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 $ε$ in $\mathrm{poly}(1/ε, \text{input dimension})$ time.

翻译：确定量子信息在有噪量子信道上传输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] 引入了该量的渐近收敛的半定规划层次外部逼近。然而，随着层次级别的增加，半定规划（SDP）的规模呈指数增长，导致其计算无法扩展。在本文中，通过利用SDP中的对称性，我们证明，对于量子信道的固定输出维度，可以在关于层次级别和输入维度的多项式时间内计算出SDP。作为我们结果的直接推论，最优保真度可在 $\mathrm{poly}(1/ε, \text{输入维度})$ 时间内以精度 $ε$ 逼近。