A Posterior MWPM Decoding Boosts the XYZ Planar Code

The minimum-weight perfect matching (MWPM) decoder is a standard decoding strategy for surface codes, but its performance degrades considerably under biased noise. In this paper, a modified surface code, termed the XYZ planar code, is introduced, and the MWPM decoder is extended to posterior MWPM (pMWPM) with almost no increase in decoding complexity. The XYZ planar code exhibits higher and more stable thresholds than the planar code under almost all bias conditions, while also achieving significantly lower logical error rates. Specifically, in the infinite-bias case, the threshold of the XYZ planar code is improved by about $36\%$ compared to that of the surface code, and it maintains comparable or higher thresholds under other biases -- for example, the threshold reaches approximately $15.5\%$ at bias $\eta = 1$ and $14.2\%$ at $\eta = 100$. Furthermore, pMWPM can be adapted to a wide range of modified surface codes, and the results presented in this work also indicate its excellent potential in other scenarios, such as configurations in which $Y$ operators involve a larger number of data qubits.