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

TL;DR

This paper constructs a high-girth quantum LDPC code family from affine-coset structures, demonstrating effective decoding performance with low error rates under the depolarizing model.

Contribution

It introduces a new quantum LDPC code construction with explicit girth properties and applies circulant permutation matrix lifts for improved decoding performance.

Findings

Base code has parameters [[512,174,8]] with girth 8.

CPM-lifted code with P=32 has parameters [[16384,4142,≤40]].

Decoder achieved frame error rate ~10^{-8} at p=0.085.

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 $[[16384,4142,\le 40]]$, under the code-capacity depolarizing model. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$, and one observed logical residual of weight $40$ gives a decoder-derived upper bound $d\le 40$.