Analog Quantum Error Correction with Encoding a Qubit into an Oscillator

To implement fault-tolerant quantum computation with continuous variables, Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important technological element. However, the analog outcome of GKP qubits, which includes beneficial information to improve the error tolerance, has been wasted, because the GKP qubits have been treated as only discrete variables. In this Letter, we propose a hybrid quantum error correction approach that combines digital information with the analog information of the GKP qubits using a maximum-likelihood method. As an example, we demonstrate that the three-qubit bit-flip code can correct double errors, whereas the conventional method based on majority voting on the binary measurement outcome can correct only a single error. As another example, we show that a concatenated code known as Knill’s $C_4/C_6$ code can achieve the hashing bound for the quantum capacity of the Gaussian quantum channel (GQC). To the best of our knowledge, this approach is the first attempt to draw both digital and analog information to improve quantum error correction performance and achieve the hashing bound for the quantum capacity of the GQC.

Figure 1

Introduction of a likelihood function. (a) Measurement outcome and deviation from the peak value in $q$ quadrature. The dotted line shows the measurement outcome $q_m$ equal to $(2t+k)\sqrt{\pi }+{\mathrm{\Delta }}_m$ $(t=0,\pm 1,\pm 2,\dots ,k=0,1)$, where $k$ is defined as the bit value that minimizes the deviation ${\mathrm{\Delta }}_m$. The red areas indicate the area that yields code word ($k+1$) mod 2, whereas the white area denotes the area that yields the codeword $k$. (b) and (c) Gaussian distribution functions as likelihood functions of the true deviation value $\overline{\mathrm{\Delta }}$ represented by the arrows. (b) refers to the case of the correct decision, where the amplitude of the true deviation value is $|\overline{\mathrm{\Delta }}|<\sqrt{\pi }/2$, whereas (c) the case of the incorrect decision $\sqrt{\pi }/2<|\overline{\mathrm{\Delta }}|<\sqrt{\pi }$.

Figure 2

Simulation results for the failure probabilities of the three-qubit bit-flip code using the conventional (blue line with open circles) and proposed methods (red line with filled circles).

Figure 3

Simulation results for the failure probabilities of the $C_4/C_6$ code using the conventional and proposed method. The failure probabilities using the conventional method (blue line) and proposed method (red line) are represented for the concatenated level 1 (solid), level 2 (dashed), level 3 (dashed-dotted), level 4 (open circles), and level 5 (filled circles).