Enhancing multi-hop remote implementation of partially unknown operations under the noisy environment

Quantum multi-hop network plays a crucial role in expanding the communication distance, improving network flexibility and boosting reliability. In view of the practical requirements of quantum remote control, we put forward two deterministic multi-hop protocols for implementing single-qudit operations belonging to the restrict sets. One uses generalized Bell states as the entanglement resource, and the other takes generalized GHZ-Bell states as the quantum channel. With the assistance of the intermediate nodes, the sender can execute the partially unknown operation on the distant receiver’s state. It should be emphasized that the recovery operation is derived from a general expression that explicitly indicates the dependence on the measurement outcomes. Furthermore, we explore the impact of amplitude-damping noise and enhance the fidelities by means of environment-assisted measurement.