Achieving the Heisenberg limit under general Markovian noise using quantum error correction without ancilla

Practical quantum metrology is generally hindered by environment noise which can demolish the advantage of breaching the shot-noise limit provided by quantum mechanics. Recently, quantum error-correction (QEC) codes have been developed for restoring the quantum advantage against Markovian noise for quantum metrology. This has been done by assuming either the existence of noiseless ancilla or specific noise types such as the spatially correlated noise and the so-called commuting noise. We consider the situation where all probing systems for metrology are infested by general uncorrelated Markovian noise; namely, there is neither noiseless ancilla nor spatial correlation of the noise. Under such circumstances, we have discussed the error-correction conditions in parallel metrology scheme where more than three probes are employed. We have shown the superiority of parallel schemes over sequential schemes in fighting Markovian noise by ancilla-free QEC with an explicit example. If we employ qubits for parameter sensing, it can be shown that one can restore Heisenberg limit (HL) via full and fast control without any ancilla, as long as the signal Hamiltonian of the probe system has nonvanishing component perpendicular to the noise. This is exactly the same requirement for restoring HL with the assistance of noiseless ancilla.