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We critically examine the internal consistency of a set of minimalassumptions entering the theory of fault-tolerant quantum error correction forMarkovian noise. These assumptions are: fast gates, a constant supply of freshand cold ancillas, and a Markovian bath. We point out that these assumptionsmay not be mutually consistent in light of rigorous formulations of theMarkovian approximation. Namely, Markovian dynamics requires either thesingular coupling limit (high temperature), or the weak coupling limit (weaksystem-bath interaction). The former is incompatible with the assumption of aconstant and fresh supply of cold ancillas, while the latter is inconsistentwith fast gates. We discuss ways to resolve these inconsistencies. As part ofour discussion we derive, in the weak coupling limit, a new master equation fora system subject to periodic driving.

Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum Markovian limit