## 4 Results
### Quantum State Tomography for Suzuki-Trotter Decomposition of Quantum Circuits  
In this section, Suzuki-Trotter decomposition is used to count the fidelity as a function of iterations of the quantum circuit. The circuit was iterated for \(n=1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), and \(10\). Quantum state tomography [54] was used to obtain the theoretical and experimental density matrices, thus calculating the fidelity [55; 56; 57] of the circuit as shown in Equation (18). \[F(\rho,\sigma)=tr\sqrt{\sqrt{\rho}\;\sigma\sqrt{\rho}} \tag{18}\]  
The general form of quantum state tomography is given by  
\[\rho=\frac{1}{2^{N}}\sum_{d=i_{1},i_{2},\ldots,i_{N}=0}^{2}T_{i_{1}i_{2}, \ldots,i_{N}}\sigma_{i_{1}}\otimes\sigma_{i_{2}}\otimes\ldots\sigma_{i_{N}} \tag{19}\]  
In particular, quantum state tomography reconstructs the density matrix of a quantum state by preparing the state many times and measuring each in a tomographically complete basis of measurement operators. In this paper, the circuit was measured with measurements of all qubits in each of the X, Y, and Z Pauli bases. For the 2-qubit system, these basis are II, IX, IY, IZ, XI, YI, ZI, XX, XY, XZ, YX, YY, YZ, ZX, ZY, and ZZ. The probabilities of the states were taken by the experiment in the IBM real superconducting chip- IBMq Lima and the density matrix was constructed by using the formula given in Equation (19). The experimental matrices of the circuits were computed through the density matrix as shown above. The experimental density matrix and the theoretical density matrix were compared and the fidelity of the quantum circuits was thus calculated. The whole circuit was simulated in \"IBMQ-Lima\". It is important to know the parameters of the qubit system that is used in the experimental process. These parameters are shown in Figure 10, which shows a map view of the qubits, T1 (relaxation time) and T2 (dephasing time), and the frequency of each qubit [58; 59]. Here, the T1 (or longitudinal) relaxation process involves a loss of energy of the qubit. It is a relaxation from the excited to the ground state (hence, longitudinal) and the corresponding timescale is the T1 time, while the T2 (or transverse) relaxation time or phase coherence time, describes how long the phase information of the superposition state of the qubit can be preserved. Probabilistic simulations and calculations for the quantum systems are usually carried out using the Monte Carlo method on a classical computer. Such methods were used to overcome the setback from the exponentially growing phase spaces. Meanwhile, a quantum simulator [60] is usually very much under the control of the experimenter, and it is capable of simulating the dynamical behaviors of the physical models. Irrespective of the degree of internal correlations or entanglement between the degrees of freedom of the model, it can perform its job efficiently [61]. In this section, the observations corresponding to the different circuits designed during quantum simulation of the pairing Hamiltonian in superconductivity are explained. Figures 4-7 display the theoretical and experimental density matrices, and the fidelity as a function of iterations for the Heisenberg, XY, transverse Ising, and longitudinal Ising models. The plots of the experimental density matrix of 2 and 10 iterations were selected to show the extremities of the iterations that were conducted. This shows how the density matrices are changing toward the theoretical density matrix as we increase the number of iterations. This increase is similar to the theoretical density matrix, as the iteration increases which is a consequence of the Trotter decomposition. It can be seen in Figure 4 that, for the Heisenberg model, the experimental density matrix deviates clearly from the theoretical density matrix. Changes in the experimental density matrix can also be seen as the number of iterations is increased. The fidelity was thus found to be oscillating until 10 iterations.