# Four-Qubit Cluster States Generation through Multi-Coin Quantum Walk
## 3 Generating Cluster States Using Quantum Gates Formed by Multi-Coin Quantum Walk
### Hadamard Gate  
In order to build Hadamard gate, conditional shift operator should be divided into two parts \(S_{-}\) and \(S_{+}\) to move the particle in left and right directions, which can be described as [15]  
\[ \begin{split} S_{-}^{i}&=\sum_{m\in Z,j}(|i\rangle \langle i|\otimes|m-1\rangle\langle m|+|j\
eq i\rangle\langle j\
eq i|\otimes |m\rangle\langle m|),\\ S_{+}^{j}&=\sum_{m\in Z,j}(|j\rangle\langle j| \otimes|m+1\rangle\langle m|+|i\
eq j\rangle\langle i\
eq j|\otimes|m \rangle\langle m|),\end{split}  \tag{26}\]  
where \(|i\rangle,|j\rangle\in\{|0\rangle,|1\rangle\}\) are the basis of Hilbert space and \(m\in\mathbb{Z}\) indicates the possible position in Hilbert space (in our case \(m\in[0,7]\)). The overall transition operator of Hadamard gate is  
\[H_{1}=I(C(0,0,\pi/4)\otimes I_{p}) \tag{27}\]  
where \(C(0,0,\pi/4)\) is the coin operator given in Equation (22). The Hadamard gate is implemented by combining \(S_{-}\) and \(S_{+}\) differently to evolve the coin state of the particle in superposition of position space which is composed of qubit 2, 3, and 4. It is a position-dependent coin operation followed by shift operators. The implementation of Hadamard gate also depends on the qubit it is performed on. For \(U_{H}^{2}\) (as shown in Figure 8a), the 8 positions in Hilbert space are marked as:  
\[ \begin{split}|000\rangle\to l&=0,\quad|100 \rangle\to l=4\\ |001\rangle\to l&=1,\quad|101\rangle\to l=5\\ |011\rangle\to l&=2,\quad|111\rangle\to l=6\\ |010\rangle\to l&=3,\quad|110\rangle\to l=7\end{split}  \tag{28}\]  
The quantum walk operations to be carried out following trace 1 (\(l=0\to 1\to 2\to 3\) or \(l=4\to 5\to 6\to 7\), shown as black arrows in Figure 8a) are:  
\[ \begin{split} H_{+}^{0}|i\rangle\otimes|m\rangle&= \sigma_{x}^{m}S_{+}^{i}(\sigma_{x}\otimes I)\\ H_{+}^{1}|i\rangle\otimes|m\rangle&=\sigma_{x}^{m}S_ {+}^{i}(\sigma_{z}\otimes I)\\ H_{-}^{0}|j\rangle\otimes|m\rangle&=\sigma_{x}^{m}S_ {-}^{i}(\sigma_{x}\otimes I)\\ H_{-}^{1}|j\rangle\otimes|m\rangle&=\sigma_{x}^{m}S_ {-}^{i}(\sigma_{z}\otimes I)\end{split}  \tag{29}\]  
Figure 7: Implementation of (**a**) \(U_{\text{ZZ}}^{12}\), (**b**) \(U_{\text{CZ}}^{23}\) and (**c**) \(U_{\text{CZ}}^{34}\) through 4-qubit quantum walk. Qubit 2, 3 and 4 are used to determine the 8 positions in Hilbert space and actual CZ operation is performed on qubit 1. The operator on the arrow indicates the operation performed on the walker if the walker occupied the starting position of the arrow. The black arrows correspond to trace 1 while the red arrows correspond to trace 2.  
where \(\sigma_{x}^{m}=\sigma_{x}\otimes|m\rangle\langle m|+I\otimes\sum\
olimits_{l\
eq m }|l\rangle\langle l|\). For \(U_{H}^{4}\) (as shown in Figure 8b), the 8 positions in Hilbert space are marked as:  
\[ \begin{split}|000\rangle\to l=0,&|001\rangle \to l=4\\ |010\rangle\to l=1,&|011\rangle\to l=5\\ |110\rangle\to l=2,&|111\rangle\to l=6\\ |100\rangle\to l=3,&|101\rangle\to l=7\\ \end{split}  \tag{31}\]  
The quantum walk operations to be carried out following trace 1 (\(l=0\to 4\to 5\to 1\) or \(l=3\to 7\to 6\to 2\), shown as black arrows in Figure 8b) are the same as Equation (30). Although the quantum walk to be carried out following trace 2 (\(l=0\to 1\to 2\to 3\) or \(l=4\to 5\to 6\to 7\), shown as red arrows in Figure 8b) are the same as Equation (29). Now we have successfully mapped the Hadamard gate to 8 positions in Hilbert space. Same as above, the directions of the arrows result from the shift operator and only indicate the movement of the walker in position space rather than state conversion. The actual operation on the cube of Hadamard gate is similar to the one of CZ gate. Position-dependent coin operator \(C\) and shift operator \(S\) will be determined by not only the position of the walker and the trace it chooses but also the targeted qubit. After the operation, Hadamard gate is performed on qubit 1 (walker), and it will move to next position according to the shift operator.