  Headline constraints on cosmological parameters from current weak lensing
surveys are derived from two-point statistics that are known to be
statistically sub-optimal, even in the case of Gaussian fields. We study the
performance of a new fast implementation of the Quadratic Maximum Likelihood
(QML) estimator, optimal for Gaussian fields, to test the performance of
Pseudo-Cl estimators for upcoming weak lensing surveys and quantify the gain
from a more optimal method. Through the use of realistic survey geometries,
noise levels, and power spectra, we find that there is a decrease in the errors
in the statistics of the recovered E-mode spectra to the level of ~20% when
using the optimal QML estimator over the Pseudo-Cl estimator on the largest
angular scales, while we find significant decreases in the errors associated
with the B-modes for the QML estimator. This raises the prospects of being able
to constrain new physics through the enhanced sensitivity of B-modes for
forthcoming surveys that our implementation of the QML estimator provides. We
test the QML method with a new implementation that uses conjugate-gradient and
finite-differences differentiation methods resulting in the most efficient
implementation of the full-sky QML estimator yet, allowing us to process maps
at resolutions that are prohibitively expensive using existing codes. In
addition, we investigate the effects of apodisation, B-mode purification, and
the use of non-Gaussian maps on the statistical properties of the estimators.
Our QML implementation is publicly available and can be accessed from GitHub.
