Efficient qubit phase estimation using adaptive measurements

Estimating correctly the quantum phase of a physical system is central problem in parameter estimation theory due to its wide range applications from metrology cryptography. Ideally, optimal estimator given by so-called Cramér-Rao bound, so any measurement strategy aims obtain estimations as close possible it. However, more often than not, current state-of-the-art methods estimate phases fail reach this bound they rely on maximum likelihood estimators non-identifiable functions. In work we thoroughly review various schemes for estimating qubit, identifying underlying which prohibits these and propose new adaptive scheme based covariant measurements circumvent problem. Our findings are carefully checked Monte Carlo simulations, showing that method both mathematically experimentally realistic efficient currently available.