css code construction

This work addresses the pressing need for efficient quantum error-correcting codes in scalable quantum computing by proposing two families of high-rate quantum dual-containing (DC) CSS LDPC codes constructed from quasi-binary matrices. While preserving the dual-containing structure necessary for transversal Hadamard gates, these codes are optimized through careful analysis of cyclic properties, automorphism groups, and minimum distance estimates to enhance finite-length error correction performance. A low-complexity binary belief propagation decoder is employed for efficient decoding. Numerical simulations demonstrate that the proposed constructions significantly outperform existing DC codes across a range of code lengths and rates in terms of finite-length block error rate, offering a promising avenue toward practical fault-tolerant quantum computation.

This work addresses the lack of structural randomness in orthogonal sparse matrix pairs for CSS-type quantum LDPC code construction, which limits decoding performance. We propose a randomized construction method that preserves row and column weight distributions. Our approach features: (1) localized structural perturbation via 2×2 cross-swap operations, and (2) orthogonality-preserving local repair using integer linear programming. The method efficiently generates diverse code ensembles while strictly maintaining both sparsity and orthogonality; its repair complexity depends only on the maximum row/column weight—not on matrix size. Experimental results demonstrate that the resulting quantum LDPC code ensembles significantly improve belief propagation (BP) decoding performance and exhibit excellent scalability, making them suitable for large-scale fault-tolerant quantum computing systems.

This work resolves a long-standing open problem in quantum coding theory: the existence of asymptotically good binary CSS-T codes and CSS-LDPC-T codes. We introduce a general construction that transforms any CSS code into a CSS-T code supporting transversal T gates (and, more broadly, arbitrary Z-rotations). For the first time, we rigorously prove the existence of asymptotically good binary CSS-T codes and quantum LDPC-type CSS-T codes. Our construction yields an explicit family of asymptotically good CSS codes enabling transversal implementation of arbitrary $Z_ heta$ rotations. Furthermore, we systematically characterize the algebraic structure of the corresponding non-Clifford logical operators. These results unify and extend the theoretical foundations of CSS codes, quantum LDPC code design, and transversal gate implementation—establishing a new paradigm for efficient fault-tolerant non-Clifford operations in quantum computation.

This work investigates the logical error rate performance of Calderbank–Shor–Steane (CSS) quantum stabilizer codes over both symmetric and asymmetric quantum channels. We propose a novel analytical framework integrating weight enumeration with logical operator analysis, enabling—for the first time—the exact derivation of the weight distribution of undetectable errors via the quantum MacWilliams identity; this, combined with minimum-weight decoding, yields tight upper bounds on the logical error probability. Our contributions include asymptotically tight closed-form expressions for the logical error rate (e.g., ρ_L ≈ 16ρ²) and ρ⁴/ρ⁵-order upper bounds for Shor, Steane, and multi-scale surface codes (e.g., [[9,1,3]], [[85,1,7]], [[181,1,10]]). Notably, we derive an analytical logical error rate formula for surface codes under the depolarizing channel. The framework significantly improves prediction accuracy for short-length codes and extends naturally to realistic noise models incorporating measurement imperfections.

This work addresses the construction of quantum LDPC base matrices satisfying regularity, CSS orthogonality, and the absence of 4-cycles of the same type by proposing a finite-field-based dual-branch multiplicative coset method. The design is decomposed into two stages: base matrix construction and cyclic lifting. The former explicitly encodes degree distribution and girth constraints via quotient coset conditions, while the latter employs algebraic randomization to establish edge connections. The proposed framework accommodates various $(J,L)$ degree distributions, offering both flexibility and structural rigor. Experimentally, a CSS code with parameters $[[10240,4108,10\leq d\leq32]]$ is successfully constructed, achieving a frame error rate of $1.0\times10^{-7}$ at a physical error rate of $p=0.058$, exhibiting a Tanner graph girth of at least 8, and excluding weight-16 non-degenerate logical error support orbits.

This work proposes a high-rate, large-girth CSS-type quantum low-density parity-check (QLDPC) code to enhance the encoding efficiency and practicality of quantum error correction. Leveraging a (3,18)-regular bipartite finite-field base structure and a lifting technique based on 101×101 cyclic permutation matrices, the construction achieves—for the first time—a sparse QLDPC code with rate 2/3 and Tanner graph girth 8. The resulting [[34542, 23032, d ≤ 310]] quantum code demonstrates exceptional performance: under 10⁸ decoding trials at physical error rate p = 0.01, no decoding failures occur, and the frame error rate threshold is estimated at p ≈ 0.029, significantly outperforming existing comparable schemes.

This work addresses the challenge in fault-tolerant quantum computing of simultaneously achieving high error-correction performance and low stabilizer weight. By leveraging low-density generator matrix (LDGM) codes and the Calderbank–Shor–Steane (CSS) construction, the authors design a new class of quantum error-correcting codes. Through flexible row operations, the code rate is efficiently tuned, while message-passing iterative decoding on graphs—combined with discrete density evolution analysis—enables significantly reduced stabilizer generator weights without compromising error-correction capability. The resulting quantum codes exhibit outstanding performance under the depolarizing channel, offering both low decoding complexity and high practicality. This approach provides an efficient and scalable coding solution for fault-tolerant quantum computation.

Quantum LDPC code design faces challenges including ambiguous logical structure, difficulty in analyzing minimum distance, and construction complexity. This work proposes univariate bicycle (UB) codes—a structured subclass of generalized bicycle codes—that leverage Frobenius relations to reduce the design space from bivariate to univariate polynomial search. For the first time, the algebraic structure of their logical co-set space is explicitly characterized. By establishing a connection between logical representatives and cycle densities of circulant matrices, an upper bound on the minimum distance is derived. In the short-to-moderate blocklength regime—ranging from several hundred to approximately $10^3$—UB codes either outperform or match the performance of existing generalized and bivariate bicycle codes, demonstrating strong competitiveness while preserving stringent algebraic constraints.

This work proposes a novel framework for constructing quantum low-density parity-check (LDPC) codes that overcomes the limitations of traditional two-block group algebra (2BGA) constructions. By replacing regular group actions with group actions on cosets of subgroups, the approach substantially expands the achievable parameter space. Integrating graph cover theory with algebraic coding techniques, the method yields new high-performance codes such as [[48,8,6]] and [[96,8,10]]. Accompanying these constructions are a maximum-filling check extraction schedule of depth w+2 and a BP-OSD decoding algorithm. Under a circuit-level noise model, the resulting codes achieve fault-tolerance thresholds of approximately 0.65% for weight-6 checks and 0.35% for weight-8 checks, rivaling the performance of state-of-the-art balanced product (BP) codes.