Continuous error correction for evolution under time-dependent Hamiltonians

We analyze the continuous operation of a three-qubit code designed to protect the coherent evolution in the code space due to an encoded Hamiltonian. Quantum error correction here requires fast detection and immediate correction of errors to avoid spurious coherent evolution in the error subspaces. To detect errors in real time, we smooth the output signals from continuous measurement of the error syndrome operators and use a double threshold protocol for error diagnosis, while correction of errors is done as in the conventional code operation. We evaluate the performance of this protocol under bit-flip errors, quantifying this in terms of fidelity and logical error rate. We show that the optimal error detection time that maximizes the final fidelity can be much shorter than that of the conventional operation, suggesting that continuous implementation is suitable for quantum error correction in the presence of encoded time-dependent Hamiltonians.