High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures

Abstract:We construct a quantum low-density parity-check code family from a length-$512$ Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are $(3,8)$-regular with girth $8$. The base code has parameters $[[512,174,8]]$. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor $P=32$, which has parameters $[[16384,4142,\le 32]]$, under the code-capacity depolarizing model. Explicit weight-$32$ non-stabilizer logical representatives on both the $X$ and $Z$ sides give $d_X,d_Z\le32$. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$; an independently observed logical residual of weight $40$ is consistent with the sharper structural bound.

arXiv:2604.20838 [quant-ph]

(or arXiv:2604.20838v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.20838 Focus to learn more arXiv-issued DOI via DataCite

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