Biased Gottesman-Kitaev-Preskill repetition code

Abstract: Continuous-variable quantum computing architectures based upon the Gottesmann-Kitaev-Preskill (GKP) encoding have emerged as a promising candidate because one can achieve fault-tolerance with a probabilistic supply of GKP states and Gaussian operations. Furthermore, by generalising to rectangular-lattice GKP states, a bias can be introduced and exploited through concatenation with qubit codes that show improved performance under biasing. However, these codes (such as the XZZX surface code) still require weight-four stabiliser measurements and have complex decoding requirements to overcome. In this work, we study the code-capacity behaviour of a rectangular-lattice GKP encoding concatenated with a repetition code under an isotropic Gaussian displacement channel. We find a numerical threshold of $\sigma = 0.599$ for the noise's standard deviation, which outperforms the biased GKP planar surface code with a trade-off of increased biasing at the GKP level. This is all achieved with only weight-two stabiliser operators and simple decoding at the qubit level. Furthermore, with moderate levels of bias (aspect ratio $\leq 2.4$) and nine or fewer data modes, significant reductions in logical error rates can still be achieved for $\sigma \leq 0.3$, opening the possibility of using GKP-biased repetition codes as a simple low-level qubit encoding for further concatenation.