  Adaptive measurements have recently been shown to significantly improve the
performance of quantum state and process tomography. However, the existing
methods either cannot be straightforwardly applied to high-dimensional systems
or are prohibitively computationally expensive. Here we propose and
experimentally implement a novel tomographic protocol specially designed for
the reconstruction of high-dimensional quantum states. The protocol shows
qualitative improvement in infidelity scaling with the number of measurements
and is fast enough to allow for complete state tomography of states with
dimensionality up to 36.
