  We numerically study coherent errors in surface codes on planar graphs,
focusing on noise of the form of $Z$- or $X$-rotations of individual qubits. We
find that, similarly to the case of incoherent bit- and phase-flips, a
trade-off between resilience against coherent $X$- and $Z$-rotations can be
made via the connectivity of the graph. However, our results indicate that,
unlike in the incoherent case, the error-correction thresholds for the various
graphs do not approach a universal bound. We also study the distribution of
final states after error correction. We show that graphs fall into three
distinct classes, each resulting in qualitatively distinct final-state
distributions. In particular, we show that a graph class exists where the
logical-level noise exhibits a decoherence threshold slightly above the
error-correction threshold. In these classes, therefore, the logical level
noise above the error-correction threshold can retain significant amount of
coherence even for large-distance codes. To perform our analysis, we develop a
Majorana-fermion representation of planar-graph surface codes and describe the
characterization of logical-state storage using fermion-linear-optics-based
simulations. We thereby generalize the approach introduced for the square
lattice by Bravyi \textit{et al}. [npj Quantum Inf. 4, 55 (2018)] to surface
codes on general planar graphs.
