# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
Similarly, our solution profiles with 3 and 4 qubits on ibm_torino are shown in Figure 17. The results obtained using ibm_torino, which is based on the Heron processor comprising 133 fixed-frequency qubits, more closely resemble the analytical solution than ibm_sherbrooke. The Heron architecture delivers a performance improvement due to reduced crosstalk, faster and more accurate two-qubit gates, and improved qubit calibration [45]. Figure 18 displays the RMSE error for 1D Poisson equation for both ibm_sherbrooke and ibm_torino using 0.1 M shots. Our results indicate that the RMSE error for Variants 1 and 2 of our proposed algorithm is lower when run on ibm_torino than on ibm_sherbrooke due to its higher decoherence time owing to its improved architecture. Figure 16: Solution profile of the 1D Poisson equation obtained using the proposed technique with 3 and 4 qubits on real quantum hardware (ibm_sherbrooke) using 0.1 M shots (**a**) 8 data points (3 qubits) (**b**) 16 data points (4 qubits). Figure 9:0 Solution profile of the 1D Poisson equation obtained using the proposed technique with 3 and 4 qubits on real quantum hardware (ibm_torino) using 0.1 M shots (**a**) 8 data points (3 qubits) (**b**) 16 data points (4 qubits). #### 5.2.2 2D Poisson Equation Results

In this section, we present the results obtained by applying our proposed algorithm to a 2D Poisson equation on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware. Noise-Free and Noisy Simulation

The solution profile for the 2D Poisson equation using noise-free simulators is illustrated in Figure 19 for 8 qubits (256 data points) and Figure 20 for 14 qubits (16,384 data points). Here, the proposed approach is compared with both classical and VQE-based techniques. The results show higher accuracy compared to the VQE-based method. However, some results for Variant 1 and the VQE-based method are not shown in Figure 20 because they could not be calculated due to the excessively long computation times. Figure 21 presents the solution profile for the 2D Poisson equation on a noisy simulator using 1 M shots with 6 qubits (64 data points), while Figure 22 shows the solution profile

Figure 9:1 Comparison of solution profiles of the 2D Poisson equation between proposed and VQE-based techniques, tested using 256 data points (8 qubits) on noisy-free (statevector) simulator. Figure 9:2 RMSE error for the 1D Poisson equation using our proposed technique on real quantum hardware (ibm_sherbrooke and ibm_torino) using 0.1M shots. Some data points are omitted due to unrealistically long execution times. for 14 qubits (16,384 data points). The solution profile results indicate that the proposed approach has a higher accuracy compared to the VQE-based approach. Note that results for Variant 1 and the VQE-based method are not included in Figure 22 due to their very long execution times. We have also compared the proposed approach with the VQE-based technique in terms of RMSE error, see Figure 23a and the total time taken, see Figure 23b using noise-free and noisy simulators for 1 M shots. The results in terms of RMSE error and total time taken demonstrate higher accuracy and scalability, along with reduced total computational time. Figure 9:3 Comparison of solution profiles of the 2D Poisson equation between proposed and VQE-based techniques, tested using 64 data points (6 qubits) on noisy (AerSimulator) simulator (1 M shots). Figure 9:4 Comparison of solution profiles of the 2D Poisson equation between proposed and VQE-based techniques, tested using 16,834 data points (14 qubits) on noisy (AerSimulator) simulator (1 M shots). Hardware Emulation

We have also evaluated the 2D Poisson equation using the proposed algorithm on a hardware emulator (FakeSherbrooke). Figure 24 presents the solution profiles obtained using 6 qubits (64 data points). Figure 25a,b presents the RMSE error and total execution time, respectively, for the proposed method using 1 M shots.