Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta \& et al., Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite program (SDP) grows exponentially with respect to the level of the hierarchy, and thus computing the SDP directly is inefficient. In this work, by exploiting the symmetries in the SDP, we show that, for fixed input and output dimensions, we can compute the SDP in polynomial time in term of level of the hierarchy. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $\epsilon$ in a time that is polynomial in $1/\epsilon$.

翻译：确定在噪声量子信道上传输量子信息的最优保真度是量子信息理论中的核心问题之一。最近，[Berta等人，《数学规划》，2021] 针对该量引入了一个渐近收敛的半定规划外层界层次结构。然而，半定规划的大小随层次级别呈指数增长，因此直接计算该半定规划效率低下。在本工作中，通过利用半定规划中的对称性，我们证明，对于固定的输入和输出维度，可以在层次级别的多项式时间内计算该半定规划。作为我们结果的直接推论，最优保真度可在关于$1/\epsilon$的多项式时间内以$\epsilon$的精度近似。