On the Capacity of Secure $K$-user Product Computation over a Quantum MAC

Inspired by a recent study by Christensen and Popovski on secure $2$-user product computation for finite-fields of prime-order over a quantum multiple access channel (QMAC), the generalization to $K$ users and arbitrary finite fields is explored. Combining ideas of batch-processing, quantum $2$-sum protocol, a secure computation scheme of Feige, Killian and Naor (FKN), a field-group isomorphism and additive secret sharing, asymptotically optimal (capacity-achieving for large alphabet) schemes are proposed for secure $K$-user (any $K$) product computation over any finite field. The capacity of modulo-$d$ ($d\geq 2$) secure $K$-sum computation over the QMAC is found to be $2/K$ computations/qudit as a byproduct of the analysis.