## 2 Model
### Nearest-Neighbor Coupling Interactions in Superconductivity
The term \(e^{-i\hat{l}_{l}\hat{\sigma^{x}}\hat{\sigma^{z}}t}\) in the second part of the operator can be written as  
\[ \begin{split}& I\cos\;J_{l}t-\hat{\sigma^{x}}\otimes\hat{ \sigma^{x}}i\sin\;J_{l}t\\ &=\begin{bmatrix}\cos(J_{l}t)&0&0&-i\sin(J_{l}t)\\ 0&\cos(J_{l}t)&-i\sin(J_{l}t)&0\\ 0&-i\sin(J_{l}t)&\cos(J_{l}t)&0\\ -i\sin(J_{l}t)&0&0&\cos(J_{l}t)\end{bmatrix}\end{split}  \tag{9}\]  
Figure 5: (**a**) Theoretical density matrix for the XY model, (**b**) experimental density matrix for 2 iterations, (**c**) experimental density matrix for 10 iterations, and (**d**) fidelity as a function of iteration. The term \(e^{-i\int_{l}\dot{\sigma}^{y}\dot{\sigma}^{y}t}\) in the second part of the operator can be written as  
\[ \begin{split}&\begin{split} I\cos\ \int_{l}t-\dot{\sigma}^{y}\otimes\dot{\sigma}^{y}i\sin\ \int_{l}t\\ =\begin{bmatrix}\cos(J_{l}t)&0&0&i\sin(J_{l}t)\\ 0&\cos(J_{l}t)&-i\sin(J_{l}t)&0\\ 0&-i\sin(J_{l}t)&\cos(J_{l}t)&0\\ i\sin(J_{l}t)&0&0&\cos(J_{l}t)\end{bmatrix}\end{split}\end{split}  \tag{10}\]  
The term \(e^{-i\int_{l}\dot{\sigma}^{y}\dot{\sigma}t}\) in the second part of the operator can be written as  
\[ \begin{split}&\begin{split} I\cos\ \int_{l}t-\dot{\sigma}^{y} \otimes\dot{\sigma}^{y}i\sin\ \int_{l}t\\ =\begin{bmatrix}e^{-l\int_{l}t}&0&0&0\\ 0&e^{l\int_{l}t}&0&0\\ 0&0&e^{l\int_{l}t}&0\\ 0&0&0&e^{-l\int_{l}t}\end{bmatrix}\end{split}\end{split}  \tag{11}\]  
Figure 6: (**a**) Theoretical density matrix for the transverse Ising model, (**b**) experimental density matrix for 2 iterations, (**c**) experimental density matrix for 10 iterations, and (**d**) fidelity as a function of iteration. Now, referring to the four Hamiltonians, the time evaluation operator can be written and then proceed to implementation. Each of the four types of Hamiltonians is a special case of Equation (7) with its parameters properly chosen. The Hamiltonian can thus be reduced to the (i) longitudinal Ising Hamiltonian for parameters \(J_{l}^{x}=J_{l}^{y}=0\) and \(J_{l}^{z}=J_{l}\); (ii) transverse Ising Hamiltonian for parameters \(J_{l}^{y}=J_{l}^{z}=0\) and \(J_{l}^{x}=J_{l}\); (iii) XY Hamiltonian for parameters \(J_{l}^{z}=0\) and \(J_{l}^{x}=J_{l}^{y}=J_{l}\); and (iv) Heisenberg Hamiltonian for parameters \(J_{l}^{x}=J_{l}^{y}=J_{l}^{z}=J_{l}\). The following step is to implement the unitary operator in a quantum circuit as obtained from the previous calculations. So, the whole calculation can be elaborated as follows. Figure 7: (**a**) Theoretical density matrix for the longitudinal Ising model, (**b**) experimental density matrix for 2 iterations, (**c**) experimental density matrix for 10 iterations, and (**d**) fidelity as a function of iteration. In General  
The Hamiltonian is given by  
\[H=\sum_{l=1}^{N}\frac{1}{2}\omega_{l}\sigma_{l}^{z}+\sum_{l=1}^{N-1}\left(J_{l}^ {x}\sigma_{l}^{x}\sigma_{l+1}^{x}+J_{l}^{y}\sigma_{l}^{y}\sigma_{l+1}^{y}+J_{l}^ {z}\sigma_{l}^{z}\sigma_{l+1}^{z}\right) \tag{12}\]  
which is simplified for a 2-qubit system:  
\[H=\frac{1}{2}\omega_{1}\sigma_{1}^{z}+\frac{1}{2}\omega_{1}\sigma_{2}^{z}+ \left(J_{1}^{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{1}^{y}\sigma_{1}^{y}\sigma_{2}^ {y}+J_{1}^{z}\sigma_{1}^{z}\sigma_{2}^{z}\right) \tag{13}\]  
For the XX operation:  
The term \(\hat{\sigma}^{x}\hat{\sigma}^{x}\) of the Hamiltonian is to be decomposed to form the circuit. Let us consider the \(U3\) gate:  
\[ \begin{split} U3(\theta,\phi,\lambda)=\begin{bmatrix}\cos\theta/ 2&e^{-i\lambda}\sin\theta/2\\ e^{i\phi}\sin\theta/2&e^{i(\phi+\lambda)}\cos\theta/2\\ U3\left(\theta,\phi=-\frac{\pi}{2},\lambda=\frac{\pi}{2}\right)=\begin{bmatrix} \cos\theta/2&-i\sin\theta/2\\ -i\sin\theta/2&\cos\theta/2\end{bmatrix}\end{split}  \tag{14}\]  
Figure 8: (**a**) Time evolution of the different models at \(\frac{\theta}{2}\) for (**a**) Heisenberg, (**b**) XY, (**c**) transverse Ising, and (**d**) longitudinal Ising models. This gate satisfies the conditions for the unitary operator.