Continuous-variable quantum error correction I: code comparison

There are currently four types of non-trivial encodings of quantum information in a single bosonic mode: cat, binomial, and numerically optimized codes are designed to protect against bosonic loss errors, while GKP codes are designed to protect against bosonic displacement errors. These four code types have yet to be compared using the same error model. We report on a numerical comparison of the entanglement fidelity of all codes with respect to the lossy bosonic channel, given an average occupation number constraint and the optimal recovery operation. GKP codes demonstrate the highest fidelities for all but the smallest values of the boson loss probability (the parameter which quantifies the strength of amplitude damping). Although designed to protect against small displacement noise, GKP codes can offer a high degree of protection against bosonic loss errors. We also examine the performance of the four code types with respect to the combination of amplitude damping and a strong Kerr non-linearity.