Deterministic hierarchical quantum operation sharing with five-qubit partially entangled states

To study quantum remote control in the form of hierarchical quantum sharing, by using a Hadamard gate, control-NOT gates and rotation gates, we construct a five-qubit partially entangled state as quantum channels of quantum remote control. By integrating the ideas of hierarchical quantum state sharing and quantum operation teleportation, we put forward two novel schemes for quadripartite hierarchical sharing a single-qubit quantum operation on a qubit in any sharer’s site with the help of the local operations and classical communication. In each scheme, there is a hierarchy among the receivers concerning powers to reconstruct the conceivable state. Owing to various unitary operations and projective measurements, the unit success probability can always be achieved irrespective of the parameters of the pre-shared partially entangled state as quantum channel. The first scheme is applicable for arbitrary unitary operations, while the other one is valid only if the operation \(\mathcal {U}_d\) (\(d=0,1\)) in question is known to belongs to some restricted sets. Consequently, the latter scheme is more economical in terms of quantum and classical resources, and the local operation complexity involved is also reduced.