# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
### Total Execution Time and Scalability

In our comparison of the proposed algorithm with the VQE-based solution in terms of total execution time on both noise-free (statevector) and noisy (AerSimulator) simulators, \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{3}{*}{**Qubits**} & \multicolumn{2}{c|}{**Variant 1**} & \multicolumn{2}{c|}{**Variant 2**} \\ \cline{2-7}  & & **Expected** & **RMSE Error on** & & **Expected** & **Mass Error on** & **Time** \\  & **Circuit Depth** & **Execution Time** & **Isom.** & **-sharebrooke** & **Circuit Depth** & **Execution Time** & **isom.** & **-sharebrooke** \\  & **(1 Shab)** & **(0.1M Shot)** & **(0.1M Shot)** & **(0.1M Shot)** & **(0.1M Shot)** & **(0.1M Shot)** & **(0.1M Shot)** \\ \hline
4 & 45,568 & 0.0000126 & 0.00236176 & 605,376 & 0.000134528 & 0.00481892 & 0.00017064 \\ \hline
6 & 250,048 & 0.00004645 & 0.00005658 & 11,46,921 & 0.000057314 & 0.000050157 & 0.00017064 \\ \hline
8 & 1,019,616 & 0.000226881 & 0.0003629968 & 183,185,902 & 0.0040707989 & \
\begin{tabular}{c} Error: \\ \end{tabular} &
\begin{tabular}{c} Error: \\ \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} 2,988,240,992 \\ long.compile time \\ \end{tabular} & \begin{tabular}{c} 2,988,240,992 \\ long.compile time \\ \end{tabular} & \
\begin{tabular}{c} 0.664053554 \\ \end{tabular} &
\begin{tabular}{c} Error: \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: 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 2D Poisson equation. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{3}{*}{**Qubits**} & \multicolumn{2}{c|}{**Variant 1**} & \multicolumn{2}{c|}{**Variant 2**} & \multicolumn{2}{c|}{**Decoherence**} \\ \cline{2-7}  & \multirow{3}{*}{**Circuit Depth**} & \begin{tabular}{c} **Expected** \\ **Execution Time** \\ **(1 Shab)** \\ \end{tabular} & \begin{tabular}{c} **RMSE Error on** \\ **(1 Shab)** \\ \end{tabular} & \begin{tabular}{c} **Fixs** **Error on** \\ **(1 Shab)** \\ \end{tabular} & \
\begin{tabular}{c} **Fixs** **Error on** \\ **(1 Shab)** \\ \end{tabular} &
\begin{tabular}{c} **Time** \\ **(1 Shab)** \\ \end{tabular} \\ \hline
6 & 250,712 & 0.000057714 & 0.001817507 & 11,428,844 & 0.002350876 & 0.00237605 & 0.00017064 \\ \hline
9 & 2,000,192 & 0.0004444857 & 0.002350428 & 725,210,400 & 0.163380089 &
\begin{tabular}{c} Error: \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 3: 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 3D Poisson equation. we found that our algorithm requires significantly less time than the VQE-based approach for solving the 1D, 2D, and 3D Poisson equations. This reason for exceptionally long execution time in variational-based algorithms is due to non-convergence issues caused by vanishing gradients, commonly referred to as _barren plateaus_. We also observed that Variant 2 executes faster than both Variant 1 and the VQE-based approach and has a scalability of up to 14 qubits. In the case of Variant 1, our scalability is only until 9 qubits, after which it takes exceptionally long execution times. Furthermore, we observed that the latest IBMQ system, ibm_torino, has a much higher decoherence time of 0.001345 s compared to ibm_sherbrooke, which has a decoherence time of 0.00017604 s. Since the architecture design of ibm_torino is much more immune to gate noise and environmental noise and has a 7.6\(\times\) higher decoherence time compared to ibm_sherbrooke, the RMSE error generated by ibm_torino is less than that generated by ibm_sherbrooke as shown in Figures 16-18 for 1D results, Figures 26-28 for 2D results, and Figures 34-36 for 3D results.