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 $\mathbf H\in\mathbb F_2^{r\times n}$ such that $\mathbf H\mathbf H^\top = \mathbf 0$. 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 $\mathbf H$ 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 $\mathbf H$, 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.

翻译：本文提出一种高效算法，用于采样随机稀疏矩阵以作为量子低密度奇偶校验（LDPC）码的校验矩阵。为简化处理，我们主要将该算法描述为一种对偶包含二进制LDPC码的采样技术，即采样满足 $\mathbf H\mathbf H^\top = \mathbf 0$ 的稀疏矩阵 $\mathbf H\in\mathbb F_2^{r\times n}$。然而如我们所示，该算法可轻松推广至非二进制有限域上的对偶包含LDPC码以及更广义的量子稳定子LDPC码的采样。尽管已有若干构造方法存在，但它们均具有某种代数特性（例如要求矩阵具有准循环结构）。相比之下，我们的算法是纯粹组合式的，仅要求 $\mathbf H$ 的行向量具有足够稀疏性。在此意义上，可将本算法视为对尽可能随机的稀疏自正交矩阵进行采样的方法。该算法在概念上极为简洁，其核心是使用信息集解码（ISD）技术逐行采样 $\mathbf H$ 的行向量。ISD的运用具有根本性意义——若缺少该技术，高效采样将无法实现。我们对算法进行了理论表征，确定了可采样的参数范围及其预期计算复杂度。数值模拟与基准测试验证了本方法的可行性与高效性。