A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support transversal non-Clifford gates. In this work, we provide a large family of such codes. The key insight is to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation. At a more abstract level, this construction is based on defining the cup product on the chain complex induced from a sheaf.
A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support transversal non-Clifford gates. In this work, we provide a large family of such codes. The key insight is to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation. At a more abstract level, this construction is based on defining the cup product on the chain complex induced from a sheaf.
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.
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.
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.
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.
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) disjoint CZ magic states with code distance of O(n), using a total number of n qubits.