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 a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $\epsilon$ in $\mathrm{poly}(1/\epsilon, \text{input dimension})$ time.

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。近期，[Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] 引入了一种渐近收敛的半定规划层级结构，用于对该量给出外部界。然而，该半定规划问题的规模随层级阶数以指数增长，导致其计算不可扩展。在本工作中，通过利用半定规划中的对称性，我们证明：对于量子信道的固定输出维度，可以在关于层级阶数和输入维度的多项式时间内计算该半定规划问题。这一结果的一个直接推论是：最优保真度可以在 $\mathrm{poly}(1/\epsilon, \text{输入维度})$ 时间内达到 $\epsilon$ 精度近似。