# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline  & \multicolumn{3}{c|}{**Variant 1**} & \multicolumn{3}{c|}{**Variant 2**} & \multicolumn{3}{c|}{**Decoherence**} \\ \cline{3-8} \multirow{-2}{*}{**Qubits**} & \multirow{-2}{*}{**Circuit Depth**} & **Expected** & **RMSE Error on** & \multirow{-2}{*}{**Circuit Depth**} & **Execution Time** & **Time** \\  & & **In Seconds** & **(0.1M shots)** & & **(1 Shorb)** & **(0.1M shots)** & **(0.1M shots)** \\ \hline
2 & 6336 & 0.000001408 & 0.020001663 & 6080 & 0.0000012351 & 0.0428295661 & 0.00017604 \\ \hline
3 & 20,384 & 0.00000350 & 0.28000118 & 109,472 & 0.0000042437 & 0.19088999 & 0.00017004 \\ \hline
4 & 51,848 & 0.000011477 & 0.370884508 & 609,654 & 0.000155848 & 0.00017504 \\ \hline
5 & 128,802 & 0.000028629 & 0.641007180 & 2,743,712 & 0.000069714 & 0.411210928 & 0.00017604 \\ \hline
6 & 276,160 & 0.000036369 & 0.410911148 & 11,453,920 & 0.020548316 & 0.744312075 & 0.00017604 \\ \hline
7 & 522,272 & 0.000116060 & 0.397430066 & 4,297,248 & 0.0102088277 & \
\begin{tabular}{c} Error \\ Ran too long \\ \end{tabular} &
\begin{tabular}{c} Error \\ Ran too long \\ \end{tabular} \\ \hline
8 & 1,057,504 & 0.000235001 & 0.5509288798 & 184,545,600 & 0.041010133 & Ran too long \\ \hline
9 & 2,026,848 & 0.000450411 & 0.441182415 & 745,152,864 & 0.165898525 &
\begin{tabular}{c} Error \\ Ran too long \\ \end{tabular} \\ \hline
10 & \begin{tabular}{c} Unrealistically \\ long compile time \\ \end{tabular} & \begin{tabular}{c} Unrealistically \\ long compile time \\ \end{tabular} & \begin{tabular}{c} Unrealistically \\ long compile time \\ \end{tabular} & \begin{tabular}{c} Unrealistically \\ long compile time \\ \end{tabular} & \
\begin{tabular}{c} Enrose \\ Ran too long \\ \end{tabular} &
\begin{tabular}{c} Enrose \\ Ran too long \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of circuit depth and execution times versus the number of qubits for Variant 1 and Variant 2 of the proposed technique on real quantum hardware (ibm_sherbrooke) for 1D Poisson equation. Execution time and Decoherence Constraint for 3D Poisson Equation

The circuit depth, expected execution time (1 shot), and RMSE error (0.1 M shots) for Variant 1 and Variant 2 of our proposed algorithm for the 3D Poisson equation are shown in Table 3. These results also exhibit a similar trend as the 1D and 2D Poisson equations. ## 6 Discussion

In this section, we discuss the results generated by our proposed algorithm designed to solve multidimensional Partial Differential Equations (PDEs). We have compared our work with the VQE-based PDE solver on both noise-free and noisy simulators. Additionally, we tested our algorithm on a hardware emulator (FakeSherbrooke) and real quantum hardware (ibm_sherbrooke and ibm_torino). ### Higher Accuracy

In this work, we compared the accuracy of our solution with classical and VQE-based methods in terms of RMSE error, using 1D, 2D, and 3D Poisson equations as case studies. We observed that the accuracy of our algorithm is comparable to that of classical solutions and significantly higher than that of the VQE-based approach on both noise-free (statevector) and noisy (AerSimulator) simulators. We presented our results using solution profiles and RMSE error plots. The main reason for the low accuracy in the VQE-based solution is its dependence on ansatz design [36], which often lacks sufficient expressibility [47]. This makes it inefficient to accurately approximate the solutions for multidimensional PDEs. Another reason for poor accuracy is the optimization process, which often encounters non-convex optimization landscapes [48]. We also tested Variant 1 and Variant 2 of our algorithm on a hardware emulator (FakeSherbrooke), and real quantum hardware (ibm_sherbrooke and ibm_torino) to assess its suitability for NISQ devices. With these tests, we observed that the accuracy of results remains good as long as the expected execution time of the circuits does not exceed the decoherence time of the hardware emulator or real quantum hardware.