AI 中文总结
研究提升积量子LDPC码中循环和吸收集相关问题，利用量子特定谱框架，通过对$H_X H_X^T$进行DFT块对角化，推导出广义双循环码的闭式4循环计数等结果，为该领域研究提供新方法和结论。
AI 中文摘要
量子低密度奇偶校验（LDPC）码在迭代解码下的有限长度性能由Tanner图的小子结构决定，主要是短循环和吸收集。对于准循环码，经典理论通过离散傅里叶变换（DFT）方法得到了很好的发展，但这些工具不能直接处理主导当前量子LDPC构造的提升积（准循环广义超图积，QC-GHP）码的两块张量结构$H_X = [\,\widetilde{H}_1 \mid I \otimes \widetilde{B}^T\,]$。本文开发了一个利用这种结构的量子特定谱框架。其核心是对$H_X H_X^T$进行DFT块对角化，将矩迹和循环计算从一个$(r_1\ell)\times(r_1\ell)$矩阵简化为$\ell$个小的$r_1\times r_1$厄米矩阵的和，第二个块仅作为标量移位进入。从这个结果出发，我们通过加法能量推导出广义双循环码的闭式4循环计数，基于Fossorier经典准则的围长6的联合Sidon特征，通过Wang-Dolecek-Wesel三角形双射得到列重为3的码中(3,3)基本吸收集数量的傅里叶表达式，以及使用扩展器混合引理得到停止集大小的下界。
英文摘要
The finite-length performance of quantum low-density parity-check (LDPC) codes under iterative decoding is governed by small substructures of the Tanner graph, principally short cycles and absorbing sets. While the classical theory of these substructures for quasi-cyclic codes is well developed through discrete Fourier transform (DFT) methods, these tools do not directly address the two-block tensor structure $H_X = [\,\widetilde{H}_1 \mid I \otimes \widetilde{B}^T\,]$ of the lifted-product (quasi-cyclic generalised hypergraph product, QC-GHP) codes that dominate current quantum LDPC constructions. In this paper we develop a quantum-specific spectral framework that exploits this structure. At its core is a DFT block-diagonalisation of $H_X H_X^T$ that reduces moment-trace and cycle computations from an $(r_1\ell)\times(r_1\ell)$ matrix to a sum of $\ell$ small $r_1\times r_1$ Hermitian matrices, with the second block entering only as a scalar shift. From this result we derive a closed-form $4$-cycle count for generalised bicycle codes via additive energies, a joint Sidon characterisation of girth $6$ in the spirit of Fossorier's classical criterion, a Fourier expression for the number of $(3,3)$ elementary absorbing sets in column-weight-$3$ codes via the Wang-Dolecek-Wesel triangle bijection, and a lower bound on stopping-set sizes using the expander mixing lemma.
