# Four-Qubit Cluster States Generation through Multi-Coin Quantum Walk
## 2 Generating Cluster States through Multi-Coin Quantum Walk
### 4-Qubit Cluster States Generation  
The 4-qubit cluster states can be described in the form below [17]:  
\[|\psi\rangle_{0123}=\frac{1}{2}(|0000\rangle+|0011\rangle+|1100\rangle-|1111 \rangle)_{0123} \tag{8}\]  
Based on the GHZ state achieved above, we can introduce another pair of Bell state to obtain 4-qubit cluster state through quantum walk, the process of which is shown in Figure 3, where the circle represents the qubit, and the straight line represents the entanglement. Again, the result is determined and we want to form cluster state between qubit 0, 2, 3, and 5 through multi-coin quantum walk in a 2-complete graph. The initial state of the system is composed of 3 pairs of Bell states and can be described as:  
\[|\psi(0)\rangle=\frac{1}{2\sqrt{2}}(|00\rangle+|11\rangle)_{01}(|00\rangle+|11 \rangle)_{23}(|00\rangle+|11\rangle)_{45}. \tag{9}\]  
We first repeat the 2-step quantum walk mentioned above to generate GHZ state (11) and (12), during which qubit 0 serves as a walker and qubit 1 and 2 as coins in turn with \(C_{1}=I\) and \(C_{2}=I\). Then, the newly introduced Bell state steps in. In the third step, we take qubit 5 as walker and qubit 4 as coin described as \(C_{3}=I\). In the fourth step, we also take qubit 5 as walker but qubit 0 as coin with \(C_{4}=H_{0}\), where \(H_{0}\) is Hadamard gate performed on qubit 0:  
\[H_{0}=\frac{1}{\sqrt{2}} \begin{bmatrix}1&1\\ 1&-1\end{bmatrix}  \tag{10}\]  
For the system state, each step of quantum walk can be described as:  
\[|\psi(1)\rangle =\frac{1}{2\sqrt{2}}(|00\rangle+|01\rangle)_{01}(|00\rangle+|11 \rangle)_{23}(|00\rangle+|11\rangle)_{45} \tag{11}\] \[|\psi(2)\rangle =\frac{1}{2\sqrt{2}}(|0\rangle+|1\rangle)_{1}(|000\rangle+|111 \rangle)_{023}(|00\rangle+|11\rangle)_{45}\] (12) \[|\psi(3)\rangle =\frac{1}{2\sqrt{2}}(|0\rangle+|1\rangle)_{1}(|000\rangle+|111 \rangle)_{023}(|00\rangle+|01\rangle)_{45}\] (13) \[|\psi(4)\rangle =\frac{1}{4}(|0000\rangle+|0011\rangle+|1100\rangle-|1111 \rangle)_{0235}|+\rangle_{1}|+\rangle_{4} \tag{14}\]  
After four steps of quantum walk, we perform X-basis measurement on qubit 1 and 4 and a 4-qubit cluster state between qubit 0, 2, 3, and 5 are achieved using only Bell states. Same as GHZ states, we also use IBM Quantum Experience to verify our method. The corresponding operations are implemented in quantum circuit and shown in Figure 4. We can see that a 4-qubit cluster state are obtained between q[0], q[2], q[3], and q[5]. Figure 3: Process of generating 4-qubit cluster state with Bell states through quantum walk. In this part, 4-qubit cluster states are generated by multi-coin quantum walk in a 2-complete graph using only Bell states and local measurements. This result can also prove that we can generate higher-dimensional entangled qubits following the the same method by adding more pairs of Bell state. Take 5-qubit cluster state as an example: In order to achieve the entangled state \(|\psi\rangle=\frac{1}{2}(|00000\rangle+|00111\rangle+|11101\rangle+|11010\rangle)\), a possible 8-step quantum walk can be implemented using 4 pairs of Bell states. The initial state of the system is  
\[|\psi(0)\rangle=\frac{1}{4}(|00\rangle+|11\rangle)_{01}(|00\rangle+|11\rangle)_ {23}(|00\rangle+|11\rangle)_{45}(|00\rangle+|11\rangle)_{67} \tag{15}\]  
Firstly, we take qubit 0 as walker and qubit 2 and 3 as coins in turn with \(C_{1}=I\) and \(C_{2}=I\). Then, Hadamard gate \(C_{3}=H\) is performed on qubit 0. Next, we take qubit 5 as walker and qubit 3, 4 and 2 as coins, in turn, with \(C_{4}=I\), \(C_{5}=I\), and \(C_{7}=I\). Finally, we take qubit 6 as walker and qubit 7 and 3 as coins with \(C_{6}=I\) and \(C_{8}=I\). The process will become more complicated as the dimension goes higher but the advantage of using only Bell states and local measurements remain the same. This method can also be used in some applications to simplify the process, for example quantum secure direct communication (QSDC) [18].