Biased Gottesman-Kitaev-Preskill repetition code

Continuous-variable quantum computing architectures based upon the Gottesman-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 generalizing 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 stabilizer measurements and have complex decoding requirements to overcome. In this work, we study the code-capacity behavior of a rectangular-lattice GKP encoding concatenated with a repetition code under an isotropic Gaussian displacement channel. We find a numerical threshold of s = 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 stabilizer operators and simple decoding at the qubit level. Furthermore, with moderate levels of bias (aspect ratio similar to 2.4) and nine or fewer data modes, significant reductions in logical error rates can still be achieved for s similar to 0.3, opening the possibility of using biased GKP repetition codes as a simple low-level qubit encoding for further concatenation.