  Surface codes based on stabilizer circuits may pave the way for large scale
fault-tolerant quantum computation. The surface code uses only single- and
two-qubit gates and the error threshold falls close to 1% for a large range of
errors. Among the most promising candidates to physically implement such
circuits and codes are superconducting qubits coupled by resonators. We
investigate a X and Z stabilizing circuit realized by two data qubits, two
ancillas and four resonators. The aim is to assess the consequences of unitary
evolution of the interacting system, in particular for given stable initial
states, on fidelities and error syndrome probabilities. We model the system
with a Jaynes-Tavis-Cummings Hamiltonian and construct the low-excitation level
evolution operators. The analysis is limited to two stable input states. We
assume an ideal system with perfect gates, perfect measurements and no
decoherence or leakage. Our analysis shows that the capture probabilities after
the execution of a single stabilizer round are not equal to 100%, but vary
between 99.2% and 99.99%. This is caused solely by the unitary evolution of the
interacting system. Two consecutive rounds of stabilizer measurements result in
capture probability values that depend heavily on the duration of the
evolution, but vary between 0% and 99%. Also due to the unitary evolution, the
final state of the data qubits leaves the the four-dimensional subspace, which
results in a state fidelity oscillating between 0 and 1. Even if an error on
the qubits is captured, the correcting operation on the qubit will not bring
the qubit to the original state. The errors induced by the Hamiltonian
evolution of the system cannot be interpreted nor classified as commonly
appearing errors. Additional or augmented quantum error correction may be
required to compensate these effects of resonator-qubit interaction.
