In many bosonic quantum error correction (QEC) schemes proposed so far, a finite-dimensional discrete-variable (DV) system (consisted of qubits) is encoded into an oscillator or into many oscillators. In such qubit(s)-into-oscillator(s) schemes, the bosonic nature of the physical oscillator modes is lost at the logical level because the error-corrected logical system is described by discrete quantum variables such as Pauli operators. Therefore, the error-corrected logical DV system is not itself tailored to continuous-variable (CV) quantum information processing tasks such as boson sampling. 

We propose a broad class of bosonic quantum error-correcting codes, GKP-stabilizer codes, that encode logical oscillator modes into physical oscillator modes. Our codes can correct Gaussian errors such as excitation losses, thermal noise and additive Gaussian noise errors. Since the bosonic nature of the physical modes is still retained at the logical level, our oscillator(s)-into-oscillators codes are tailored to CV quantum information processing tasks.     

In particular, we show that there exists a highly hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code (shown above), that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures using only two bosonic modes (one data mode and one ancilla mode).  

Our codes are the first instances of oscillator(s)-into-oscillators codes that can correct Gaussian errors. Specifically, we circumvent the established no-go theorems that state Gaussian errors cannot be corrected by Gaussian QEC schemes, by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian resources in our codes.   

The overhead cost of performing universal fault-tolerant quantum computation for large scale quantum algorithms is very high. Despite several attempts at alternative schemes, magic state distillation remains one of the most efficient schemes for simulating non-Clifford gates in a fault-tolerant way. However, since magic state distillation circuits are not fault-tolerant, all Clifford operations must be encoded in a large distance code in order to have comparable failure rates with the magic states being distilled. 

In this work, we introduce a new concept which we call redundant ancilla encoding. The latter combined with flag qubits allows for circuits to both measure stabilizer generators of some code, while also being able to measure global operators to fault-tolerantly prepare magic states, all using nearest neighbor interactions. In particular, we apply such schemes to a planar architecture of the triangular color code family. In addition to our scheme being suitable for experimental implementations, we show that for physical error rates near 10^(-4) and under a full circuit-level noise model, our scheme can produce magic states using an order of magnitude fewer qubits and space-time overhead compared to the most competitive magic state distillation schemes. Further, we can take advantage of the fault-tolerance of our circuits to produce magic states with very low logical failure rates using encoded Clifford gates with noise rates comparable to the magic states being injected. Thus, stabilizer operations are not required to be encoded in a very large distance code. Consequently, we believe our scheme to be suitable for implementing fault-tolerant universal quantum computation with hardware currently under development.

Quantum capacity of a noisy quantum channel is a fundamental quantity that quantifies the maximum number of quantum bits (per channel use) that can be transmitted faithfully through the noisy channel upon an optimal quantum error correction scheme. Gaussian thermal loss channels model energy loss and gain errors in realistic optical communication channels and microwave cavity modes. Thus, evaluation of the quantum capacity of a Gaussian thermal loss channel is of fundamental importance to the continuous-variable quantum information processing. 

The best known lower bound of the Gaussian thermal loss channel capacity was its coherent information with respect to an input single-mode thermal state, or an uncorrelated multi-mode thermal state. We found that sometimes correlated multi-mode thermal states can outperform the single-mode thermal state subject to the same average photon number constraint and thus established an improved lower bound of the Gaussian thermal loss channel capacity.  By doing so, we established the superadditivity of Gaussian thermal loss channels with respect to Gaussian input states. 

In addition to (heuristically) showing that the hexagonal-lattice GKP code is the optimal single-mode bosonic code for Gaussian thermal loss channels, we also proved that a family of multi-mode GKP codes (defined over an optimal symplectic lattice) achieves the quantum capacity of Gaussian thermal loss channels up to at most a constant number (~log_{2}e = 1.44...) of qubits per channel use.