Continuous quantum correction on Markovian and Non-Markovian models

We investigate quantum error correction through a continuous quantum-jump process, comparing performance under a Markovian error model to two distinct non-Markovian models. The first model is an X-X Hamiltonian coupling (simulating crosstalk) between the system and environment qubits subject to a "cooling" bath, which can Markovian or non-Markovian behavior. The second model uses the post-Markovian master equation (PMME), which incorporates time-correlations of the bath through an exponentially decaying memory kernel.

We applied the three models to a single-qubit system to obtain analytical solutions, and then extended the study to the three-qubit repetition code and the five-qubit "perfect" code. At short times, the fidelity in the Markovian case decays more rapidly than in the non-Markovian models, suggesting that non-Markovianity of the errors can enhance the performance of continuous quantum error correction. For instance, in the single-qubit model, fidelity exhibits a linear decay, as opposed to non-Markovian models that show a quadratic decay. Similarly, for the three-qubit and five-qubit codes, fidelity decays quadratically in the Markovian case and cubically in the non-Markovian case. We attribute this difference to the presence of a quantum Zeno regime in both non-Markovian models.