Fidelity of a quantum state protected by the surface code in the presence of a finite-temperature bosonic bath

We evaluate the fidelity of a multi-qubit quantum state protected by the surface code during a single quantum error correction cycle when qubits couple to a gapless bosonic environment. We discuss the protection of the state for different spectral functions and bath temperatures. Analytical results are supported by finite-size scaling analyses based on Monte Carlo and exact numerical calculations. Our results demonstrate a finite threshold that explicitly depends on the bath-mediated qubit-qubit interaction range and bath spectral function and temperature.