# Four-Qubit Cluster States Generation through Multi-Coin Quantum Walk
## 3 Generating Cluster States Using Quantum Gates Formed by Multi-Coin Quantum Walk  
Multi-coin quantum walk can also construct quantum gates which are the basis of quantum circuit. Therefore, we take one step further to generate 4-qubit cluster states using quantum gates formed by quantum walk in order to introduce a more practical method of cluster state generation. It is the most common case to use Controlled-Z gates (CZ gates) to generate 4-qubit cluster states: 4 qubits with an initial state of \(\left|+\rangle_{1}\left|+\rangle_{2}\right|+\rangle_{3}\left|+\rangle_{4}\right.\) can be converted to 4-qubit cluster state through 3 CZ gates and 2 Hadamard gates, which is shown in Figure 5 and system states after each operation are described as below:\[|\Phi(1)\rangle =|+\rangle_{1}|+\rangle_{2}|+\rangle_{3}|+\rangle_{4} \tag{17}\] \[|\Phi(2)\rangle =U_{\text{CZ}}^{12}|\Phi(1)\rangle=\frac{1}{\sqrt{2}}(|0\rangle|+ \rangle+|1\rangle|-\rangle)_{12}|+\rangle_{3}|+\rangle_{4}\] (18) \[|\[PERSON](3)\rangle =U_{H}^{2}|\[PERSON](2)\rangle=\frac{1}{\sqrt{2}}(|0\rangle|0\rangle+| 1\rangle|1\rangle)_{12}|+\rangle_{3}|+\rangle_{4}\] (19) \[|\[PERSON](4)\rangle =U_{\text{CZ}}^{23}|\[PERSON](3)\rangle\] (20) \[=\frac{1}{2}(|0\rangle|0\rangle+|0\rangle|0\rangle|1\rangle+|1 \rangle|1\rangle|0\rangle-|1\rangle|1\rangle|1\rangle)_{123}|+\rangle_{4}\] \[|\Phi(5)\rangle =U_{H}^{4}U_{\text{CZ}}^{34}|\Phi(4)\rangle\] (21) \[=\frac{1}{2}(|0\rangle|0\rangle|0\rangle|0\rangle+|0\rangle|0 \rangle|1\rangle|1\rangle+|1\rangle|1\rangle|0\rangle|0\rangle-|1\rangle|1 \rangle|1\rangle)_{1234}\]  
Then, we introduce how to implement CZ gate and Hadamard gate using quantum walk. Previous studies [15] describe the 2-qubit and 3-qubit systems in detail, but only give a basic idea of the 4-qubit system. Here, we propose a complete and improved method of quantum gate implementation in 4-qubit system. The idea is to indicate qubits using computational basis and to simulate general computing gates using controlled conversion operators. For 4-qubit quantum walk of a single particle, the qubit 1 is considered as walker while the other 3 qubits describe the position in Hilbert space. Therefore, the process of 4-qubit quantum walk of a single particle can be mapped onto a 3D cube (shown in Figure 6) and qubit 1 (real particle) will perform quantum walk within 8 positions according to the operation to be implemented.  
As for the operators, the coin operator of the 4-qubit quantum walk is identical to the one of the 2-qubit or 3-qubit quantum walks [15], which are generally described as:  
Figure 5: Process of generating 4-qubit cluster state using several quantum gates.  
Figure 6: Mapping 4-qubit quantum walk onto a 3D cube.  
\[C(\epsilon,\alpha,\theta)=\begin{bmatrix}e^{i\epsilon}cos(\theta)&e^{i\alpha}sin( \theta)\\ e^{-i\alpha}sin(\theta)&-e^{-i\epsilon}cos(\theta)\end{bmatrix}, \tag{22}\]  
while shift operator \(S\) is more complicated. In order to cope with the extra qubit (extra dimension in position space), shift operator \(S\) is different according to different transition traces. If qubit 2 (first part of position) is fixed, in which case the state transition takes place in the front and back side of the cube in Figure 6, \(S=I\); if qubit 4 (third part of position) is fixed, in which case the state transition takes place in the left and right side of the cube in Figure 6, \(S=-I\). Therefore, we have  
\[\left\{\begin{aligned} S=I,& when\quad|000\rangle\to|001\rangle\to|011\rangle\to|010 \rangle\\ & and\quad|100\rangle\to|101\rangle\to|111\rangle\to|110\rangle\quad( trace1),\\ S=-I,& when\quad|000\rangle\to|010\rangle\to|110\rangle\to|100 \rangle\\ & and\quad|001\rangle\to|011\rangle\to|111\rangle\to|101\rangle\quad( trace2)\end{aligned}\right. \tag{23}\]  
Based on this, we can build the required CZ gate and Hadamard gate.