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 for more efficient computation regardless of the numerical algorithm being used, and provides insight into the quantum channels which obtain the optimal rate-distortion tradeoff. 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.

翻译：量子率畸变函数在量子信息论中具有基础性作用，但目前尚无实用算法能够高效计算中等信道维度下的高精度结果。本文展示了对称性约化如何显著简化纠缠辅助量子率畸变问题的常见实例。无论使用何种数值算法，该方法均能提升计算效率，并揭示获得最优率畸变权衡的量子信道特性。此外，我们提出一种非精确镜像下降算法变体，用于以可证明的亚线性收敛速率计算量子率畸变函数。我们阐明了该镜像下降算法与信息论中先前用于解决相似问题的Blahut-Arimoto算法及期望最大化方法之间的关联。基于这些技术，我们首次开展了多量子比特量子率畸变函数的数值实验，结果表明与现有方法相比，本文算法具有更快的求解速度与更高的计算精度。