Heuristic Search for Minimum-Distance Upper-Bound Witnesses in Quantum APM-LDPC Codes

2026-04-16

AI Summary

This work addresses the challenge of precisely characterizing the minimum distance of quantum APM-LDPC codes by proposing an efficient method to construct tight, certifiable upper bounds. The approach constructs low-weight non-stabilizer logical operators and verifies that they belong to the dual code’s kernel yet lie outside the row space of the stabilizer matrix, thereby yielding rigorous upper bounds. Innovatively integrating multiple witness-construction strategies, the method introduces a block-compression criterion enabling exact verification in certain cases, and combines heuristic search, kernel and row-space exclusion tests, fiber selection, CRT strips, 8-cycle trapping sets, and decoding failure residual analysis. Evaluated on numerous APM-LDPC code instances, the resulting bounds significantly outperform existing results across a broad parameter range and are formally certified.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• quantum LDPC codes
• minimum distance
• upper-bound witnesses
• CS-S codes
• APM-LDPC

Innovation

• quantum LDPC codes
• minimum distance upper bound
• heuristic search
• non-stabilizer logical operators
• certified witnesses