Quantum secret sharing (QSS) schemes can be constructed from quantum error-correcting codes, but characterizing their access structures and implementation costs becomes challenging for finite-length, high-rate quantum low-density parity-check (qLDPC) codes. We develop an exact computational framework for QSS induced by CSS qLDPC codes. The framework characterizes authorized, forbidden, and intermediate coalitions using restricted parity-check matrices and binary linear algebra, and introduces the accessible logical dimension \(κ(A)\), defined through the symplectic rank of locally supported logical classes, to refine the characterization of intermediate coalitions. We further incorporate physical network constraints through separate Tanner-incidence, Steiner-routing, and scheduling-depth costs, and treat the partitioning of physical qubits into participants as an explicit design variable. Numerical experiments on four CSS qLDPC codes verify the predicted complement duality and monotonicity properties and show substantial variation in access transitions and network costs. For the codes studied, the bivariate bicycle instance exhibits the sharpest transition, while network embedding costs vary by more than an order of magnitude across physical topologies. We also show that participant partitioning can substantially improve block-level security and reconstruction thresholds. These results provide a practical bridge between the algebraic structure of CSS qLDPC codes, the graded accessibility of quantum secrets, and the physical constraints of networked QSS implementations.
Quantum secret sharing (QSS) schemes can be constructed from quantum error-correcting codes, but characterizing their access structures and implementation costs becomes challenging for finite-length, high-rate quantum low-density parity-check (qLDPC) codes. We develop an exact computational framework for QSS induced by CSS qLDPC codes. The framework characterizes authorized, forbidden, and intermediate coalitions using restricted parity-check matrices and binary linear algebra, and introduces the accessible logical dimension (κ(A)), defined through the symplectic rank of locally supported logical classes, to refine the characterization of intermediate coalitions. We further incorporate physical network constraints through separate Tanner-incidence, Steiner-routing, and scheduling-depth costs, and treat the partitioning of physical qubits into participants as an explicit design variable. Numerical experiments on four CSS qLDPC codes verify the predicted complement duality and monotonicity properties and show substantial variation in access transitions and network costs. For the codes studied, the bivariate bicycle instance exhibits the sharpest transition, while network embedding costs vary by more than an order of magnitude across physical topologies. We also show that participant partitioning can substantially improve block-level security and reconstruction thresholds. These results provide a practical bridge between the algebraic structure of CSS qLDPC codes, the graded accessibility of quantum secrets, and the physical constraints of networked QSS implementations.
Quantum low-density parity-check (qLDPC) codes can encode multiple logical qubits using sparse parity checks, yet searching for useful finite-length instances remains a challenging design problem because code performance must be optimized while satisfying practical constraints. Motivated by recent advances in artificial-intelligence agents for scientific discovery, we develop a multi-agent framework for discovering practical qLDPC codes. The framework combines specialist proposal and review, persistent scientific memory, long-horizon evolution of executable programs, and deterministic construction and evaluation within a closed-loop search. These programs instantiate coset-orbit balanced-product codes, providing a search space that includes bicycle and lifted-product constructions as well as non-normal subgroup actions. To incorporate practical constraints, we restrict the search to binary CSS codes with block length n≤400 and overall weight w≤10. Within this regime, the framework discovers codes with leading or competitive rate--distance performance in every weight class considered, with representative instances including [[288,16,18]] at w=7, [[288,18,18]] at w=9, and [[234,28,18]] at w=10. The search also uncovers structurally distinct, high-performing constructions, including a [[336,12,≤24]] candidate and a [[368,18,16]] code, both of which are genuine balanced-product constructions with non-normal subgroup actions. When evaluated under code-capacity depolarizing noise using a common BP-OSD decoding protocol, the discovered codes also exhibit low logical failure rates. Together, these results provide hardware-relevant finite-length candidates for further experimental evaluation and show how structured agentic search can contribute to scientific discovery.
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.
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.
Quantum low-density parity-check (qLDPC) codes are promising candidates for realizing large-scale fault-tolerant quantum computing. Although many codes with favorable theoretical parameters have been developed, their practical adoption must take hardware implementability into account. For mainstream quantum platforms such as superconductors and neutral atoms, the connectivity, the length of non-local couplings, and the complexity of wiring or atom rearrangement are key factors that dictate the difficulty of hardware realization. Here, we propose a new family of qLDPC codes, termed routing codes. Within this family, we find explicit instances whose encoding rates are competitive to those of bivariate bicycle (BB) codes, while systematically reducing qubit connectivity, shortening the length of non-local couplings, and, crucially, making all non-local couplings mutually parallel. This parallelism translates into quantifiable benefits, substantially reducing wiring crossings in superconducting multi-layer architectures and simplifying the scheduling of atom movement in neutral-atom arrays. The weight-7 routing codes maintain a high threshold of 0.7\% and reduce the physical qubit overhead by approximately a factor of 8, compared to surface codes achieving a same logical error rate. Further increasing the weight to 11 yields a [[200, 24, 14]] and [[200, 16, 17]] codes with an encoding rate over 23 times that of the surface code, at the cost of a higher logical error rate. These results establish routing codes as a hardware-centric qLDPC family that bridges the gap between theoretical optimality and near-term physical feasibility.
Quantum low-density parity-check (QLDPC) codes with good parameters are promising candidates for low-overhead fault-tolerant quantum computing, but their non-local stabilizers require long-range connectivity and frequent qubit movement, introducing practical challenges. Prior work has studied the networked implementation of topological codes, where each node only holds one or a few qubits of the entire code, and demonstrated competitive performance under practical constraints such as the quality of network-provided entanglement. However, since these codes are already geometrically local, such a networked setting might not be essential. In this work, we propose and study the networked implementation of better QLDPC codes, specifically bivariate bicycle codes due to their similarity to surface codes and the controlled amount of long-range connections in their stabilizers. We begin by recreating networked surface codes in Stim, with one code qubit per node, and provide additional insights into their circuit-level noise performance. We then extend this approach to bipartitions of bivariate bicycle codes, using balanced min-cut partitioning on their combined X-Z Tanner graph to identify optimal qubit splits. For stabilizers spanning nodes, we implement teleported CNOTs and vary the Bell pair fidelity enabling these gates. Through circuit-level noise simulations with BP-OSD decoding, we provide the first insights into networked realizations of these codes and compare their performance with monolithic implementations. We conclude by outlining advantages, limitations, and future directions.
We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths N<105, the constructions introduced here span guaranteed rates from approximately 0.35% to 25.8% and certified quantum distances from 12 to 69, with stabilizer weights between 8 and 16. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate RQ and minimum distance D satisfying RQ=1−O(loglogN1),D=Ω(N1/4), and stabilizer weight O(logNloglogN). Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.