  Quantum error correction is essential for robust quantum information
processing with noisy devices. As bosonic quantum systems play a crucial role
in quantum sensing, communication, and computation, it is important to design
error correction codes suitable for these systems against various different
types of noises. While most efforts aim at protecting qubits encoded into the
infinite dimensional Hilbert space of a bosonic mode, [Phys. Rev. Lett. 125,
080503 (2020)] proposed an error correction code to maintain the
infinite-dimensional-Hilbert-space nature of bosonic systems by encoding a
single bosonic mode into multiple bosonic modes. Enabled by
Gottesman-Kitaev-Preskill states as ancilla, the code overcomes the no-go
theorem of Gaussian error correction. In this work, we generalize the error
correction code to the scenario with general correlated and heterogeneous
Gaussian noises, including memory effects. We introduce Gaussian pre-processing
and post-processing to convert the general noise model to an independent but
heterogeneous collection of additive white Gaussian noise channels and then
apply concatenated codes in an optimized manner. To evaluate the performance,
we develop a theory framework to enable the efficient calculation of the noise
standard deviation after the error correction, despite the non-Gaussian nature
of the codes. Our code provides the optimal scaling of the residue noise
standard deviation with the number of modes and can be widely applied to
distributed sensor-networks, network communication and composite quantum memory
systems.
