# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
Figure 34 shows the solution profile obtained using the proposed algorithm for 6 qubits (64 data points) on quantum hardware ibm_sherbrooke. Similarly, Figure 35 shows the solution profile obtained using the proposed algorithm for 6 qubits (64 data points) on ibm_torino. The RMSE error obtained by solving the 3D Poisson equation using ibm_sherbrooke and ibm_torino is shown in Figure 11:5 The solution profile and the RMSE results on the hardware emulator (FakeSherbrooke) have a similar trend to real quantum hardware. Figure 11:6 Solution profile of the 3D Poisson equation obtained using the proposed technique with 64 data points (6 qubits) on real quantum hardware (ibm_sherbrooke) using 0.1 M shots. Figure 11:7 Solution profile of the 3D Poisson equation obtained using the proposed technique with 64 data points (6 qubits) on real quantum hardware (ibm_torino) using 0.1 M shots. Figure 11:8 RMSE error for the 3D Poisson equation using our proposed technique on real quantum hardware (ibm_sherbrooke and ibm_torino). Some data points are omitted due to unrealistically long execution times. ### Execution Time and Decoherence Constraint for Multidimensional Poisson on Real Quantum Hardware

In this section, we present the circuit depth, expected execution time (1 shot), and the RMSE error (0.1 M shots) for Variant 1 and Variant 2 of the proposed algorithm using a multidimensional Poisson equation as a case study on real quantum hardware (ibm_sherbrooke). These experimental results are generated by varying the number of qubits. The decoherence time used here is the median of the T2 times [24]. We have not presented similar results, such as circuit depth, expected execution time, and the RMSE error for ibm_torino because it lacks the pulse-level calibration data necessary for certain low-level quantum operations [46]. Execution time and Decoherence Constraint for 1D Poisson Equation

This section presents a comparison of circuit depth, expected execution time (1 shot), and the RMSE error (0.1M shots) for Variant 1 and Variant 2 of our proposed algorithm applied to 1D Poisson equation, see Figure 11:9 As expected, the output of the quantum circuits generated using the proposed algorithm should be affected by noise when the expected circuit execution time exceeds the decoherence time. However, the T2 time of some qubits on quantum hardware (ibm_sherbrooke) is significantly lower than the decoherence time (refer to Section 2.4). Due to this variation, the quantum circuits generated are affected by decoherence noise even before the expected execution time exceeds the decoherence time limit. As can be seen from Table 1, the quantum circuits generated using Variant 1 of our proposed algorithm are more depth-optimized than those generated using Variant 2 and hence are less affected by decoherence noise. Also, the expected execution time exceeds the decoherence time after 7 qubits in Variant 1 and after 4 qubits in Variant 2, resulting in a higher RMSE error for Variant 2. Execution time and Decoherence Constraint for 2D Poisson Equation

This section presents a comparison of circuit depth, expected execution time (1 shot), and the RMSE error (0.1M shots) for Variant 1 and Variant 2 of our proposed algorithm for 2D Poisson equation, see Figure 12:0 As shown in Table 2, the circuit depth and the expected execution time increase with the number of qubits. Also, the RMSE error, expected execution time, and circuit depth have a similar trend as that observed for the 1D Poisson equation.