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.
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.
We exhibit nontrivial transversal logical multi-controlled-Z gates on [[N,Θ(N),Θ~(N)]] quantum low-density parity-check (qLDPC) codes with soundness 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-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.
We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-Z gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-Z gate parallelizability. More precisely, on input a good quantum AG code over the alphabet Fq with transversal CmZ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet Fq2 supporting a transversal CmZ gate as well as addressable and parallelizable Cm−1Z gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.
We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical gates on any triplet of logical qubits with their logical- X membranes having a Z 2 triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the Z 2 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate k / n = O ( 1 / log ( n ) ) and O ( log ( n ) ) distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with O ( 1 / log 1 2 ( n ) ) rate and O ( log 1 2 ( n ) ) distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- X measurements.
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1Z gate, for every constant r≥3. In particular, we obtain length-n quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/r supporting transversal Cr−1Z gates on a close-to-linear number k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n) logical Cr−1Z gates, though with stabilizer weight and physical circuit depth nε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits.
To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
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 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 CZ gates with 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 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.