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.

中文翻译：

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量子率失真函数的高效计算

量子率失真函数在量子信息论中发挥着基础作用，但目前还没有实用的算法可以在中等通道维度下高效地高精度计算该函数。在本文中，我们展示了对称性约简如何显着简化纠缠辅助量子率失真问题的常见情况。这使我们能够更好地理解获得最佳率失真权衡的量子通道的属性，同时还允许更有效地计算量子率失真函数，而不管使用何种数值算法。此外，我们提出了镜像下降算法的不精确变体，以计算具有可证明的亚线性收敛速率的量子率失真函数。我们展示了这种镜像下降算法与 Blahut-Arimoto 和先前用于解决信息论中类似问题的期望最大化方法的关系。使用这些技术，我们提出了第一个计算多量子位量子率失真函数的数值实验，并表明与现有方法相比，我们提出的算法求解速度更快且精度更高。