# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 1 Introduction  
Partial Differential Equations (PDEs) can model complex systems across various fields, such as finance, physics, and mathematics [1, 2]. Although there are advanced classical algorithms for solving PDEs [3] due to the curse of dimensionality, solving them is highly computationally intensive, especially for multidimensional PDEs [4]. As classical semiconductor technology is reaching its limits, there is a need to find alternative technologies for performing compute-intensive tasks [5]. In order to reduce computational time, researchers are exploring various technologies for solving PDEs [6, 7, 8]. Owing to its great potential to revolutionize computational speed, quantum computing in the last decade has emerged as a promising technology to solve even highly computationally intensive practical problems[9, 10]. This is because it supports inherent parallelism due to the quantum mechanical properties of superposition and entanglement [11].  
Another key advantage of quantum computing is its ability to handle large datasets using only a logarithmic number of quantum bits (qubits) relative to the data size. Because of these benefits, solving PDEs using quantum computers relevant to various practical applications is becoming an active area of research [12]. Recently, researchers have also started exploring the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve nonlinear partial differential equations [13].  
Most of the currently used quantum computing-based PDE-solvers are either based on the Harrow-Hassidim-Lloyd (HHL) algorithm or on Variational Quantum Algorithms (VQAs) [6]. The PDEs can be solved using the HHL algorithm [14] after discretizing it into linear equations using the Finite Difference Method (FDM) [15]. Although the HHL algorithm can provide exponential speedup over classical methods for solving linear systems with sparse coefficient matrices, it requires error-corrected systems with a large number of qubits [16]. Because of the limitations of error-correction in Noisy-Intermediate Scale Quantum (NISQ) devices, researchers are currently investigating alternative techniques such as VQAs to solve PDEs [17, 18, 19, 20]. The VQAs used to solve PDEs are either based on Variational Quantum Eigensolver (VQE) or Variational Quantum Linear Solver (VQLS) [17, 18, 19, 20]. However, the VQA-based PDE solvers suffer from high execution time, limited scalability, and very low accuracy.  
In this work, we propose a time-efficient, scalable, and accurate quantum algorithm for solving multidimensional PDEs on Quantum-Centric High-Performance Computing (QC-HPC) systems. Our algorithm is categorized into two variants, i.e., Variant 1 and Variant 2. Variant 1 uses the Finite Difference Method (FDM), Classical-to-Quantum (C2Q) data encoding [21], and numerical instantiation [22], whereas Variant 2 uses FDM, C2Q encoding, and Column-by-Column Decomposition (CCD) [23, 24]. For our experiments, we have used multidimensional Poisson equations as a case study. Our experimental results showcase favorable results in terms of accuracy, scalability, and execution time in comparison to VQA-based solvers on noise-free and noisy simulators, hardware emulators, and real quantum hardware from IBM [24].  
The remaining sections of this paper are structured as follows: Section 2 provides background information, Section 3 highlights related work, and Section 4 discusses the proposed methodology in detail. The experimental results are presented in Section 5. Section 6 provides a discussion about the experimental outcomes. Finally, Section 7 concludes the paper with potential future work.