Threshold Behavior of ZX and ZY Surface Codes Under Circuit-Level Biased and Crosstalk Noise

Abstract: Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing $Z$-type stabilizers with $Y$-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a $ZY$ surface code by replacing the $X$-type stabilizers with $Y$-type stabilizers. We compare it with the standard $ZX$ surface code under circuit-level Pauli-$X$ biased noise, with and without an additional gate-based $XX$ crosstalk noise. We find that for the $ZX$ surface code, the $X$-memory threshold increases monotonically with bias while the $Z$-memory threshold decreases and saturates. For the $ZY$ surface code, the $Y$-memory threshold is nearly constant across all bias values. The $Z$-memory thresholds of the $ZX$ and $ZY$ codes are consistent within the uncertainty. Adding $XX$ crosstalk reduces the $Z$-memory threshold beyond the fitting uncertainty while leaving the $X$ memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.