  Conventional methods for computing maximum-likelihood estimators (MLE) often
converge slowly in practical situations, leading to a search for simplifying
methods that rely on additional assumptions for their validity. In this work,
we provide a fast and reliable algorithm for maximum likelihood reconstruction
that avoids this slow convergence. Our method utilizes the state-of-the-art
convex optimization scheme---an accelerated projected-gradient method---that
allows one to accommodate the quantum nature of the problem in a different way
than in the standard methods. We demonstrate the power of our approach by
comparing its performance with other algorithms for n-qubit state tomography.
In particular, an 8-qubit situation that purportedly took weeks of computation
time in 2005 can now be completed in under a minute for a single set of data,
with far higher accuracy than previously possible. This refutes the common
claim that MLE reconstruction is slow, and reduces the need for alternative
methods that often come with difficult-to-verify assumptions. In fact, recent
methods assuming Gaussian statistics or relying on compressed sensing ideas are
demonstrably inapplicable for the situation under consideration here. Our
algorithm can be applied to general optimization problems over the quantum
state space; the philosophy of projected gradients can further be utilized for
optimization contexts with general constraints.
