Quantum enhanced metrology with noise assistance and error correction

Quantum metrology has seen various important applications in science and engineering, ranging from atomic frequency estimation to gravitational wave detection. It has been shown that quantum resources can outperform their classical counterparts as regard to improving the precision of parameter estimation. However, there is inevitable difficulty in manipulating a quantum system because the system keeps leaking out information to the environment that it is coupled to, making superiority brought by the ``quantumness" disappear beyond decoherence time. Therefore, it is critical to protect the system from noise, which can be achieved by means of adding control signals.
Most of the previous work has a focus on time-invariant parameter-independent Hamiltonian and noise. By contrast, in this paper we provide a sufficient and necessary condition under which one can employ control to recover the Heisenberg precision limit for systems with time-invariant parameter-dependent Hamiltonian and noise. We further consider the case where both Hamiltonian and noise are time-varying and parameter-dependent. We show a detailed technique via quantum error correction to harvest, or even beat the optimal quantum Fisher information in the noiseless case.