  Quantum relative entropy programs are convex optimization problems which
minimize a linear functional over an affine section of the epigraph of the
quantum relative entropy function. Recently, the self-concordance of a natural
barrier function was proved for this set. This has opened up the opportunity to
use interior-point methods for nonsymmetric cone programs to solve these
optimization problems. In this paper, we show how common structures arising
from applications in quantum information theory can be exploited to improve the
efficiency of solving quantum relative entropy programs using interior-point
methods. First, we show that the natural barrier function for the epigraph of
the quantum relative entropy composed with positive linear operators is
optimally self-concordant, even when these linear operators map to singular
matrices. Compared to modelling problems using the full quantum relative
entropy cone, this allows us to remove redundant log determinant expressions
from the barrier function and reduce the overall barrier parameter. Second, we
show how certain slices of the quantum relative entropy cone exhibit useful
properties which should be exploited whenever possible to perform certain key
steps of interior-point methods more efficiently. We demonstrate how these
methods can be applied to applications in quantum information theory, including
quantifying quantum key rates, quantum rate-distortion functions, quantum
channel capacities, and the ground state energy of Hamiltonians. Our numerical
results show that these techniques improve computation times by up to several
orders of magnitude, and allow previously intractable problems to be solved.
