On the Minimum Distances of Finite-Length Lifted Product Quantum LDPC Codes
Nithin Raveendran, David Declercq, Bane Vasić
TL;DR
This work addresses the practical, finite-length design of degenerate LP-QLDPC quantum codes by deriving combinatorial constraints on type-I quasi-cyclic base matrices to guarantee a quantum minimum distance $d_{\min}^{\mathcal{Q}}$ exceeding the stabilizer weight $w_{\text{stab}}$. The authors formulate generalized row/column partition constraints (RCPC) that ensure $d_{\min}^{\mathcal{Q}}=\min(d_{\min}^{\mathcal{C}},\,m+n)$ for LP-QLDPC codes constructed from base matrices, and demonstrate how degeneracy (i.e., $d_{\min}^{\mathcal{Q}} > w_{\text{stab}}$) improves decoding performance under belief-propagation with order-statistic decoding. Through concrete examples and exhaustive base-matrix searches, they show that many RCPC-satisfying configurations exist and that larger circulant sizes and careful base-code choices can yield degenerate codes with favorable minimum distances. The findings offer a principled design framework for practically implementable finite-length QLDPC codes with improved error-correction capabilities for near-term quantum hardware.
Abstract
Quantum error correction (QEC) is critical for practical realization of fault-tolerant quantum computing, and recently proposed families of quantum low-density parity-check (QLDPC) code are prime candidates for advanced QEC hardware architectures and implementations. This paper focuses on the finite-length QLDPC code design criteria, specifically aimed at constructing degenerate quasi-cyclic symmetric lifted-product (LP-QLDPC) codes. We describe the necessary conditions such that the designed LP-QLDPC codes are guaranteed to have a minimum distance strictly greater than the minimum weight stabilizer generators, ensuring superior error correction performance on quantum channels. The focus is on LP-QLDPC codes built from quasi-cyclic base codes belonging to the class of type-I protographs, and the necessary constraints are efficiently expressed in terms of the row and column indices of the base code. Specifically, we characterize the combinatorial constraints on the classical quasi-cyclic base matrices that guarantee construction of degenerate LP-QLDPC codes. Minimal examples and illustrations are provided to demonstrate the usefulness and effectiveness of the code construction approach. The row and column partition constraints derived in the paper simplify the design of degenerate LP-QLDPC codes and can be incorporated into existing classical and quantum code design approaches.