  Constrained optimization plays a crucial role in the fields of quantum
physics and quantum information science and becomes especially challenging for
high-dimensional complex structure problems. One specific issue is that of
quantum process tomography, in which the goal is to retrieve the underlying
quantum process based on a given set of measurement data. In this paper, we
introduce a modified version of stochastic gradient descent on a Riemannian
manifold that integrates recent advancements in numerical methods for
Riemannian optimization. This approach inherently supports the physically
driven constraints of a quantum process, takes advantage of state-of-the-art
large-scale stochastic objective optimization, and has superior performance to
traditional approaches such as maximum likelihood estimation and projected
least squares. The data-driven approach enables accurate, order-of-magnitude
faster results, and works with incomplete data. We demonstrate our approach on
simulations of quantum processes and in hardware by characterizing an
engineered process on quantum computers.
