  We derive the effective channel for a logical qubit protected by an arbitrary
quantum error-correcting code, and derive the map between channels induced by
concatenation. For certain codes in the presence of single-bit Pauli errors, we
calculate the exact threshold error probability for perfect fidelity in the
infinite concatenation limit. We then use the control theory technique of
balanced truncation to find low-order non-asymptotic approximations for the
effective channel dynamics.
