uremath{\left|0\right\rangle}}_L$. This kind of gates is useful for quantum error correction [@Reed2012Realization] and serves as a central element for implementation of the quantum search algorithm [@Nielsen]. In addition to implementation of quantum gates, our method can be used for stabilizing the parity of a cat state: When an environmentally-induced parity jump occurs, one can correct for it through the combination of a conditional qubit $2\pi$ rotation and two displacement operations.

We thank valuable discussions with Chen Wang. This work was supported by National Key Research and Development Program of China No.2017YFA0304303 and the National Natural Science Foundation of China under Grants No.11474177, No. 11874114, No. 11674060, and No. 11875108.

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