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-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for fixed input and output dimensions of the given quantum channel, we can compute the SDP in polynomial time in terms of the 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的核心问题之一。近期，[Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] 引入了一种渐近收敛的半定规划层次结构，用于该量的外部界。然而，半定规划（SDP）的规模随层次级别呈指数增长，导致计算难以扩展。在本工作中，通过利用SDP中的对称性，我们证明了对于给定量子信道的固定输入和输出维度，可以在多项式时间内按层次级别计算该SDP。作为我们结果的直接推论，最优保真度可以在$1/\epsilon$的多项式时间内以$\epsilon$精度逼近。