  As current experiments already realize small quantum circuits on error
corrected qubits, it is important to fully understand the effect of physical
errors on the logical error channels of these fault-tolerant circuits. Here, we
investigate a lattice-surgery-based CNOT operation between two surface code
patches under phenomenological error models. (i) For two-qubit logical Pauli
measurements -- the elementary building block of the CNOT -- we optimize the
number of stabilizer measurement rounds, usually taken equal to $d$, the size
(code distance) of each patch. We find that the optimal number can be greater
or smaller than $d$, depending on the rate of physical and readout errors, and
the separation between the code patches. (ii) We fully characterize the
two-qubit logical error channel of the lattice-surgery-based CNOT. We find a
symmetry of the CNOT protocol, that results in a symmetry of the logical error
channel. We also find that correlations between X and Z errors on the logical
level are suppressed under minimum weight decoding.
