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 "$N$-sum box", a black box generalization of a two-sum protocol of Song \emph{et al.} with recent applications to $N$-server private information retrieval. The $N$-sum box has a 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. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe $N$-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.

翻译：量子多对一通信网络中的线性计算，可通过结合发射机间量子纠缠以实现超密编码增益的方案（如干扰对齐等经典技术）来降低通信开销。若能将底层量子功能通过经典黑箱模型转化为合适的抽象方法，此类问题将具有更广泛的普适性。本文以"$N$元求和盒"的形式形式化了一种抽象方法——该黑箱推广了Song等人提出的二元求和协议，并已应用于$N$服务器私有信息检索。$N$元求和盒的通信开销为$N$个量子比特，经典输出为一个由$N$个$q$进制数字构成的向量，该向量通过一个$N \times 2N$的传输矩阵线性依赖于分布在$N$个发射机上的$2N$个经典输入。我们刻画了在发射机与接收机处是否允许额外本地可逆经典操作的条件下，该构造所能实现的传输矩阵特征。此外，我们提供了跨子空间对齐方案的应用实例，以获取高效的量子私有信息检索与量子安全分布式批矩阵乘法实例。我们首先基于最大稳定子描述了$N$元求和盒，随后采用非最大稳定子构造方案实现量子对称私有信息检索实例。