Heuristic Search for Minimum-Distance Upper-Bound Witnesses in Quantum APM-LDPC Codes
For researchers working on quantum LDPC codes, this work provides practical methods to obtain certified minimum-distance upper bounds, though the results are incremental and domain-specific.
This paper develops a unified framework for constructing low-weight non-stabilizer logical representatives to obtain certified upper bounds on the minimum distance of quantum APM-LDPC codes. The methods sharpen previously reported upper bounds and provide concrete certified values across the explored parameter range.
This paper investigates certified upper bounds on the minimum distance of an explicit family of Calderbank-Shor-Steane quantum LDPC codes constructed from affine permutation matrices. All codes considered here have active Tanner graphs of girth eight. Rather than attempting to prove a general lower bound for the full code distance, we focus on constructing low-weight non-stabilizer logical representatives, which yield valid upper bounds once they are verified to lie in the opposite parity-check kernel and outside the stabilizer row space. We develop a unified framework for such witnesses arising from latent row relations, restricted-lift subspaces including block-compressed, selected-fiber, and CRT-stripe constructions, cycle- 8 elementary trapping-set structures, and decoder-failure residuals. In every case, search is used only to generate candidates; the reported bounds begin only after explicit kernel and row-space exclusion tests have been passed. For the latent part, we also identify a block-compression criterion under which the certification becomes exact. Applying these methods to representative APM-LDPC codes sharpens previously reported upper bounds and provides concrete certified values across the explored parameter range.