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 $d_X,d_Z\le 32$. 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.

v3: Corrected the P=32 lift distance bound from d<=40 to d<=32 by adding explicit weight-32 X- and Z-type non-stabilizer logical representatives and rank checks; clarified that the observed weight-40 residual is decoder-derived evidence