  Accurate quantum tomography is a vital tool in both fundamental and applied
quantum science. It is a task that involves processing a noisy measurement
record in order to construct a reliable estimate of an unknown quantum state,
and is central to quantum computing, metrology, and communication. To date,
many different approaches to quantum state estimation have been developed, yet
no one method fits all applications, and all fail relatively quickly as the
dimensionality of the unknown state grows. In this work, we suggest that
projected gradient descent is a method that can evade some of these
shortcomings. We present three novel tomography algorithms that use projected
gradient descent and compare their performance with state-of-the-art
alternatives, i.e. the diluted iterative algorithm and convex programming. Our
results find in favour of the general class of projected gradient descent
methods due to their speed, applicability to large states, and the range of
conditions in which they perform.
