  Typical studies of quantum error correction assume probabilistic Pauli noise,
largely because it is relatively easy to analyze and simulate. Consequently,
the effective logical noise due to physically realistic coherent errors is
relatively unknown. Here, we prove that encoding a system in a stabilizer code
and measuring error syndromes decoheres errors, that is, causes coherent errors
to converge toward probabilistic Pauli errors, even when no recovery operations
are applied. Two practical consequences are that the error rate in a logical
circuit is well quantified by the average gate fidelity at the logical level
and that essentially optimal recovery operators can be determined by
independently optimizing the logical fidelity of the effective noise per
syndrome.
