$N$-Sum Box: An Abstraction for Linear Computation over Many-to-one Quantum Networks

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.