Spectral and Additive Combinatorial Methods for Cycles and Absorbing Sets in Lifted-Product Quantum LDPC Codes

Abstract

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 $H_X=[{\tilde{H}}_1\mid I\otimes {\tilde{B}}^T]$ 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.

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