Improving the Precision of Multi-Parameter Estimation by Quantum Error Correction Based on Continuous Detection Error

In this paper, a quantum error correction scheme combined with quantum feedback control is proposed, which can achieve the Heisenberg limit even if the Hamiltonian-Not-in-Lindblad-Span condition in the optimal error correction protocol is not satisfied. Compared with other error correction schemes, our scheme not only does not require the ancilla system to be noiseless, but also has a lower dimensional code space. In particular, our scheme can achieve the highest estimation precision for the three components of a magnetic field simultaneously with only a 2-dimensional code space, while at least 4-dimensional code space is required in the optimal error correction protocol