Fig. 1: Schematic of the QEC procedure with the lowest-order binomially encoded logical qubit.

The auxiliary qubit is first encoded to the logical qubit in an oscillator with \(\{\left|{0}_{{\rm{L}}}\right\rangle =\left(\left|0\right\rangle +\left|4\right\rangle \right)/\sqrt{2},\left|{1}_{{\rm{L}}}\right\rangle =\left|2\right\rangle \}\). Once a single-photon-jump error occurs, the logical qubit state falls out of the code space to the error space with the basis states: \(\{\left|{0}_{{\rm{E}}}\right\rangle =\left|3\right\rangle ,\left|{1}_{{\rm{E}}}\right\rangle =\left|1\right\rangle \}\). After repetitive error detecting and correcting, the logical qubit state is protected against single-photon-jump errors. Finally, quantum state is decoded back to the auxiliary qubit for a final state characterization. The cardinal point states in the Bloch spheres of the code and error spaces are defined as \(\left|+{Z}_{{\rm{L}}({\rm{E}})}\right\rangle =\left|{0}_{{\rm{L}}({\rm{E}})}\right\rangle ,\left|+{X}_{{\rm{L}}({\rm{E}})}\right\rangle =(\left|{0}_{{\rm{L}}({\rm{E}})}\right\rangle +\left|{1}_{{\rm{L}}({\rm{E}})}\right\rangle )/\sqrt{2}\) and \(\left|+{Y}_{{\rm{L}}({\rm{E}})}\right\rangle =(\left|{0}_{{\rm{L}}({\rm{E}})}\right\rangle +i\left|{1}_{{\rm{L}}({\rm{E}})}\right\rangle )/\sqrt{2}\), respectively.