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

Concatenation of a bosonic code with a qubit code is one of the promising ways to achieve fault-tolerant quantum computation. As one of the most important bosonic codes, Gottesman-Kitaev-Preskill (GKP) code is proposed to correct small displacement error in phase space. If the noise in phase space is biased, square-lattice GKP code can be concatenated with XZZX surface code or repetition code that promises a high fault-tolerant threshold to suppress the logical error. In this work, we study the performance of GKP repetition codes with physical ancillary GKP qubits in correcting biased noise. We find that there exists a critical value of noise variance for the ancillary GKP qubit such that the logical Pauli error rate decreases when increasing the code size. Furthermore, one round of GKP error correction has to be performed before concatenating with repetition code. Our study paves the way for practical implementation of error correction by concatenating GKP code with low-level qubit codes.