Continuous-variable error correction for general Gaussian noises*

Quantum error correction mainly concerns the protection of quantum information encoded in discrete-variable systems modelled as qubits. However, many applications, e.g. quantum sensing and communication, require continuous-variable systems modelled as oscillators. Recently, Gottesman-Kitaev-Preskill (GKP) states are shown to enable the protection of continuous-variable systems, via encoding a single oscillator into multiple oscillators. Here we extend the study to multiple oscillators under independent but inhomogeneous Gaussian noises. To get the minimum reduced noise, we optimize over the code structure in terms of the ordering of the multiple oscillators. Asymptotic analyses and numerical simulations show that both the GKP-two-mode-squeezing code and GKP-squeezing-repetition code can reduce the noise standard deviation to the order of the geometric mean of the standard deviations of the heterogeneous noises, under global optimization of the codes. Our approach applies to general correlated Gaussian noises, which can be unraveled to independent Gaussian noises through Gaussian operations. As an example, we apply our strategies to memory channels.