# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
Also, we observed that the circuits generated using Variant 2 have significantly greater depth than those generated using Variant 1. Due to this, Variant 2 is more susceptible to decoherence noise compared to Variant 1. These findings suggest that as quantum hardware technology advances, both the accuracy and scalability of our proposed algorithm will also improve on real quantum hardware. ### Complexity Comparison of Proposed Approach with Classical Technique

We also conducted a circuit depth analysis of both variants of our proposed algorithm and compared it with the time complexity of the best-known classical technique [49]. Variant 1 of our proposed algorithm consists of Classical-to-Quantum (C2Q) [21] encoding and a numerical instantiation technique [22]. We have used the numerical instantiation proposed by BQSKit [22], which is a highly depth-optimized technique. Since we could not find any generalized depth complexity value for the circuits generated using BQSKit, we obtained its value through experiments using various test cases. The depth complexity of the circuit generated using BQSKit is \(\mathcal{O}(N)\), where \(N=2^{n}\) represents the number of states and \(n\) is the number of qubits. The depth complexity of the C2Q encoding technique is \(\mathcal{O}(N)\)[21]. The overall depth complexity of Variant 1 of the proposed technique is \(\mathcal{O}(N)+\mathcal{O}(N)\approx\mathcal{O}(N)\). Variant 2 of our proposed algorithm consists of C2Q encoding and Column-by-Column Decomposition (CCD) [23]. The depth complexity of CCD is \(\mathcal{O}(N^{2})\)[23], and the depth complexity of C2Q is \(\mathcal{O}(N)\). Therefore, the overall depth complexity of Variant 2 is \(\mathcal{O}(N)+\mathcal{O}(N^{2})\approx\mathcal{O}(N^{2})\). Table 1 shows that the depth of Variant 1 after around 6 qubits roughly doubles with each additional qubit. This doubling behavior corresponds to the overall depth complexity of \(\mathcal{O}(N)\). In contrast, for Variant 2, the depth grows by fourfold for every additional qubit after around 6 qubits, indicating an overall depth complexity of \(\mathcal{O}(N^{2})\). In our analysis of the time complexity of solving a multidimensional Poisson equation using the most optimized classical computing approach of Fast Fourier Transform (FFT)-based solvers, the time complexity number is \(\mathcal{O}(NlogN)\)[49], where \(N\) is the number of grid points. However, with ongoing research in data encoding techniques and continuously improving compilers, we anticipate that the depth of the quantum circuits produced by the synthesis algorithms will be further optimized. ### FDM Coefficient Matrix for Multidimensional Poisson Equation

We have also provided the mathematical description and procedure to convert the continuous Poisson equation into a finite difference equation for up to four dimensions. Additionally, we have also included an algorithm to compute the coefficient matrix and the vector representing the source of the Poisson equation up to four dimensions. With minor modifications, this algorithm can be extended to handle any number of dimensions. This description is included to make it easier to implement our proposed algorithm and its extension to higher dimensions. ## 7 Conclusions

In this work, we present a generalized quantum algorithm designed to efficiently and accurately solve multidimensional Partial Differential Equations (PDEs) using Quantum-Centric High-Performance Computing (QC-HPC). The algorithm is developed in two distinct forms: the first variant employs the Finite Difference Method (FDM), numerical instantiation, and Classical-to-Quantum (C2Q) encoding whereas the second variant uses FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). To demonstrate the effectiveness of our technique, we have used the multidimensional Poisson equation as a case study.