  We compare the effect of single qubit incoherent and coherent errors on the
logical error rate of the Steane [[7,1,3]] quantum error correction code by
performing an exact full-density-matrix simulation of an error correction step.
We find that the effective 1-qubit process matrix at the logical level reveals
the key differences between the error models and provides insight into why the
Pauli twirling approximation is a good approximation for incoherent errors and
a poor approximation for coherent ones. Approximate channels composed of
Clifford operations and Pauli measurement operators that are pessimistic at the
physical level result in pessimistic error rates at the logical level. In
addition, we observe that the pseudo-threshold can differ by a factor of five
depending on whether the error is calculated using the fidelity or the
distance.
