clifford group on any number of blocks of the 2D toric code, including within a single block, and addressable logical CCZ gates on any triple of logical qubits on any number of blocks of the 3D toric code. Beyond manifold codes, we find addressable non-Clifford gates on codes with many encoded qubits. The chain map hierarchy naturally encompasses and extends other constructions of logical gadgets, for instance providing a universal parameterization of cup products. As such, our work provides a unified, useful, and intuitive language for computing with qLDPC codes.
We systematically construct and classify fault-tolerant logical gates implemented by constant-depth circuits for quantum codes using cohomology operations and symmetry. These logical gates are obtained from unitary operators given by symmetry-protected topological responses, which correspond to generators of group cohomology and can be expressed explicitly on the lattice using cohomology operations including cup product, Steenrod squares and new combinations of higher cup products called higher Pontryagin powers. Our study covers most types of the cohomology operations in the literature. This hence gives rise to logical Cn−1ZC^{n-1}ZCn−1Z gates in nnn copies of quantum codes via the nnn-fold cup product in the usual color code paradigm, as well as several new classes of diagonal and non-diagonal logical gates in increasing Clifford hierarchies beyond the color code paradigm, including the logical RkR_kRk​ and multi-controlled CmRkC^m R_kCmRk​ gates for codes defined in projective spaces. Implementing these gates could make it more efficient to compile specific types of quantum algorithms such as Shor's algorithm. We further extend the construction to quantum codes with boundaries, which generalizes the folding approach in color codes. We also present a formalism for addressable and parallelizable logical gates in LDPC codes via higher-form symmetries. We further construct logical Clifford gates in expander-based codes including the asymptotically good LDPC codes and hypergraph-product codes. As a byproduct, we find new topological responses of finite higher-form symmetries using higher Pontryagin powers.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.