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.
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 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 topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the 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), constant rate K=Θ(N), and polynomial distance 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) exceeding the 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) magic states in parallel in a single round.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for n-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of n-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (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 gauge theory (equivalent to D4 topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d) rounds via code switching. In 3D, we construct a transversal logical 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 gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.
Quantum error correction is necessary to perform large-scale quantum computation but requires extremely large overheads in both space and time. High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. In this work, we design fast and parallelizable logical gates for qLDPC codes and demonstrate their utility for key algorithmic subroutines such as the quantum adder. Our gate gadgets utilize transversal logical s between a data qLDPC code and a suitably constructed ancilla code to perform parallel Pauli product measurements (PPMs) on the data logical qubits. For hypergraph product codes, we show that the ancilla can be constructed by simply modifying the base classical codes of the data code, achieving parallel PPMs on a subgrid of the logical qubits with a lower space-time cost than existing schemes for an important class of circuits. Generalizations to 3D and 4D homological product codes further feature fast PPMs in constant depth. While prior work on qLDPC codes has focused on individual logical gates, we initiate the study of fault-tolerant compilation with our expanded set of native qLDPC code operations, constructing algorithmic primitives for preparing k -qubit Greenberger-Horne-Zeilinger states and distilling or teleporting k magic states with O ( 1 ) space overhead in O ( 1 ) and O ( k log k ) logical cycles, respectively. We further generalize this to key algorithmic subroutines, demonstrating the efficient implementation of quantum adders using parallel operations. Our constructions are naturally compatible with reconfigurable architectures such as neutral atom arrays, paving the way to large-scale quantum computation with low space and time overheads.
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.
Transversal gates are the ideal gates in a fault-tolerant scenario; relatively easy to implement, and minimally error propagating. Their availability will maximise fault tolerant thresholds, enabling universal quantum computation in a wider range of noisy hardware. Transversal gates in quantum low density parity check (qLDPC) codes are largely unstudied, with the early results of Burton & Browne suggesting that transversal non-Clifford gates may be impossible. In this paper, we contradict this expectation with constructions for both hypergraph product codes and balanced product codes, although these first examples have weak properties. We find qLDPC codes with transversal phase gates that have a number of logical qubits that grows linearly with n, the number of physical qubits. The distance is highly asymmetric; while the distance against bit flip errors also grows (almost) linearly in n, the distance against phase flip errors is limited to O(1). Moreover, existing distance rebalancing techniques are one-sided; we show that they only preserve the transversality when rebalancing to increase the asymmetry, not decrease it. We also collate a toolbox of techniques that identify a single transversal gate from which we can individuate a large range of different transversal gates. This is critical when addressing the question of what a transversal phase gate truly means when there are many logical qubits in the system.