Logical Error Rates of XZZX and Rotated Quantum Surface Codes

Surfacecodesareversatilequantumerror-correcting codes known for their planar geometry, making themideal for practical implementations. While the original proposalused PauliXor PauliZoperators in a square structure, thesecodes can be improved by rotating the lattice or incorporating amix of generators in the XZZX variant. However, a comprehen-sive theoretical analysis of the logical error rate for these variantshas been lacking. To address this gap, we present theoreticalformulas based on recent advancements in understanding theweight distribution of stabilizer codes. For example, over anasymmetric channel with asymmetryA= 10and a physicalerror rate p -> 0, we observe that the logical error rateasymptotically approaches pL -> 10p(2 ) for the rotated[[9,1,3]]XZZX code and pL -> 18.3p2for the[[13,1,3]] surfacecode. Additionally, we observe a particular behavior regardingrectangular lattices in the presence of asymmetric channels.Our findings demonstrate that implementing both rotation andXZZX modifications simultaneously can lead to suboptimal performance. Thus, in scenarios involving a rectangular lattice,it is advisable to avoid using both modifications simultaneously