Performance analysis of GKP error correction

TL;DR

This paper provides a comprehensive analysis of GKP error correction methods, deriving analytical expressions for errors, comparing schemes, and highlighting the advantages of the Knill approach with qunaught states for optical implementations.

Contribution

It offers the first detailed analytical comparison of GKP error correction schemes, emphasizing the practical benefits of the Knill approach with qunaught states.

Findings

Knill approach with qunaught states achieves superior GKP squeezing.

Flexibility exists in choosing entangling gates for the Knill scheme.

Knill error correction is simpler to implement experimentally in optics.

Abstract

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.