Performance analysis of GKP error correction

Quantum error correction is essential for achieving fault-tolerant quantum computing. Gottesman-Kitaev-Preskill (GKP) codes are particularly effective at correcting continuous noise, such as Gaussian noise and loss, and can significantly reduce overhead when concatenated with qubit error-correcting codes like surface codes. GKP error correction can be implemented using either a teleportation-based method, known as Knill error correction, or a quantum non-demolition-based approach, known as Steane error correction. In this work, we conduct a comprehensive performance analysis of these established GKP error correction schemes, deriving an analytical expression for the post-correction GKP squeezing and displacement errors. Our results show that there is flexibility in choosing the entangling gate used with the teleportation-based Knill approach. Furthermore, when implemented using the recently introduced qunaught states, the Knill approach not only achieves superior GKP squeezing compared to other variants but is also the simplest to realize experimentally in the optical domain.