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arXiv:2511.04634

Random Construction of Quantum LDPC Codes

Koki Okada, Kenta Kasai

quant-phcs.ITSubmitted Nov 6, 2025• Announced Nov 7, 2025• 0 citations• Revised Aug 14, 2026 (v2)
v1 · Nov 6, 2025v2 · Aug 14, 2026
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Abstract

We propose a method for modifying orthogonal sparse matrix pairs used in CSS codes while preserving their matrix row and column weight distributions, which play a crucial role in determining the performance of belief-propagation decoding. Unlike simple row or column permutations that merely reorder existing elements, the proposed local modification introduces genuine structural randomness through small $2\times2$ cross-swap operations followed by integer-linear-program-based local repairs that restore orthogonality. By applying this procedure repeatedly in a random manner, ensembles of randomized quantum LDPC codes can be constructed. The computational complexity of each repair depends only on the maximum row and column weights and is independent of the overall matrix size, ensuring scalability to large code blocks.

Comments
v2: Corrected two entries in the displayed repair matrix Delta and added exact verification of the final 12x32 pair: ranks 12, k=8, d_X=4, d_Z=2, hence [[32,8,2]], with explicit minimum-weight representatives

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We propose a method for modifying orthogonal sparse matrix pairs used in CSS codes while preserving their matrix row and column weight distributions, which play a crucial role in determining the performance of belief-propagation decoding. Unlike simple row or column permutations that merely reorder existing elements, the proposed local modification introduces genuine structural randomness through small 2×22\times22×2 cross-swap operations followed by integer-linear-program-based local repairs that restore orthogonality. By applying this procedure repeatedly in a random manner, ensembles of randomized quantum LDPC codes can be constructed. The computational complexity of each repair depends only on the maximum row and column weights and is independent of the overall matrix size, ensuring scalability to large code blocks.
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We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.
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In this paper, we present an efficient algorithm to sample random sparse matrices to be used as check matrices for quantum Low-Density Parity-Check (LDPC) codes. To ease the treatment, we mainly describe our algorithm as a technique to sample a dual-containing binary LDPC code, hence, a sparse matrix H∈F2r×n\mathbf H\in\mathbb F_2^{r\times n}H∈F2r×n​ such that HH⊤=0\mathbf H\mathbf H^\top = \mathbf 0HH⊤=0. However, as we show, the algorithm can be easily generalized to sample dual-containing LDPC codes over non binary finite fields as well as more general quantum stabilizer LDPC codes. While several constructions already exist, all of them are somewhat algebraic as they impose some specific property (e.g., the matrix being quasi-cyclic). Instead, our algorithm is purely combinatorial as we do not require anything apart from the rows of H\mathbf HH being sparse enough. In this sense, we can think of our algorithm as a way to sample sparse, self-orthogonal matrices that are as random as possible. Our algorithm is conceptually very simple and, as a key ingredient, uses Information Set Decoding (ISD) to sample the rows of H\mathbf HH, one at a time. The use of ISD is fundamental as, without it, efficient sampling would not be feasible. We give a theoretical characterization of our algorithm, determining which ranges of parameters can be sampled as well as the expected computational complexity. Numerical simulations and benchmarks confirm the feasibility and efficiency of our approach.
We propose a method of constructing quantum LDPC codes from multiplicative groups of order p - 1, where p = 4n + 1 is a prime for some positive integer n. The proposed quantum LDPC codes of non-CSS structure are constructed from a pair of classical regular quasi-cyclic (QC) LDPC codes. We show that these classical regular QC-LDPC codes without cycles of length 4 are orthogonal with respect to the symplectic inner product. Moreover, the proposed construction method yields a large number of new quantum LDPC codes with various code lengths and rates. The performance of the proposed quantum LDPC codes over quantum depolarizing channels with an iterative belief-propagation decoding algorithm is evaluated by simulations.