  Gaussian quantum information processing with continuous-variable (CV) quantum
information carriers holds significant promise for applications in quantum
communication and quantum internet. However, applying Gaussian state
distillation and quantum error correction (QEC) faces limitations imposed by
no-go results concerning local Gaussian unitary operations and classical
communications. This paper introduces a Gaussian QEC protocol that relies
solely on local Gaussian resources. A pivotal component of our approach is CV
gate teleportation using entangled Gaussian states, which facilitates the
implementation of the partial transpose operation on a quantum channel.
Consequently, we can efficiently construct a two-mode noise-polarized channel
from two noisy Gaussian channels. Furthermore, this QEC protocol naturally
extends to a nonlocal Gaussian state distillation protocol.
