  When a logical qubit is protected using a quantum error-correcting code, the
net effect of coding, decoherence (a physical channel acting on qubits in the
codeword) and recovery can be represented exactly by an effective channel
acting directly on the logical qubit. In this paper we describe a procedure for
deriving the map between physical and effective channels that results from a
given coding and recovery procedure. We show that the map for a concatenation
of codes is given by the composition of the maps for the constituent codes.
This perspective leads to an efficient means for calculating the exact
performance of quantum codes with arbitrary levels of concatenation. We present
explicit results for single-bit Pauli channels. For certain codes under the
symmetric depolarizing channel, we use the coding maps to compute exact
threshold error probabilities for achievability of perfect fidelity in the
infinite concatenation limit.
