  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.
