  The quantum rate-distortion function plays a fundamental role in quantum
information theory, however there is currently no practical algorithm which can
efficiently compute this function to high accuracy for moderate channel
dimensions. In this paper, we show how symmetry reduction can significantly
simplify common instances of the entanglement-assisted quantum rate-distortion
problems. This allows us to better understand the properties of the quantum
channels which obtain the optimal rate-distortion trade-off, while also
allowing for more efficient computation of the quantum rate-distortion function
regardless of the numerical algorithm being used. Additionally, we propose an
inexact variant of the mirror descent algorithm to compute the quantum
rate-distortion function with provable sublinear convergence rates. We show how
this mirror descent algorithm is related to Blahut-Arimoto and
expectation-maximization methods previously used to solve similar problems in
information theory. Using these techniques, we present the first numerical
experiments to compute a multi-qubit quantum rate-distortion function, and show
that our proposed algorithm solves faster and to higher accuracy when compared
to existing methods.
