Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates. As examples, we construct$[[{98,6,12}]]$and$[[{162, 8, 12}]]$BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block group algebra codes, which might be of independent interest.
Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates. As examples, we construct[[98,6,12]]and[[162,8,12]]BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block group algebra codes, which might be of independent interest.
Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.
Quantum low-density parity-check codes are promising candidates for quantum error correcting codes as they might offer more resource-efficient alternatives to surface code architectures. In particular, bivariate bicycle codes have recently gained attention due to their 2D-local structure, high encoding rate, and promising performance under simulation. In this work, we will explore how one can transform bivariate bicycle codes defined on lattices with periodic boundary conditions to codes with the same locality properties on a 2D lattice with open boundary conditions. For this, we introduce the concept of pruning quantum codes. We explain how pruning bivariate bicycle codes is always possible when the codes are hypergraph products of two classical cyclic codes. We also indicate that this might be possible for more general bivariate bicycle codes by constructing explicit examples. Finally, we investigate fault-tolerant quantum computation using the constructed pruned codes by describing fold-transversal gates.
Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions 6 ≤ k ≤ 13 , e.g., ⟦ 78 , 6 , 6 ⟧ , ⟦ 107 , 7 , 7 ⟧ , ⟦ 268 , 8 , 12 ⟧ , ⟦ 405 , 9 , 15 ⟧ , ⟦ 348 , 10 , 13 ⟧ , ⟦ 450 , 11 , 15 ⟧ , ⟦ 386 , 12 , 12 ⟧ , and ⟦ 362 , 13 , 11 ⟧ , all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a ⟦ 282 , 12 , 14 ⟧ code, exhibiting an efficiency metric ( k d 2 / n ) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.
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⌉ 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 CrZ 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.
Abstract Fault-tolerant quantum computation critically depends on architectures uniting high encoding rates with physical implementability. Quantum low-density parity-check (qLDPC) codes, including bivariate bicycle (BB) codes, achieve dramatic reductions in qubit overhead, yet their logical operations remain a key challenge under planar hardware constraints. Here, we introduce code craft, a framework for designing fault-tolerant logical operations on planar BB codes within a translationally invariant, two-dimensional qubit lattice. By systematically deforming codes through local modifications---stretching, cutting, and painting---we enable the manipulation of logical qubits using strictly planar operations. We establish fault tolerance through numerical optimization of code distances and show that logical operations, including controlled-NOT gates, state transfers, and Pauli measurements, can be efficiently implemented within this framework to assemble an individually addressable logical qubit network. Universal quantum computation can then be realized by coupling just one BB-code logical qubit to a surface-code block. By combining the high encoding efficiency of qLDPC codes with geometric locality, our approach offers a practical and resource-efficient path to fault-tolerant quantum computation.
Bivariate bicycle codes are promising candidates for high-threshold, low-overhead fault-tolerant quantum memories. Meanwhile, color codes are the most prominent self-dual CSS codes, supporting transversal Clifford gates that have been demonstrated experimentally. In this work, we combine these advantages and introduce a broad family of self-dual bivariate bicycle codes. These codes achieve higher encoding rates than surface and color codes while admitting transversal CNOT, Hadamard, and S gates. In particular, we enumerate weight-8 self-dual bivariate bicycle codes with up to n ⩽ 200 physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality. Representative examples include codes with parameters [[ n , k , d ]]: [[16, 4, 4]], [[40, 6, 6]], [[56, 6, 8]], [[64, 8, 8]], [[120, 8, 12]], [[152, 6, 16]], and [[160, 8, 16]].