  Flexible characterization techniques that identify and quantify experimental
imperfections under realistic assumptions are crucial for the development of
quantum computers. Gate set tomography is a characterization approach that
simultaneously and self-consistently extracts a tomographic description of the
implementation of an entire set of quantum gates, as well as the initial state
and measurement, from experimental data. Obtaining such a detailed picture of
the experimental implementation is associated with high requirements on the
number of sequences and their design, making gate set tomography a challenging
task even for only two qubits.
  In this work, we show that low-rank approximations of gate sets can be
obtained from significantly fewer gate sequences and that it is sufficient to
draw them randomly. Such tomographic information is needed for the crucial task
of dealing with coherent noise. To this end, we formulate the data processing
problem of gate set tomography as a rank-constrained tensor completion problem.
We provide an algorithm to solve this problem while respecting the usual
positivity and normalization constraints of quantum mechanics by using
second-order geometrical optimization methods on the complex Stiefel manifold.
Besides the reduction in sequences, we demonstrate numerically that the
algorithm does not rely on structured gate sets or an elaborate circuit design
to robustly perform gate set tomography. Therefore, it is more flexible than
traditional approaches. We also demonstrate how coherent errors in shadow
estimation protocols can be mitigated using estimates from gate set tomography.
