  The capacity of noisy quantum channels characterizes the highest rate at
which information can be reliably transmitted and it is therefore of practical
as well as fundamental importance. Capacities of classical channels are
computed using alternating optimization schemes, called Blahut-Arimoto
algorithms. In this work, we generalize classical Blahut-Arimoto algorithms to
the quantum setting. In particular, we give efficient iterative schemes to
compute the capacity of channels with classical input and quantum output, the
quantum capacity of less noisy channels, the thermodynamic capacity of quantum
channels, as well as the entanglement-assisted capacity of quantum channels. We
give rigorous a priori and a posteriori bounds on the estimation error by
employing quantum entropy inequalities and demonstrate fast convergence of our
algorithms in numerical experiments.
