Fig. 3: Circuit representation of the standard QEC procedure of generalized NP codes.

(a–d) are the Wigner functions of noisy NP codes in different error correction stages. Here, we choose the logical state \({\left\vert+\right\rangle }_{L}\) of a D-NP code \([s=2;\,f=1/2;\,r=-0.1;\,{\bar{n}}_{{{{\rm{code}}}}}=9]\) as an example. The noisy state \({{{\mathcal{N}}}}(\hat{\rho })\) [(a)] has an overlap  ~ 0.37 with the ideal logical state, which is obtained by the evolution in the error model Eq. (16) with γt = 0.1 and κt = 0.01; (b) is obtained with the single-photon loss event of code state is identified by the number parity measurement. The Wigner function (c) is obtained after performing the interface gates \({\hat{U}}_{f}^{{{\dagger}} }\), which is back to the R-NP structure, and the shallower peaks are present due to the NP vortex effect. The state (d), corrected via a perfect QEC cycle, has an overlap  ~ 0.97 with the ideal logical state. (e) and (f) are the circuit representations of the modular number-parity measurement and the teleportation-based QEC, defined in Eqs. (12) and (14), respectively. The double lines indicate the transmission of classical measurement outcomes for the recovery feedback.