Secret-shared teleportation of a two-qudit gate with a deferred recipient
An unknown quantum operation, like an unknown quantum state, cannot be perfectly copied or stored for later use. A specific gate can be embedded in an entangled resource and teleported, but its destination is fixed when the resource is created. Here we show that the capability to apply a two-qudit entangling gate can instead be secret-shared among many parties: the shares are distributed in advance, and a later key release lets exactly one of them, chosen after distribution, exercise it. A provider encrypts half of a Bell pair held with a user into $n$ shares, one per candidate node. The selected node, given the key qudits, completes an exact non-local $\mathtt{CSUM}^{k}$ gate with the user's data, in every measurement branch and for any qudit dimension, with the power $k$ chosen by the user alone. Any coalition of fewer than $n$ candidate nodes, or the key register alone, holds no information, unselected nodes cannot tell whether a selection occurred, and the key cannot decrypt twice. For two candidates the registers form a $((3,5))$ threshold scheme. Encoder and decoder are explicit $O(n)$ circuits. The construction supports jointly authorized and anonymous gate application, advancing distributed quantum computing and quantum network capabilities.