Graphical Analysis of Lifted Product Code Constructions

2026-06-10

AI Summary

This study investigates the structure of X- and Z-type parity-check matrices in lifted product quantum LDPC codes and their impact on decoding performance and error floor behavior. By employing graph-theoretic analysis and Tanner graph modeling, the work rigorously establishes—for the first time—that the Tanner graphs associated with \(H_X\) and \(H_Z\) are isomorphic, and derives necessary and sufficient conditions for their connectivity. Building on absorbing set theory, the authors further derive upper and lower bounds on the size of minimal absorbing sets. These results uncover the key combinatorial structures governing error floor phenomena, thereby providing a theoretical foundation for understanding and optimizing the decoding performance of lifted product codes.

Abstract

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.

Problem

• lifted product codes
• quantum LDPC codes
• Tanner graphs
• parity-check matrices
• decoding performance

Innovation

• lifted product codes
• QLDPC
• Tanner graph isomorphism
• absorbing sets
• graph connectivity