Hierarchical quantum operation sharing of single- and two-qubit partially unknown quantum operations

In this paper, we deal with two problems of quantum operation sharing, which is the remote implementation of quantum operations on qubits held at distant places by one of the many possible receivers. It is a combination of quantum secret sharing and quantum operator teleportation. We perform it on single-qubit and two-qubit systems. The operations are partially unknown and are collected from specific sets of unitary operators. We use maximally entangled states as our quantum resources. Moreover, there is a hierarchy of receivers by which the securities of the protocols are enhanced. We have investigated the performance of one of our protocols under various quantum noise channels, including amplitude damping (AD), phase damping (PD), bit-flip (BF), phase-flip (PF) and depolarizing (DP) channels. The fidelity of the protocol is calculated for each noise type, showing maximum values of 1 in all cases, while the minimum fidelity varies: 0 for AD, BF and PF noise; 0.5 for PD noise; and 0.25 for DP noise. We have simulated one of our protocols on the Qiskit platform. A statistical analysis of the results obtained from the simulation indicates that the standard deviation of the probabilities \(P(|0\rangle )\) and \(P(|1\rangle )\) decreases with increased number of shots.