  Constructing a fast and optimal estimator for the $B$-mode power spectrum of
cosmic microwave background (CMB) is of critical importance for CMB science.
For a general CMB survey, the Quadratic Maximum Likelihood (QML) estimator for
CMB polarization has been proved to be the optimal estimator with minimal
uncertainties, but it is computationally very expensive. In this article, we
propose two new QML methods for $B$-mode power spectrum estimation. We use the
Smith-Zaldarriaga approach to prepare pure-$B$ mode map, and $E$-mode recycling
method to obtain a leakage free $B$-mode map. We then use the scalar QML
estimator to analyze the scalar pure-$B$ map (QML-SZ) or $B$-mode map (QML-TC).
The QML-SZ and QML-TC estimators have similar error bars as the standard QML
estimators but their computational cost is nearly one order of magnitude
smaller. The basic idea is that one can construct the pure $B$-mode CMB map by
using the $E$-$B$ separation method proposed by Smith and Zaldarriaga (SZ) or
the one considering the template cleaning (TC) technique, then apply QML
estimator to these scalar fields. By simulating potential observations of
space-based and ground-based detectors, we test the reliability of these
estimators by comparing them with the corresponding results of the traditional
QML estimator and the pure $B$-mode pseudo-$C_{\ell}$ estimator.
