{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T07:52:46Z","timestamp":1768031566196,"version":"3.49.0"},"reference-count":53,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T00:00:00Z","timestamp":1742428800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/a18030176","type":"journal-article","created":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T04:58:38Z","timestamp":1742533118000},"page":"176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits"],"prefix":"10.3390","volume":"18","author":[{"given":"Manu","family":"Chaudhary","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-5028-5596","authenticated-orcid":false,"given":"Kareem","family":"El-Araby","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5440-923X","authenticated-orcid":false,"given":"Alvir","family":"Nobel","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vinayak","family":"Jha","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dylan","family":"Kneidel","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ishraq","family":"Islam","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manish","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sunday","family":"Ogundele","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ben","family":"Phillips","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kieran","family":"Egan","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sneha","family":"Thomas","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Devon","family":"Bontrager","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serom","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4575-1049","authenticated-orcid":false,"given":"Esam","family":"El-Araby","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"C667","DOI":"10.1137\/140997907","article-title":"Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations","volume":"37","author":"Klawonn","year":"2015","journal-title":"Siam J. Sci. Comput."},{"key":"ref_2","unstructured":"Evans, L.C. (2022). Partial Differential Equations, American Mathematical Society."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1002\/nme.5938","article-title":"A multilevel FETI-DP method and its performance for problems with billions of degrees of freedom","volume":"116","author":"Toivanen","year":"2018","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1017\/S0962492911000055","article-title":"Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs","volume":"20","author":"Schwab","year":"2011","journal-title":"Acta Numer."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1038\/530144a","article-title":"The chips are down for Moore\u2019s law","volume":"530","author":"Waldrop","year":"2016","journal-title":"Nat. News"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Tosti Balducci, G., Chen, B., M\u00f6ller, M., Gerritsma, M., and De Breuker, R. (2022). Review and perspectives in quantum computing for partial differential equations in structural mechanics. Front. Mech. Eng., 8.","DOI":"10.3389\/fmech.2022.914241"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Anderson, J., Kayraklioglu, E., Imani, H.R., Miscuglio, M., Sorger, V.J., and El-Ghazawi, T. (2020, January 1\u20133). Virtualizing analog mesh computers: The case of a photonic PDE solving accelerator. Proceedings of the 2020 International Conference on Rebooting Computing (ICRC), Atlanta, GA, USA.","DOI":"10.1109\/ICRC2020.2020.00008"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Smith, J.D., Severa, W., Hill, A.J., Reeder, L., Franke, B., Lehoucq, R.B., Parekh, O.D., and Aimone, J.B. (2020, January 28\u201330). Solving a steady-state PDE using spiking networks and neuromorphic hardware. Proceedings of the International Conference on Neuromorphic Systems 2020, Oak Ridge, TN, USA.","DOI":"10.1145\/3407197.3407202"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"El-Araby, E., Chaudhary, M., Islam, I.U., Levy, D., Kneidel, D., Jeng, M., Nobel, A., and Jha, V. (2024). Practical Applications of Quantum Computing. Quantum Computing: Innovations and Applications in Modern Research, IntechOpen.","DOI":"10.5772\/intechopen.1003033"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"015004","DOI":"10.1103\/RevModPhys.94.015004","article-title":"Noisy intermediate-scale quantum algorithms","volume":"94","author":"Bharti","year":"2022","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","unstructured":"Nielsen, M.A., and Chuang, I.L. (2010). