  An outstanding challenge for quantum information processing using bosonic
systems is Gaussian errors such as excitation loss and added thermal noise
errors. Thus, bosonic quantum error correction (QEC) is essential. Most bosonic
QEC schemes encode a finite-dimensional logical qubit or qudit into noisy
bosonic oscillator modes. In this case, however, the infinite-dimensional
bosonic nature of the physical system is lost at the error-corrected logical
level. On the other hand, there are several proposals for encoding an
oscillator mode into many noisy oscillator modes. However, these
oscillator-into-oscillators encoding schemes are in the class of Gaussian
quantum error correction. Therefore, these codes cannot correct practically
relevant Gaussian errors due to the established no-go theorems which state that
Gaussian errors cannot be corrected by using only Gaussian resources. Here, we
circumvent these no-go results and show that it is possible to correct Gaussian
errors by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian
resources. In particular, we propose a non-Gaussian oscillator-into-oscillators
code, the two-mode GKP-repetition code, and demonstrate that it can correct
additive Gaussian noise errors. In addition, we generalize the two-mode
GKP-repetition code to an even broader class of non-Gaussian oscillator codes,
namely, GKP-stabilizer codes. Specifically, we show that there exists a highly
hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code, that
can quadratically suppress additive Gaussian noise errors in both the position
and momentum quadratures up to a small logarithmic correction. Moreover, for
any GKP-stabilizer code, we show that logical Gaussian operations can be
readily implemented by using only physical Gaussian operations. We also show
that our oscillator codes can correct practically relevant excitation loss and
thermal noise errors.
