Continuous Error Correction for Markovian and Non Markovian models

We study continuous quantum error correction under both Markovian and non-Markovian noise. Two non-Markovian models are considered: a system-environment XX interaction with a cooling bath, and the post-Markovian master equation (PMME) with an exponential memory kernel. We compare their performance against the Markovian case using a single qubit, the three-qubit repetition code, and the five-qubit perfect code, finding improved fidelity in the non-Markovian cases due to a quantum Zeno effect. Additionally, we benchmark various quantum master equations—including second- and fourth-order time-convolutionless (TCL) expansions and several Markovian approximations—against the exact solution of the damped Jaynes-Cummings model across multiple spectral densities, for different temperature regimes. We also simulate dynamics using the Hierarchical Equations of Motion (HEOM). Our results can explain trapped ions and superconducting 1/f noise and emphasize the need for accurate open-system modeling in non-Markovian environments.