  For a generic set of Markovian noise models, the estimation precision of a
parameter associated with the Hamiltonian is limited by the $1/\sqrt{t}$
scaling where $t$ is the total probing time, in which case the maximal possible
quantum improvement in the asymptotic limit of large $t$ is restricted to a
constant factor. However, situations arise where the constant factor
improvement could be significant, yet no effective quantum strategies are
known. Here we propose an optimal approximate quantum error correction (AQEC)
strategy asymptotically saturating the precision lower bound in the most
general adaptive parameter estimation scheme where arbitrary and frequent
quantum controls are allowed. We also provide an efficient numerical algorithm
finding the optimal code. Finally, we consider highly-biased noise and show
that using the optimal AQEC strategy, strong noises are fully corrected, while
the estimation precision depends only on the strength of weak noises in the
limiting case.

Quantum Physics