# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
Favorable solution profiles and RMSE error results have been achieved for up to 4 qubits. Beyond 4 qubits, the circuits generated by the proposed approach exceed the decoherence time limit of the hardware emulator (FakeSherbrooke). Figure 9:5 RMSE error and total execution time for the 2D Poisson equation using our technique versus the VQE-based technique on noise-free (statevector) and noisy (KerSimulator) simulators for 1M shots. Some data points are omitted due to unrealistically long execution times (a) RMSE error (b) Total time taken. Figure 9:6 Solution profile of the 2D Poisson equation obtained using the proposed technique with 6 qubits on hardware emulator (FakeSherbrooke) using 1 M shots. The solution profiles of the 2D Poisson equation obtained using 4 qubits on real quantum hardware (ibm_sherbrooke) using 0.1 M shots are shown in Figure 9:7

Similarly, Figure 27 shows the solution profiles obtained using 4 qubits on ibm_torino. Figure 28 presents the RMSE error for the 2D Poisson equation on ibm_sherbrooke and ibm_torino using 0.1M shots. In the case of the 2D Poisson equation experiments on real quantum hardware, the execution time of the circuits generated by Variant 2 of the proposed approach exceeds the decoherence time limit of the quantum hardware (ibm_sherbrooke) after 4 qubits. In the case of Variant 1, this limit is exceeded after 7 qubits. Figure 9:8 Solution profile of the 2D Poisson equation obtained using the proposed technique with 16 data points (4 qubits) on real quantum hardware (ibm_sherbrooke) using 0.1 M shots. Figure 9:9 Solution profile of the 2D Poisson equation obtained using the proposed technique with 16 data points (4 qubits) on real quantum hardware (ibm_torino) using 0.1 M shots. #### 5.2.3 3D Poisson Equation Results

In this section, we present the results obtained by applying our proposed algorithm to a 3D Poisson equation using noise-free and noisy simulators, as well as hardware emulators and real quantum hardware. Noise-Free and Noisy Simulation

The solution profile results obtained by our approach for 6 qubits (64 data points) and 12 qubits (4096 data points), respectively, on the noise-free simulator are shown in Figures 29 and 30. The results demonstrate the effectiveness of our technique, as the solution profiles closely match those produced by classical methods. We have also investigated the efficacy of our algorithm on noisy simulators. Figures 31 and 32 show the solution profile for 6 qubits (64 data points) and 12 qubits (4096 data points), respectively, using 1 M shots. Additionally, Figure 33a and Figure 33b compares the RMSE error and the total time taken, respectively, with the VQE-based approach. These results further confirm the high accuracy, reduced total time, and scalability of our approach. Figure 11:0 Comparison of solution profiles of the 3D Poisson equation between proposed and VQE-based techniques, tested using 4,096 data points (12 qubits) on noisy-free (statevector) simulator. Figure 11:1 Comparison of solution profiles of the 3D Poisson equation between proposed and VQE-based techniques, tested using 64 data points (6 qubits) on noisy-free (statevector) simulator. Figure 11:2 RMSE error and total execution time for the 3D Poisson equation using our technique versus the VQE-based technique on noise-free (statevector) and noisy (AerSimulator) simulators for 1M shots. Some data points are omitted due to unrealistically long execution times (**a**) RMSE error (**b**) total time taken. Figure 11:3 Comparison of solution profiles of the 3D Poisson equation between proposed and VQE-based techniques, tested using 64 data points (6 qubits) on noisy (AerSimulator) simulator (1 M shots). Figure 11:4 Comparison of solution profiles of the 3D Poisson equation between proposed and VQE-based techniques, tested using 4096 data points (12 qubits) on noisy (AerSimulator) simulator (1 M shots). We also investigated the efficiency of our proposed technique on the 3D Poisson equation using real quantum hardware.