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In this paper, a four-party scheme is presented for remotely sharing a single-qubit operation with the help of LOCC and the generalized seven-qubit Brown state [Formula: see text]. Besides, we consider a theoretical scheme for four-party QOS by using the generalized [Formula: see text]-qubit Brown state [Formula: see text] proposed by Muralidharan and Panigrahi (Phys. Rev. A 77 (2008) 032321). Furthermore, some concrete discussions are made to study its important features, including the scheme determinacy, the sharer symmetry, the scheme security, the nowaday’s experimental feasibility as well as the intrinsic efficiency.

A new concept of bidirectional quantum operation teleportation (BQOT) is proposed, which is essentially a union of the idea of quantum operation teleportation (QOT) and bidirectional quantum teleportation (QT). This means that Alice can transmit an unknown single-qubit unitary operation [Formula: see text] on the remote Bob’s quantum system; and at the same time, Bob can also transmit an arbitrary single-qubit unitary operation [Formula: see text] on Alice’s quantum system. In this paper, we present three BQOT schemes via different quantum channels. Furthermore, using these quantum channels, we investigate the BQOT in noisy environments, such as phase-damping noise and amplitude-damping noise. We considered the influence of the noises on the process of these three BQOT protocols through analytical derivation of the fidelity. Moreover, these three schemes are amply compared with each other from five aspects, i.e. quantum resource consumption, operation complexity, classical resource consumption, success probability and efficiency. It is found that these schemes can be realized deterministically and the first scheme is better than the other two schemes.

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