Heuristic Search for Minimum-Distance Upper-Bound Witnesses in Quantum APM-LDPC Codes
2026-04-16 • Information Theory
Information Theory
AI summaryⓘ
The authors study a specific type of quantum error-correcting codes called Calderbank-Shor-Steane quantum LDPC codes that are built using affine permutation matrices. Instead of proving the minimum error they can detect is large, they look for special kinds of errors that are small and valid to find upper limits on this minimum distance. They create a comprehensive method to identify these small errors using various structural patterns and tests to confirm their validity. Applying their approach to example codes improves previous estimates and provides reliable upper bounds for code performance.
Quantum LDPC codesCalderbank-Shor-Steane codesAffine permutation matricesMinimum distanceTanner graphsLogical operatorsParity-check kernelStabilizer codesTrapping setsBlock compression
Authors
Kenta Kasai
Abstract
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.