Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an \qmarks{$N$-sum box}, a black box generalization of a two-sum protocol of Song \emph{et al.} with recent applications to $N$-servers private information retrieval. The $N$-sum box has communication cost of $N$ qudits and classical output of a vector of $N$ $q$-ary digits linearly dependent (via an $N \times 2N$ transfer matrix) on $2N$ classical inputs distributed among $N$ transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers.

翻译：量子多对一通信网络中的线性计算通过利用发送方之间的量子纠缠实现超密编码增益的方案，并结合干扰对齐等经典技术，为通信成本改进提供了机会。若能为基本量子功能找到合适的经典黑盒模型抽象，该问题的研究将变得更具普适性。本文以“$N$-求和盒”形式形式化这一抽象概念——它是Song等人双和协议的黑盒推广，并已应用于$N$服务器私有信息检索问题。$N$-求和盒的通信成本为$N$个量子比特，其经典输出为$N$个$q$进制数字构成的向量，该向量通过$N\times 2N$传输矩阵线性依赖于分布在$N$个发送方处的$2N$个经典输入。我们刻画了哪些传输矩阵可通过本文构造实现，并分别考虑发送方和接收方能否额外应用本地可逆经典操作的情形。