Continuous error correction for quantum metrology

Protecting the information in quantum metrology by quantum error correction has been studied intensively in recent years. The usual assumption in quantum error correction is that the syndrome measurements and correction operations are done frequently enough so that errors generated by the noise are correctable with high probability. Here, we analyze how the rate of continuous-time quantum error correction affects quantum metrology. We show that if the rate of error correction is finite, the Heisenberg limit in quantum metrology can only be sustained for a finite period. Moreover, in contrast to quantum metrology without noise (or with sufficiently fast quantum error correction), a longer evolution time does not always produce more information. There exists an optimal time at which the Fisher information reaches the maximum. We use the simple three-qubit bit-flip code to illustrate this result.