# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
A very small statistical noise as observed in Figure 10 can be further reduced by increasing the number of shots (samples). Figure 11 demonstrates that as we increase the shot count from 1 M to 32 M, the RMSE error further decreases. Figure 10: Comparison of solution profiles of the 1D Poisson equation between proposed techniques and VQE-based technique, tested using 8 and 14 qubits on noisy (AerSimulator) simulator (1 M shots). (a) 256 data points (8 qubits) (b) 16,834 data points (14 qubits). Figure 9: Comparison of solution profiles of the 1D Poisson equation between proposed techniques and VQE-based technique, tested using 8 and 14 qubits on a noise-free (statevector) simulator. (a) 256 data points (8 qubits) (b) 16,834 data points (14 qubits). We also compared the RMSE error and total time taken, see Figure 11(a),b, which further confirms the accuracy, scalability, and the reduced total time of our approach compared to VQE-based technique. Hardware Emulation

The results of our proposed approach on a hardware emulator are shown in Figure 13, with experiments using 3 qubits (8 data points) and 4 qubits (16 data points). Figure 13(a),b compares the RMSE error and total time taken by our proposed technique with 1M shots. We achieved accurate solution profile results up to 4 qubits with Variant 2. However, beyond 4 qubits, circuits generated by Variant 2 exceeded the decoherence time limit of the hardware emulator (FakeSherbrooke). In the case of Variant 1, we exceeded the decoherence time beyond 8 qubits. We also tested the hardware emulator approach with the Pauli Twirling [44] noise mitigation technique (1 M shots). The resulting RMSE and total computation time for solving the 1D Poisson equation are presented in Figure 14(a) and Figure 14(b), respectively. Applying Pauli Twirling noise mitigation to the hardware emulator (FakeSherbrooke) did not provide any perceptible benefits in terms of reducing the RMSE error. Our conclusion from these experiments is that noise mitigation techniques are not mature enough to significantly reduce errors and are still an active area of research. Figure 11: Comparison of solution profiles of the 1D Poisson equation between proposed technique and VQE-based technique, tested using 8 and 14 qubits on noisy (AerSimulator) simulator (32 M shots) (**a**) 256 data points (8 qubits) (**b**) 16,834 data points (14 qubits). Figure 12: RMSE error and total execution time for the 1D Poisson equation using our technique versus the VQE-based technique on noise-free (statevector) and noisy (AerSimulator) simulators (1 M shots). Some data points are omitted due to unrealistically long execution times. (**a**) RMSE error (**b**) total time taken. Figure 14: RMSE error and total execution time for the 1D Poisson equation using our proposed technique on quantum emulator (FakeSherbrooke) using 1 M shots. Some data points are omitted due to unrealistically long execution times. (**a**) RMSE error (**b**) Total time taken. Figure 13: Solution profile of the 1D Poisson equation obtained using the proposed technique with 3 and 4 qubits on hardware emulator (FakeSherbrooke) using 1 M shots (**a**) 8 data points (3 qubits) (**b**) 16 data points (4 qubits). Figure 15: RMSE error and the total time taken by the proposed technique for 1D Poisson equation on hardware emulator with noise mitigation (Pauli Twirling) for 1 M shots (**a**) RMSE error (**b**) total time taken. The proposed approach solution profile results of the 1D Poisson equation for real quantum hardware (ibm_sherbrooke) are presented in Figure 16. The results, shown for 3 qubits (8 data points) and 4 qubits (16 data points) on ibm_sherbrooke closely match those obtained with the hardware emulator (FakeSherbrooke). Up to 4 qubits, we achieved favorable results but exceeded the decoherence time limit of ibm_sherbrooke in Variant 2 beyond 4 qubits. However, in Variant 1, we exceeded the decoherence time limit after 7 qubits.