# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
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24:Compute physical step sizes in each dimension
25:\(\Delta_{x}\leftarrow\frac{X_{diff}}{(N_{x}-1)}\)
26:





## 5 Results

### Experimental Setup

This section is divided into three parts that focus on the description of the hardware, software, and datasets used to evaluate different multidimensional PDEs. #### 5.1.1 Hardware Testbed

In our experiments, we used both the noise-free (statevector) and noisy (AerSimulator) simulators to evaluate the accuracy, scalability, and total execution time of the proposed algorithm. Moreover, we have also used a hardware emulator (FakeSherbrooke) and real quantum hardware (ibm_sherbrooke and ibm_torino). Quantum system ibm_sherbrooke has 127 qubits with a decoherence time of 0.00017604 s, whereas ibm_torino has 133 qubits with a decoherence time of 0.001345 s. For all the experiments, we have used a cluster node from the University of Kansas, Lawrence, USA, equipped with a 48-core Intel Xeon Gold 6342 CPU, three NVIDIA A100 80 GB GPUs (CUDA version 11.7) with PCIe 4.0 connectivity, and 256 GB of 3200 MHz DDR4 RAM. #### 5.1.2 Software Frameworks

All the variants of the proposed algorithm were evaluated using Qiskit SDK (1.1.0) from IBM [26], utilizing the simulators, emulators, and real quantum hardware. As mentioned in Section 4.1, for the circuit synthesis of matrix \(A\), we have used BQSKit [22] in Variant 1 and QuantumCircuit.unitary() [40] function of the Qiskit [26] in Variant 2. In both variants, normalized \(\vec{b}\) is encoded using the highly depth-optimized classical-to-quantum (C2Q) encoding technique. For the VQE-based PDE solver experiments, we have used Qiskit SDK (0.46.0) from IBM [26] to perform tests on both noise-free and noisy environments. To convert the discretized PDE into its Pauli-based representation, the Qiskit MatrixOp [41] function is used. The ansatz used in the VQE-based PDE solver is EfficientSU2 [42] and the optimizer is ADAM [43]. #### 5.1.3 Test Case Datasets

As a case study, we tested the multidimensional Poisson equation, i.e., 1D, 2D, and 3D with the specific source functions and boundary values (SVs) as defined below:

* 1D Poisson Equation: A source function \(f(x)=10\), distance \(d=X_{max}-X_{min}=1\) with the boundary values \(u_{X_{-1}}=-0.6\) and \(u_{X_{X_{x}}}=-0.7\). * 2D Poisson Equation: A source function \(f(x,y)=xy\), distance \(d=1\) with boundary values (SVs) = 0, on both \(x\) and \(y\) axes. * 3D Poisson Equation: A source \(f(x,y,z)=xyz\), distance \(d=1\) with boundary values (SVs) = 0, in all three axes. ### Experimental Results

We benchmark our proposed algorithm with the multidimensional Poisson equation. In addition, we compared its solution profile, total execution times, and accuracy results to the VQE-based method on both noise-free and noisy simulators. Moreover, we have also obtained these results using a hardware emulator and real quantum hardware. #### 5.2.1 1D Poisson Equation Results

In this section, we present the results obtained by applying our proposed algorithm to the 1D Poisson equation using noise-free and noisy simulators, as well as hardware emulators and real quantum hardware. Noise-Free and Noisy Simulation

We conducted experiments on noise-free and noisy simulators ranging from 4 data points represented by 2 qubits up to 16,384 points represented by 14 qubits. The solution profile results from these experiments are displayed alongside the VQE-based approach in Figure 9 for 8 qubits and 14 qubits using the noise-free simulator. A strong correlation between the solution profile obtained using the noise-free simulator and the analytical solution indicates the accuracy of our proposed approach. Figure 10 presents the solution profiles obtained using our approach along with the VQE-based approach for 8 qubits and 14 qubits on a noisy simulator, with 1 M shots (samples). The disparity between our approach and the VQE-based approach is more evident from the solution profile results on the noisy simulator, further confirming the effectiveness of our technique.