Modeling error correction with Lindblad dynamics and approximate channels

We analyze the performance of a quantum error-correction code subject to physically motivated noise modeled by a Lindblad master equation. We consider dissipative and coherent single-qubit terms and two-qubit crosstalk, studying how different approximations of the noise capture the success rate of a code. Focusing on the five-qubit code and adapting it to partially correct two-qubit errors in relevant parameter regimes according to the noise model, we find that a composite-channel approximation where every noise term is considered separately captures the behavior in many physical cases up to long timescales, eventually failing due to the effect of noncommuting terms. In contrast, we find that single-qubit approximations do not properly capture the error-correction dynamics with two-qubit noise, even for short times. A Pauli approximation going beyond a single-qubit channel is sensitive to the details of the noise, state, and decoder and succeeds at short timescales relative to the noise strength, beyond which it fails. Furthermore, we point out a mechanism for a Pauli model failure where it underestimates the success rate of a code even with frequent syndrome projection and correction cycles. These results shed light on the performance of error correction in the presence of realistic noise and can advance the ongoing efforts towards useful quantum error correction.