Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.

翻译：提升积码是一类重要的量子低密度奇偶校验（QLDPC）码，因为它们是首个被证明具有渐近优良性的QLDPC码族。理解其奇偶校验矩阵$H_{\mathsf{X}}$和$H_{\mathsf{Z}}$的结构以及相关的Tanner图，对于分析其译码行为和错误平底性能至关重要。本文证明了$H_{\mathsf{X}}$和$H_{\mathsf{Z}}$的Tanner图实际上是同构的，并研究了它们的图论结构。我们建立了确保这些图连通性的条件，并给出了它们最小吸收集的界，从而为影响译码性能的组合结构提供了新的见解。