In some quantum computing architectures, Pauli noise is highly biased. Tailoring quantum error-correcting codes to the biased noise may benefit reducing the physical qubit overhead without reducing the logical error rate. In this paper, we propose a family of quantum $XYZ$ cyclic codes, which is the only family of quantum cyclic codes with code distance increasing with code length to the best of our knowledge. We exploit fully decoupled belief propagation combined with ordered statistics decoding (FDBP-OSD) and Tesseract decoder (TD) to explore the error-correcting performance of the codes. Our simulation results show that the quantum $XYZ$ cyclic codes have $50\%$ code-capacity thresholds for all three types of pure Pauli noise. Under depolarizing noise, we observe $13\%$ code-capacity threshold obtained by FDBP-OSD, while the results obtained by TD indicates that the optimal code-capacity threshold may be approximately $19\%$. In the finite-bias regime, when the noise is biased towards Pauli $Z$ errors with noise bias ratios ${\eta }_Z=1000$, the results obtained by FDBP-OSD and TD both indicate the quantum $XYZ$ cyclic codes have around $49\%$ code-capacity threshold. When ${\eta }_Z=10$ and 100, TD indicates that the corresponding code-capacity thresholds may be around $29\%$ and $32\%$, respectively. In addition, we show that to reach the same code distance, the physical qubit overhead of $XYZ$ cyclic code is much less than that of the XZZX surface code.

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• Quantum computation
• Quantum error correction