  The realistic coherent errors could induce very different behaviors compared
with their stochastic counterparts in the quantum error correction (QEC) and
fault tolerant quantum computation. Their impacts are believed to be very
subtle, more detrimental and hard to analyze compared to those ideal stochastic
errors. In this paper, we study the independent coherent error due to the
imperfect unitary rotation on each physical qubit of the toric code. We find
that the surface code under coherent error satisfies generalized Knill-Laflamme
(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, and then show that the generalized K-L criterion
approaches the normal K-L criterion when the code distance becomes large. In
addition, we also find that if the code with a fixed distance d is
$\epsilon$-correctable, the value of $\epsilon$ describing the accuracy of the
approximate QEC cannot be smaller than a lower bound. 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.
