Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
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 c...
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MDPI AG
2025-03-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/18/3/176 |
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| author | Manu Chaudhary Kareem El-Araby Alvir Nobel Vinayak Jha Dylan Kneidel Ishraq Islam Manish Singh Sunday Ogundele Ben Phillips Kieran Egan Sneha Thomas Devon Bontrager Serom Kim Esam El-Araby |
| author_facet | Manu Chaudhary Kareem El-Araby Alvir Nobel Vinayak Jha Dylan Kneidel Ishraq Islam Manish Singh Sunday Ogundele Ben Phillips Kieran Egan Sneha Thomas Devon Bontrager Serom Kim Esam El-Araby |
| author_sort | Manu Chaudhary |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b90ae477ce1543e6bca0d4efaa0f5b4f |
| institution | OA Journals |
| issn | 1999-4893 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj-art-b90ae477ce1543e6bca0d4efaa0f5b4f2025-08-20T02:11:12ZengMDPI AGAlgorithms1999-48932025-03-0118317610.3390/a18030176Solving Multidimensional Partial Differential Equations Using Efficient Quantum CircuitsManu Chaudhary0Kareem El-Araby1Alvir Nobel2Vinayak Jha3Dylan Kneidel4Ishraq Islam5Manish Singh6Sunday Ogundele7Ben Phillips8Kieran Egan9Sneha Thomas10Devon Bontrager11Serom Kim12Esam El-Araby13Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USADepartment of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USAQuantum 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.https://www.mdpi.com/1999-4893/18/3/176numerical instantiationclassical-to-quantum (C2Q) encodingpolar decompositionfinite difference method (FDM)variational quantum algorithm (VQA)column-by-column decomposition (CCD) |
| spellingShingle | Manu Chaudhary Kareem El-Araby Alvir Nobel Vinayak Jha Dylan Kneidel Ishraq Islam Manish Singh Sunday Ogundele Ben Phillips Kieran Egan Sneha Thomas Devon Bontrager Serom Kim Esam El-Araby Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits Algorithms numerical instantiation classical-to-quantum (C2Q) encoding polar decomposition finite difference method (FDM) variational quantum algorithm (VQA) column-by-column decomposition (CCD) |
| title | Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits |
| title_full | Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits |
| title_fullStr | Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits |
| title_full_unstemmed | Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits |
| title_short | Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits |
| title_sort | solving multidimensional partial differential equations using efficient quantum circuits |
| topic | numerical instantiation classical-to-quantum (C2Q) encoding polar decomposition finite difference method (FDM) variational quantum algorithm (VQA) column-by-column decomposition (CCD) |
| url | https://www.mdpi.com/1999-4893/18/3/176 |
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