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.