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Concatenation of a bosonic code with a qubit code is one of the promisingways to achieve fault-tolerant quantum computation. As one of the mostimportant bosonic codes, Gottesman-Kitaev-Preskill (GKP) code is proposed tocorrect small displacement error in phase space. If the noise in phase space isbiased, square-lattice GKP code can be concatenated with XZZX surface code orrepetition code that promises a high fault-tolerant threshold to suppress thelogical error. In this work, we study the performance of GKP repetition codeswith physical ancillary GKP qubits in correcting biased noise. We find thatthere exists a critical value of noise variance for the ancillary GKP qubitsuch that the logical Pauli error rate decreases when increasing the code size.Furthermore, one round of GKP error correction has to be performed beforeconcatenating with repetition code. Our study paves the way for practicalimplementation of error correction by concatenating GKP code with low-levelqubit codes.

Correcting biased noise using Gottesman-Kitaev-Preskill repetition code with noisy ancilla