Surface codes with coherent errors: optimal decoding and finite-size crossovers

arXiv:2611.13098v1 Announce Type: new Abstract: We study the surface code under uniform coherent $Z$ rotations $\exp{(i\theta Z)}$ followed by perfect syndrome measurements, a setting known to exhibit strong finite-size effects that complicate the extraction of the asymptotic decoding threshold. We identify finite-size behavior of both Pauli decoding measures and the coherent information with two characteristic lengths: an algebraic ballistic-crossover scale $d_{\mathrm{ba}}\sim 1/\beta_R^{\alpha}$ ($\alpha=1$ for odd $d$ and $\alpha=2$ for even $d$) and a much larger renormalization-group scale $d_\mathrm{RG}\sim \exp{(C/\beta^{4}_R)}$, where $\beta_R=-\tfrac{1}{2}\ln\tan\theta$ and $d$ is the system size.