Quantum error correction with the color-Gottesman-Kitaev-Preskill code

The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP only protects against small shift errors in $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ and \^{}\fi{}}{q}$ quadratures, it necessary to concatenate with a stabilizer for larger error correction. In this paper, we consider concatenation single-mode two-dimensional (2D) color (color-GKP code) on square-octagon lattice. We use Steane-type scheme maximum-likelihood estimation correction show its advantage concatenation. our main minimum-weight perfect matching algorithm applied decode color-GKP Complemented continuous-variable information from code, threshold 2D improved. If data qubits are noisy, reaches $\ensuremath{\sigma}\ensuremath{\approx}0.59(\overline{p}\ensuremath{\approx}13.3%)$ compared $\overline{p}=10.2%$ normal measurements also introduce generalized restriction decoder three-dimensional space-time graph decoding. $\ensuremath{\sigma}\ensuremath{\approx}0.46$ when noiseless, $\ensuremath{\sigma}\ensuremath{\approx}0.24$ all noisy. Lastly, good performance shown giving at $3.1%$ under phenomenological model.