Consequences of unitary evolution of coupled qubit-resonator systems for stabilizing circuits in surface codes

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.