# A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment
## 2 Four-Party Controlled Cyclic Asymmetrical RSP Protocol of Sequentially Increasing Qubits States
### Description of Four-Party Controlled Cyclic Asymmetrical RSP Protocol  
In this scheme, the four participants including three players and one controller, and each of the players acts both as a sender and as a receiver. From the schematic given in Figure 3, the procedure of the four-party controlled cyclic asymmetrical RSP protocol is detailed as follows:  
**Step 1**: [PERSON] selects an appropriate measurement basis according to the known quantum state to perform a single qubit measurement on his qubits. After completing the measurement operations, [PERSON] announces his outcomes to the [PERSON] via the classical channels. [PERSON] performs a single qubit measurement on qubit 9 based on the following basis:  
\[\left\{ \begin{array}{l}|\Phi^{+}\rangle_{9}=(a_{0}|0\rangle+a_{1}|1\rangle)_ {9}\\ |\Phi^{-}\rangle_{9}=(a_{1}|0\rangle-a_{0}|1\rangle)_{9}\end{array} \right. \tag{5}\]  
Figure 3: Diagram of four-party controlled cyclic asymmetrical RSP protocol, where SM is short for single-qubit measurement. Figure 2: The construction of the target channel. (**a**) Quantum circuit illustrating the construction of target channel. (**b**) The histogram shows theoretical and experimental results of the quantum state’s mean probability distribution. The x-axis of this histogram is arranged in the order of \((c^{\prime\prime},b^{\prime},\cdots 2,1)\) from the bottom to the top, showing the average probability and percentage error of each state in the target channel of 13 qubits. The measurement result has two possibilities, and the remaining qubits collapse with a probability of 0.5 to a new state as follows:  
(6)  
[PERSON] performs a single qubit measurement on an auxiliary qubit \(a^{\prime}\). For \({}_{9}\langle\Phi^{+}|\,\varphi^{{}^{\prime}}\rangle_{123456789,10,a^{\prime}b^ {\prime\prime}c^{\prime\prime\prime}}\), the basis are  
\[\left\{ \begin{array}{l}\left|\Psi^{+}\right\rangle_{a^{\prime}}=\frac{1}{ \sqrt{2}}(e^{-i\theta_{0}}|0\rangle+e^{-i\theta_{1}}|1\rangle)_{a^{\prime}}\\ \left|\Psi^{-}\right\rangle_{a^{\prime}}=\frac{1}{\sqrt{2}}(e^{-i\theta_{0}}|0 \rangle-e^{-i\theta_{1}}|1\rangle)_{a^{\prime}}.\end{array} \right. \tag{7}\]  
The remaining qubits collapse with a probability of 0.5 to a new state as follows:  
(8)  
For \({}_{9}\langle\Phi^{-}\,\left|\,\varphi^{{}^{\prime}}\right\rangle_{12345678910a^ {\prime}b^{\prime\prime}c^{\prime\prime\prime}}\), the basis are  
\[\left\{ \begin{array}{l}\left|\Omega^{+}\right\rangle_{a^{\prime}}=\frac{1}{ \sqrt{2}}(e^{-i\theta_{1}}|0\rangle+e^{-i\theta_{0}}|1\rangle)_{a^{\prime}} \\ \left|\Omega^{-}\right\rangle_{a^{\prime}}=\frac{1}{\sqrt{2}}(e^{-i\theta_{1}} |0\rangle-e^{-i\theta_{0}}|1\rangle)_{a^{\prime}}.\end{array} \right. \tag{9}\]  
The remaining qubits collapse with a probability of 0.5 to a new state as follows:  
(10)  
**Step 2:** [PERSON] selects an appropriate measurement basis according to the known quantum state to perform a single qubit measurement on his qubits. After completing the measurement, [PERSON] announces his outcomes to the [PERSON] via the classical channels. \[