Coherent error and stochastic error in surface code

We compare single qubit coherent error model with stochastic error model obtained by Pauli twirl approximation in surface code. We find that the surface code under coherent error satisfies generalized K-L criterion and falls into the category of approximate QEC. The extra term in the generalized K-L criterion corresponds to the coherent part of the error channel at logical level. It is already known that, when code distance is large and coherent rotation angle is small, the coherent error channel converges towards probabilistic quantum channel at the logical level after syndrome measurement. Then, we conclude that the generalized K-L criterion of this model also approaches normal K-L criterion in that situation. We then study the success probability of QEC under such coherent errors, and confirm that the exact success probability under coherent error is smaller than the results using Pauli twirling approximation at physical level.