# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 3 Multi-Party Controlled Cyclic Asymmetrical RSP Protocol of Sequentially Increasing Qubits States  
The controlled cyclic asymmetrical RSP protocol of sequentially increasing qubits states can be extended to multiparty scenario easily. Suppose there are n players \([PERSON]_{1}\), \([PERSON]_{2}\),... and \([PERSON]_{n}\), and controller [PERSON]. \([PERSON]_{i}\) can prepare an i-qubit state to his neighbor \([PERSON]_{i+1}\) under the control of [PERSON]. The entangled channel share among participants is \(\left|\varphi(\sum\limits_{i=1}^{n}{(i+1)+1)}\right\rangle\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Runs** & \(\left|\mathbf{000000}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{000001}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{000110}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{111000}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{111001}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{111110}\right\rangle_{\mathbf{234678}}\) & \(\left|\mathbf{111111}\right\rangle_{\mathbf{234678}}\) \\ \hline
1 & 0.125 & 0.117 & 0.126 & 0.118 & 0.118 & 0.138 & 0.130 & 0.128 \\ \hline
2 & 0.132 & 0.125 & 0.122 & 0.121 & 0.129 & 0.131 & 0.119 & 0.121 \\ \hline
3 & 0.120 & 0.128 & 0.124 & 0.127 & 0.129 & 0.118 & 0.126 & 0.127 \\ \hline
4 & 0.125 & 0.123 & 0.130 & 0.127 & 0.123 & 0.127 & 0.120 & 0.125 \\ \hline
5 & 0.132 & 0.123 & 0.129 & 0.122 & 0.124 & 0.118 & 0.125 & 0.127 \\ \hline
6 & 0.128 & 0.129 & 0.122 & 0.125 & 0.124 & 0.122 & 0.129 & 0.121 \\ \hline
7 & 0.124 & 0.126 & 0.124 & 0.129 & 0.122 & 0.119 & 0.132 & 0.123 \\ \hline
8 & 0.121 & 0.121 & 0.123 & 0.130 & 0.127 & 0.126 & 0.124 & 0.127 \\ \hline
9 & 0.120 & 0.131 & 0.122 & 0.129 & 0.121 & 0.125 & 0.120 & 0.132 \\ \hline
10 & 0.123 & 0.130 & 0.126 & 0.125 & 0.125 & 0.124 & 0.124 & 0.124 \\ \hline A & 0.125 & 0.1253 & 0.1248 & 0.1253 & 0.1242 & 0.1248 & 0.1249 & 0.1255 \\ \hline S & 0.0044 & 0.0044 & 0.0029 & 0.0039 & 0.0035 & 0.0063 & 0.0045 & 0.0039 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data analysis of the output state, where A is short for the average value, S is for standard deviation. Figure 5: Histogram of output results. (**a**) Histogram of output results after running once in the ‘ibmq_qsam_simulator’ quantum processor. The x-axis of this histogram is arranged in the order of \((8,\cdots 2,1)\) from the bottom to the top. (**b**) Histogram of output results after running ten times in Jupyter Notebook. The x-axis of this histogram is arranged in the order of \((c^{\prime\prime},b^{\prime\prime},\cdots 2,1)\) from the bottom to the top. \[ \begin{array}{l}\left|\rho(\sum\limits_{i=1}^{n}(i+1)+1)\right\rangle\\ =\frac{1}{\sqrt{2}}[(|0\cdots 0\rangle+|1\cdots 1\rangle)_{A_{n}^{1},A_{1}^{1}, \cdots,A_{1}^{n}}\otimes(|0\cdots 0\rangle+|1\cdots 1\rangle)_{A_{n-1}^{1},A_{n}^{2}, \cdots,A_{n}^{n}}\cdots\otimes(|0\cdots 0\rangle\\ +|1\cdots 1\rangle)_{A_{i}^{1},A_{i+1}^{2},\cdots A_{i+1}^{i+1}}\cdots\otimes(|00 \rangle+|111\rangle)_{A_{i}^{1},A_{2}^{2},A_{3}^{3}}\otimes(|00\rangle+|11 \rangle)_{A_{i}^{1+1},A_{2}^{2}}\otimes|0\rangle_{(\sum\limits_{i}^{n}(i+1) +1)}\\ +(|0\cdots 1\rangle-|1\cdots 0\rangle)_{A_{n}^{1},A_{1}^{1},\cdots,A_{1}^{n}} \otimes(|0\cdots 1\rangle-|1\cdots 0\rangle)_{A_{n-1}^{1},A_{n}^{2},\cdots,A_{n}^{n}} \cdots\otimes(|0\cdots 1\rangle\\ -|1\cdots 0\rangle)_{A_{i}^{1},A_{i+1}^{2},\cdots A_{i+1}^{i+1}}\cdots\otimes(|0 11\rangle-|100\rangle)_{A_{i}^{1},A_{2}^{2},A_{3}^{3}}\otimes(|01\rangle-|10 \rangle)_{A_{1}^{n+1},A_{2}^{2}}\otimes|1\rangle_{(\sum\limits_{i=1}^{n}(i+1) +1)}],\end{array}  \tag{21}\]  
where \(A_{1}^{n}\) represents the nth qubit owned by \([PERSON]_{1}\), \(A_{n}^{n}\) represents the nth qubit owned by \([PERSON]_{n}\), and \(A_{i+1}^{i}\) represents the \(i\)-th qubit owned by \([PERSON]_{i+1}\), and \(i\in\{1,2,\cdots,n\}\). [PERSON] holds control qubit labeled by \(\sum\limits_{i=1}^{n}(i+1)+1\). For example, if \(n=3\), the entangled channel shared should have the form as Equation (1).