Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts

2026-06-25

AI Summary

This work proposes a high-rate, large-girth CSS-type quantum low-density parity-check (QLDPC) code to enhance the encoding efficiency and practicality of quantum error correction. Leveraging a (3,18)-regular bipartite finite-field base structure and a lifting technique based on 101×101 cyclic permutation matrices, the construction achieves—for the first time—a sparse QLDPC code with rate 2/3 and Tanner graph girth 8. The resulting [[34542, 23032, d ≤ 310]] quantum code demonstrates exceptional performance: under 10⁸ decoding trials at physical error rate p = 0.01, no decoding failures occur, and the frame error rate threshold is estimated at p ≈ 0.029, significantly outperforming existing comparable schemes.

Abstract

We construct a rate-$2/3$ quantum low-density parity-check (LDPC) code from a $(3,18)$-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree $P=101$. The resulting code is a Calderbank-Shor-Steane (CSS) code with parameters $[[34542,23032,d\le 310]]$. We do not regard this upper bound as an estimate of the true minimum distance; rather, $d\le310$ is the tightest upper bound currently obtained from structural lifts and decoder-produced logical errors. The construction has row weight 18 and column weight 3, and the Tanner graphs of $H_X$ and $H_Z$ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in $10^8$ trials at $p=0.01$, and a finite-length frame error rate (FER) sweep estimates the transition near $p=0.029$.

Problem

• quantum LDPC codes
• high rate
• girth
• regular code
• CSS code

Innovation

• quantum LDPC codes
• girth-8 Tanner graph
• CPM lift
• finite-field base construction
• high-rate CSS code