# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 5 Comparison and Conclusions  
Efficiency [39] is a main factor utilizing for weighing the superiority of RSP protocol. In comparison with the other cyclic RSP schemes, we calculate the scheme's efficiency (\(\eta\)) with the following manner. \[\eta=\frac{q_{i}}{q_{s}+q_{c}}, \tag{37}\]  
where \(q_{i}\) denotes the number of states to be transmitted, \(q_{s}\) indicates the quantum resource consumption, and \(q_{c}\) means the classical resource consumption. From Table 3, the following conclusions can be drawn:  
(1) The efficiency of our scheme is 32%, which is relatively higher than those of the schemes in [16; 23; 24; 25; 30; 32]. Besides, our scheme has a larger transmission capacity than the other schemes because the sequentially increasing qubits could carry more information via the transmission in the scheme. (2) Due to the influence of the real environment, the initial maximally entangled state easily evolves into a non-maximally entangled state or a mixed state. Compared with the [25], we employ a ten-qubit non-maximally entangled state as quantum channel which is more realistic than the maximally entangled state. (3) In comparison with [25], we increase the number of states to be transmitted in the four-party cyclic remote preparation scheme of an arbitrary single-particle state, an arbitrary two-qubit state and an arbitrary three-qubit state, which is more efficient and realistic. (4) Another remarkable advantage of our scheme is that the protocol can be employed to send multiple quantum information among multiple players either in an asymmetric manner. Compared with [16; 23; 24; 25; 30; 32], our scheme can realize the system having N \(>\) 3 observers. When the value of N is specified, we can construct anyone of all possible quantum channels. In this paper, the most significant innovation of the proposed scheme is reflected in realizing the remote state preparation with sequentially increasing qubits among multi-party, which is not only cyclically asymmetric but also bidirectional. We introduce a specific case of our scheme concretely, namely, the four-party controlled cyclic asymmetrical RSP protocol. Then the four-party controlled cyclic asymmetrical RSP protocol is demonstrated on the IBM quantum computer by designing appropriate quantum circuits using single-qubit and two-qubit quantum gates. The desired quantum states are reconstructed by adopting appropriate unitary transformations with a success probability of 100%. Furthermore, arbitrary qubit transmission channels with different circulation directions are designed and constructed. Considering the mutual coupling between the open quantum system and the environment, the feasibility of the scheme under different noise environments is analyzed respectively. In our subsequent research, we will consider a case where the input state is a mixed state and the output state is also a mixed state. Besides, the phase transition between the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Scheme** & **Type** & \(\mathbf{q_{i}}\) & \(\mathbf{q_{s}}\) & \(\mathbf{q_{c}}\) & **Operation** & **Efficiency** \\ \hline
[25] & CCJRSP & 3 (three arbitrary single-qubit) & 10 & 6 & 7 SM & 19\% \\ \hline
[32] & CJRSP & 3 (three arbitrary single-qubit) & 9 & 9 & 6 SM & 17\% \\ \hline
[30] & CCJRSP & 2 (An arbitrary two-qubit state) & 7 & 5 & 5 SM & 17\% \\ \hline
[24] & CBRSP & 4(Two arbitrary two-qubit ) & 9 & 8 & 9 SM & 24\% \\ \hline
[16] & CBRSP and QT & 2 (two arbitrary single-qubit) & 7 & 4 & 1 BSM\({}_{3}\) SM & 18\% \\ \hline
[23] & CCRSP & 3 (three arbitrary single-qubit) & 7 & 6 & 6 SM & 23\% \\ \hline Ours & CCARSP & 6(A single-qubit, a two-qubit, a three-qubit) & 13 & 6 & 7 SM & 32\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison with other RSP schemes, where BSM is short for Bell-state measurement and SM for single-qubit measurement. output state and the input state in the scheme can be discussed by simulating it in IBM Quantum Experience. Conceptualization, N.Z. and T.W. ; methodology, N.Z. and T.W. ; software, T.W. and Y.Y.