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"130503","DOI":"10.1103\/PhysRevLett.113.130503","article-title":"Quantum support vector machine for big data classification","volume":"113","author":"Rebentrost","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Tennie, F., Laizet, S., Lloyd, S., and Magri, L. (2025). Quantum computing for nonlinear differential equations and turbulence. Nat. Rev. Phys., 1\u201311.","DOI":"10.1038\/s42254-024-00799-w"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.103.150502","article-title":"Quantum algorithm for linear systems of equations","volume":"103","author":"Harrow","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","unstructured":"Nagel, J.R. (2011). Solving the Generalized Poisson Equation Using the Finite-Difference Method (FDM), Department of Electrical and Computer Engineering, University of Utah. Lecture Notes."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","article-title":"Quantum computing in the NISQ era and beyond","volume":"2","author":"Preskill","year":"2018","journal-title":"Quantum"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"052409","DOI":"10.1103\/PhysRevA.104.052409","article-title":"Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation","volume":"104","author":"Sato","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Trahan, C.J., Loveland, M., Davis, N., and Ellison, E. (2023). A variational quantum linear solver application to discrete finite-element methods. Entropy, 25.","DOI":"10.3390\/e25040580"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"014054","DOI":"10.1103\/PhysRevApplied.20.014054","article-title":"Performance study of variational quantum algorithms for solving the Poisson equation on a quantum computer","volume":"20","author":"Ali","year":"2023","journal-title":"Phys. Rev. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"022418","DOI":"10.1103\/PhysRevA.104.022418","article-title":"Variational quantum algorithm for the Poisson equation","volume":"104","author":"Liu","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1109\/TC.2023.3248276","article-title":"Towards complete and scalable emulation of quantum algorithms on high-performance reconfigurable computers","volume":"72","author":"Mahmud","year":"2023","journal-title":"IEEE Trans. Comput."},{"key":"ref_22","unstructured":"Younis, E., Iancu, C.C., Lavrijsen, W., Davis, M., and Smith, E. (2021). Berkeley Quantum Synthesis Toolkit (bqskit) v1, Lawrence Berkeley National Laboratory (LBNL). Technical Report."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"032318","DOI":"10.1103\/PhysRevA.93.032318","article-title":"Quantum circuits for isometries","volume":"93","author":"Iten","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_24","unstructured":"(2025, February 28). IBM Quantum. Available online: https:\/\/quantum-computing.ibm.com."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shende, V.V., Bullock, S.S., and Markov, I.L. (2005, January 18\u201321). Synthesis of Quantum Logic Circuits. Proceedings of the 2005 Asia and South Pacific Design Automation Conference, Shanghai, China.","DOI":"10.1145\/1120725.1120847"},{"key":"ref_26","unstructured":"IBM Quantum (2025, February 28). Qiskit: An Open-Source Framework for Quantum Computing. Available online: https:\/\/www.ibm.com\/quantum\/qiskit."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"014003","DOI":"10.1088\/2058-9565\/ab8e92","article-title":"t| ket>: A retargetable compiler for NISQ devices","volume":"6","author":"Sivarajah","year":"2020","journal-title":"Quantum Sci. Technol."},{"key":"ref_28","unstructured":"(2025, February 28). Cirq, a Python Framework for Creating, Editing, and Invoking Noisy Intermediate Scale Quantum (NISQ) Circuits. Available online: https:\/\/github.com\/quantumlib\/Cirq."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"032315","DOI":"10.1103\/PhysRevA.69.032315","article-title":"Optimal quantum circuits for general two-qubit gates","volume":"69","author":"Vatan","year":"2004","journal-title":"Phys. Rev. A\u2014At. Mol. Opt. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"130502","DOI":"10.1103\/PhysRevLett.93.130502","article-title":"Quantum circuits for general multiqubit gates","volume":"93","author":"Vartiainen","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Younis, E., and Iancu, C. (2022, January 18\u201323). Quantum circuit optimization and transpilation via parameterized circuit instantiation. Proceedings of the 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), Broomfield, CO, USA.","DOI":"10.1109\/QCE53715.2022.00068"},{"key":"ref_32","first-page":"1","article-title":"Leap: Scaling numerical optimization based synthesis using an incremental approach","volume":"4","author":"Smith","year":"2023","journal-title":"Acm Trans. Quantum Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2019.10.001","article-title":"Quantum decoherence","volume":"831","author":"Schlosshauer","year":"2019","journal-title":"Phys. Rep."