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.

24 pages, 9+2 figures