  Optimal rates for achieving an information processing task are often
characterized in terms of regularized information measures. In many cases of
quantum tasks, we do not know how to compute such quantities. Here, we exploit
the symmetries in the recently introduced $D^{\#}$ in order to obtain a
hierarchy of semidefinite programming bounds on various regularized quantities.
As applications, we give a general procedure to give efficient bounds on the
regularized Umegaki channel divergence as well as the classical capacity and
two-way assisted quantum capacity of quantum channels. In particular, we obtain
slight improvements for the capacity of the amplitude damping channel. We also
prove that for fixed input and output dimensions, the regularized sandwiched
R\'enyi divergence between any two quantum channels can be approximated up to
an $\epsilon$ accuracy in time that is polynomial in $1/\epsilon$.