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1109\/MIC.2021.3133845","article-title":"Decoherence-optimized circuits for multidimensional and multilevel-decomposable quantum wavelet transform","volume":"26","author":"Mahmud","year":"2021","journal-title":"IEEE Internet Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"134519","DOI":"10.1103\/PhysRevB.72.134519","article-title":"Decoherence in a superconducting quantum bit circuit","volume":"72","author":"Ithier","year":"2005","journal-title":"Phys. Rev. B\u2014Condens. Matter Mater. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1038\/s42254-021-00348-9","article-title":"Variational quantum algorithms","volume":"3","author":"Cerezo","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1920","DOI":"10.1137\/16M1087072","article-title":"Quantum algorithm for systems of linear equations with exponentially improved dependence on precision","volume":"46","author":"Childs","year":"2017","journal-title":"SIAM J. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1137\/1035134","article-title":"On the early history of the singular value decomposition","volume":"35","author":"Stewart","year":"1993","journal-title":"SIAM Rev."},{"key":"ref_39","unstructured":"Chaudhary, M., Islam, I., Levy, D., Nobel, A., Kneidel, D., Jha, V., and El-Araby, E. (2024, January 22\u201325). An Efficient Quantum Solver for Multidimensional Partial Differential Equations. Proceedings of the International Conference on Emergent Quantum Technologies (ICEQT 2024), Las Vegas, NV, USA."},{"key":"ref_40","unstructured":"IBM (2025, February 28). Quantumcircuit.Unitary. Available online: https:\/\/docs.quantum.ibm.com\/api\/qiskit\/qiskit.circuit.library.UnitaryGate."},{"key":"ref_41","unstructured":"IBM (2025, February 28). MatrixOp Class. Available online: https:\/\/docs.quantum.ibm.com\/api\/qiskit\/0.32\/qiskit.aqua.operators.primitive_ops.MatrixOp.to_pauli_op."},{"key":"ref_42","unstructured":"IBM (2025, February 28). EfficientSU2 Circuit. Available online: https:\/\/docs.quantum.ibm.com\/api\/qiskit\/qiskit.circuit.library.EfficientSU2."},{"key":"ref_43","unstructured":"IBM (2025, February 28). ADAM Optimizer. Available online: https:\/\/docs.quantum.ibm.com\/api\/qiskit\/0.28\/qiskit.algorithms.optimizers.ADAM."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"045005","DOI":"10.1103\/RevModPhys.95.045005","article-title":"Quantum error mitigation","volume":"95","author":"Cai","year":"2023","journal-title":"Rev. Mod. Phys."},{"key":"ref_45","unstructured":"(2025, February 28). IBM. IBM Debuts Next-Generation Quantum Processor & IBM Quantum System Two, Extends Roadmap to Advance Era of Quantum Utility IBM Press Release, Available online: https:\/\/newsroom.ibm.com\/2023-12-04-IBM-Debuts-Next-Generation-Quantum-Processor-IBM-Quantum-System-Two%2C-Extends-Roadmap-to-Advance-Era-of-Quantum-Utility."},{"key":"ref_46","unstructured":"Almeida, D.G., Ferris, K., Kanazawa, N., Johnson, B., and Davis, R. (2025, February 28). New fractional gates reduce circuit depth for utility-scale workloads, IBM Research Blog, Available online: https:\/\/www.ibm.com\/quantum\/blog\/fractional-gates."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"010313","DOI":"10.1103\/PRXQuantum.3.010313","article-title":"Connecting ansatz expressibility to gradient magnitudes and barren plateaus","volume":"3","author":"Holmes","year":"2022","journal-title":"PRX Quantum"},{"key":"ref_48","unstructured":"Grimsley, H.R., Barron, G.S., Barnes, E., Economou, S.E., and Mayhall, N.J. (2022). Adapt-vqe is insensitive to rough parameter landscapes and barren plateaus. arXiv."},{"key":"ref_49","unstructured":"Qiang, J., and Paret, S. (2014). Poisson solvers for self-consistent multi-particle simulations. arXiv."},{"key":"ref_50","unstructured":"Williams, C.P. (2010). Explorations in Quantum Computing, Springer Science & Business Media."},{"key":"ref_51","unstructured":"Buffington, G. (2014). Polar Decomposition of a Matrix, University of Puget Sound. Available online: http:\/\/buzzard.ups.edu\/courses\/2014spring\/420projects\/math420-UPS-spring-2014-buffington-polar-decomposition.pdf."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1137\/0907079","article-title":"Computing the polar decomposition\u2014With applications","volume":"7","author":"Higham","year":"1986","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_53","unstructured":"Ghosh, K. (2021). Encoding classical data into a quantum computer. arXiv."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/3\/176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:57:33Z","timestamp":1760029053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/3\/176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,20]]},"references-count":53,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["a18030176"],"URL":"https:\/\/doi.org\/10.3390\/a18030176","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,20]]}}}