  Quantum state tomography, which aims to find the best description of a
quantum state -- the density matrix, is an essential building block in quantum
computation and communication. Standard techniques for state tomography are
incapable of tracking changing states and often perform poorly in the presence
of environmental noise. Although there are different approaches to solve these
problems theoretically, experimental demonstrations have so far been sparse.
Our approach, matrix-exponentiated gradient tomography, is an online tomography
method that allows for state tracking, updates the estimated density matrix
dynamically from the very first measurements, is computationally efficient, and
converges to a good estimate quickly even with noisy data. The algorithm is
controlled via a single parameter, its learning rate, which determines the
performance and can be tailored in simulations to the individual experiment. We
present an experimental implementation of matrix-exponentiated gradient
tomography on a qutrit system encoded in the transverse spatial mode of
photons. We investigate the performance of our method on stationary and
evolving states, as well as significant environmental noise, and find
fidelities of around 95% in all cases.
