Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.
Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.
Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. This problem is important in practice, since low-weight measurements are necessary for reliable implementations of quantum error correction on physical hardware. It is also important theoretically, as it can be used to construct constant-locality versions of quantum locally testable codes, which may be relevant to the quantum PCP conjecture. We give a streamlined geometric procedure for quantum weight reduction based entirely on coning and treating X and Z checks symmetrically, which simplifies Hastings' previous approach. In particular, given an arbitrary [[n,k,d]] quantum code with weight w, our method produces a code with parameters [[O(nw2logw),k,Ω(dw)]], check weight 5, and qubit weight 6; these bounds are optimal or close to optimal within the current geometric framework. Applied to random dense CSS codes, our procedure yields explicit quantum codes surpassing the square-root distance barrier, with parameters [[n,Ω~(n1/3),Ω~(n2/3)]]. These codes also admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound, recovering the layer-code result. Our weight reduction technique also improves fault-tolerant logical-operator measurements by reducing the required number of ancilla qubits.
Quantum low density parity check (LDPC) codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any Calderbank-Shor-Steane code with a degree-δ Tanner graph, we design stabilizer measurement circuits with depth at most (2δ+2) using at most ⌈δ/2⌉ layers. We observe a circuit-noise threshold of 0.28% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of 10^{-4}, this family reaches a logical error rate of 10^{-15} using fourteen times fewer physical qubits than the surface code.
Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.
Realizing practical quantum computers requires quantum error correction, but the most widely implemented approach, the planar surface code, demands a substantial physical qubit overhead. Here, we demonstrate that partitioning quantum processors into manageable, flat modules with non-local inter-module connections provides a natural solution. By wiring sparse, static boundary connections between Euclidean planar modules, we construct modular hyperbolic surface and color codes, based on new families of semi-hyperbolic codes. Using modular syndrome extraction circuits and efficient neural network and matching decoders, our circuit-level simulations show that this modular memory design can reduce the physical qubit overhead by tenfold or more compared to the surface code, even under elevated inter-module seam error rates. Projecting to larger system sizes, we find modular codes with encoding rate k/n=1/16 and distance d≥22, exceeding both the rate and distance of the [[288,12,18]] two-gross bivariate bicycle code while using mostly nearest-neighbor gates within planar modules, and implying an over 30× overhead reduction relative to surface codes. We develop fault-tolerant walking circuits for implementing logic via code automorphisms, which we co-design with low-weight, highly symmetrical logical bases and a modular extractor system that is ≈4.5× smaller than the code itself. Combining these methods, our modular memory can be used as dense storage in a universal fault-tolerant architecture, unlocking the efficiency of high-rate codes without sacrificing the fabrication advantages of planar modules.
It is widely accepted that quantum error correction is essential for realizing large-scale fault-tolerant quantum computing. Recent experiments have demonstrated error correction codes operating below threshold, primarily using local planar codes such as the surface code and color code. In parallel, theoretical advances in quantum low-density parity-check (LDPC) codes promise significantly lower overheads, albeit at the cost of requiring non-local parity checks. While these results are encouraging, implementing such codes remains challenging for near-term experiments, creating obstacles to holistic benchmarking of hardware architectures capable of supporting long-range couplers. In this work, we present a simple construction recipe for small quantum LDPC codes based on recent developments in the field. Our codes are approximately twice as efficient as comparable surface codes, yet require only weight-four parity checks, which simplifies experimental realization compared to other quantum LDPC codes. We provide concrete proposals for implementations with superconducting qubits in flip-chip architectures and with semiconductor spin qubits using shuttling-based approaches.
Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.