## 4 Results
### Quantum State Tomography for Suzuki-Trotter Decomposition of Quantum Circuits
This change corresponds to the increase in probability of a state as the number of iterations increases. This also took a toll on the fidelity of the system as a whole. The error for every iteration was checked and a graph was plotted to check the trend of variation. The variation of fidelity in the system shows the noise in the existing quantum processor. The average fidelity obtained for the system was around 0.55 for the Heisenberg model. In Figure 5, for the XY model, it was seen that the experimental density matrix changed as the number of iterations increased. This is again due to the change in probability of occurrence of each state at a certain iteration. As per the fidelity vs. iteration plot, the fidelity increased from 1 to 2 iterations, then decreased until 4 iterations. The fidelity then increased from 5 iterations until 7 iterations, decreased at 8 iterations, and again increased at 9 iterations and then decreased afterward. The graph shows the trend of these changes. Variations in noise in the system may give such erratic results, although an average value of the fidelity can be obtained to be within 0.5-0.55. For the transverse Ising model (Figure 6), the experimental density matrix figures also changed with increasing number of iterations. The fidelity of the circuit increased from 1 iteration to 10 iterations but showed an erratic pattern. The average value of fidelity for this model was within 0.5-0.6. Finally, for the longitudinal Ising model (Figure 7), the experimental density matrix changed with increasing number of iterations from 2 to 10. The fidelity of the circuit followed the same pattern as the other Hamiltonians, showing an erratic pattern. The average fidelity fell between 0.4-0.55 for this model. The variation in the fidelity shown for the four models can be attributed to the inclusion of noise from the real chip quantum computer. The noise parameter followed a periodic pattern which affected the fidelity of the circuits, thus showing a near periodic variation in the fidelity. Other than the noise parameter which certainly affected the fidelity, there were a few more findings which we could interpret from that. As per the Trotter decomposition, as we increase the number of iterations, the fidelity of the circuit should increase. But again, in the real experimental circuit inclusion of higher numbers of gates as we increased the number of iterations, it would affect the fidelity so that it decreases. The fluctuations in the fidelity that we see is the effect of both of these parameters. It should also be mentioned that the data points in the graph are low, and thus we can see such fluctuations. But if  
Figure 10: Properties of the quantum processor used: (**a**) map view of the IBMQ Lima quantum processor, (**b**) T1 (thermal relaxation time) of the qubits from qubit 0–4, (**c**) T2 (dephasing time) of the qubits from qubit 0–4, and (**d**) frequency of operation of the different number of qubits. we increase the iterations to higher values to fit the graph, we would see that the fidelity would decrease and converge to zero. In theory, if the number of gates increases in a circuit due to inclusion of more noise, it is expected that the fidelity will decrease. But in the experimental regime, this trend may be affected and can also give a higher fidelity for systems with more gates, as seen in some of the models above. Figure 8 shows the variation in probability of possible states with respect to different values at \(\theta=\frac{\pi}{2}\) and as the time varies for \(t=0,\ldots,4\). For the Heisenberg model, it can be seen that the state \(|01>\) is dominant at \(t=0\) s, which decreases to the lowest level at \(t=4\) s. For the XY model, the dominant state is \(|01>\) at time \(t=0\) s and state \(|00>\) at \(t=4\) s. For the transverse Ising model, the dominant state at \(t=0\) s is \(|00>\) while at \(t=4\) s, the dominant state is \(|01>\). For the longitudinal Ising model, the dominant state at \(t=0\) s is \(|00>\) while at \(t=4\) s, the dominant state is \(|00>\).