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, allowing for more efficient computation regardless of the numerical algorithm being used. For some of these problem instances, symmetry reduction allows us to derive closed-form expressions for the quantum rate-distortion function. 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：量子速率失真函数在量子信息理论中扮演着基本角色，然而目前尚无实用算法能在中等信道维度下高效且高精度地计算该函数。本文展示了对称性约化如何显著简化常见的纠缠辅助量子速率失真问题实例，使得无论采用何种数值算法都能进行更高效的计算。针对部分问题实例，对称性约化使我们能够推导出量子速率失真函数的闭式表达式。此外，我们提出了一种非精确的镜像下降算法变体，以可证明的次线性收敛速率计算量子速率失真函数。我们展示了该镜像下降算法与先前用于信息论中类似问题的Blahut-Arimoto方法和期望最大化方法之间的关联。利用这些技术，我们首次开展了多量子比特量子速率失真函数的数值实验，并证明所提算法相比现有方法求解速度更快且精度更高。