# Four-Qubit Cluster States Generation through Multi-Coin Quantum Walk
## 2 Generating Cluster States through Multi-Coin Quantum Walk
### 4-Qubit Cluster States Generation
A brief procedure of QSDC based on our method can be described as follows:  
Figure 4: Generating cluster states through 4-step multi-coin quantum walk: (**a**) Quantum circuit generating cluster states and (**b**) simulation results. \[ \begin{split}\left|\phi^{+}\right\rangle&=\frac{1}{\sqrt{2}}( \left|00\right\rangle+\left|11\right\rangle)\to 00\\ \left|\phi^{-}\right\rangle&=\frac{1}{\sqrt{2}}( \left|00\right\rangle-\left|11\right\rangle)\to 11\\ \left|\psi^{+}\right\rangle&=\frac{1}{\sqrt{2}}( \left|01\right\rangle+\left|10\right\rangle)\to 01\\ \left|\psi^{+}\right\rangle&=\frac{1}{\sqrt{2}}( \left|01\right\rangle-\left|10\right\rangle)\to 10\\ \end{split}  \tag{16}\]  
Provided that the encoding scheme in Equation (16) is used by the sender [PERSON] and the receiver [PERSON]:  
Firstly, [PERSON] prepared an ordered sequence of Bell states of \(\left|\phi^{+}\right\rangle\), where every 3 Bell states are seen as a group. For each group, qubit 0, 2, and 4 are kept by [PERSON] while qubit 1, 3, and 5 are sent to [PERSON], creating sequence \(P_{024}\) and \(P_{135}\), respectively. After [PERSON] receives sequence \(P_{135}\), [PERSON] and [PERSON] carry out 4-step quantum walk using our method to generate cluster states between qubit 0, 2, 3, and 5. X-basis measurement are made on qubit 1 and 4, which serves as the first security check. If both measurement results turn out to be \(\left|0\right\rangle\), the channel is secure and the corresponding sequences \(P_{02}^{{}^{\prime}}\) and \(P_{35}^{{}^{\prime}}\) (the ordered sequences after measurements) are kept, otherwise they are discarded. Secondly, sufficient groups of qubits are chosen from \(P_{02}^{{}^{\prime}}\) as checking qubits and one of the operations in \(\left\{I,\sigma_{x},\sigma_{y},\sigma_{z}\right\}\) is performed on each of the checking qubit. The information to be sent is then encoded onto the remaining groups in \(P_{02}^{{}^{\prime}}\) and sent to [PERSON]. Extra disturbing qubits are added during this process and the positions of disturbing qubits and checking qubits are also told to [PERSON], as well as the operations carried out on checking qubits and the corresponding results. Next, [PERSON] removes the disturbing qubits after receiving the sequence \(P_{02}^{{}^{\prime}}\) and then measure the checking qubits and compare the result to what [PERSON] has told [PERSON], which is the second security check. If the results are the same, the channel is determined to be secure and the checking qubits can also be removed. Otherwise, the channel is discarded and another one needs to be established. Finally, [PERSON] can restore the information from cluster states in \(P_{02}^{{}^{\prime\prime}}\) and \(P_{35}^{{}^{\prime\prime}}\) (the sequences after second security check) according to the encoding scheme. Compared with original QSDC, cluster states can now be generated during the process rather than beforehand and QSDC can be performed without adding extra qubits as checking qubit or disturbing qubit for the first security check because there are qubits left (qubit 1 and 4) after generating the cluster states. In addition, only X-basis measurements are required in the first security check which slightly simplifies the process.