Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding

Determining the quantum capacity of a noisy quantum channel is an important problem in the field of quantum communication theory. In this work, we consider the Gaussian random displacement channel $\mathcal{N}{\sigma}$, and attempt to make progress on determining its quantum capacity by analyzing the error-correction performance of several families of multi-mode Gottesman-Kitaev-Preskill (GKP) codes. In doing so we analyze the surface-square GKP codes using an efficient and exact maximum likelihood decoder (MLD) up to a large code distance of d=39. We find that the error threshold of the surface-square GKP code is remarkably close to $\sigma=1/\sqrt{e}\simeq 0.6065$ at which the best-known lower bound of the quantum capacity of $\mathcal{N}{\sigma}$ vanishes. We also analyze the performance of color-hexagonal GKP codes up to a code distance of d=13 using a tensor-network decoder serving as an approximate MLD. Our work shows that GKP codes can achieve non-zero quantum state transmission rates for a Gaussian random displacement channel $\mathcal{N}_{\sigma}$ at larger values of $\sigma$ than previously demonstrated, which reduces the gap between the quantum communication theoretic bounds and the performance of explicit GKP codes.