Utilizing Quantum Computing for Solving Multidimensional Partial Differential Equations

Quantum computing has the potential to revolutionize computational problem-solving by leveraging the quantum mechanical phenomena of superposition and entanglement, which allows for processing a large amount of information simultaneously. This capability is significant in the numerical solution of complex and/or multidimensional partial differential equations (PDEs), which are fundamental to modeling various physical phenomena. There are currently many quantum techniques available for solving partial differential equations (PDEs), which are mainly based on variational quantum circuits. However, the existing quantum PDE solvers, particularly those based on variational quantum eigensolver (VQE) techniques, suffer from several limitations. These include low accuracy, high execution times, and low scalability on quantum simulators as well as on noisy intermediate-scale quantum (NISQ) devices, especially for multidimensional PDEs.

 In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated and evaluated our algorithm using the multidimensional Poisson equation as a case study. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQE-based solvers on noise-free and noisy quantum simulators from IBM. Additionally, we validated our approach on hardware emulators and actual quantum hardware, employing noise mitigation techniques. We will also focus on extending these techniques to PDEs relevant to computational fluid dynamics and financial modeling, further bridging the gap between theoretical quantum algorithms and practical applications.