Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits

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Abstract
Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs.

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