Quantum Information & Computation
Volume 25, Issue 3
RESEARCH ARTICLE Open access

Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization

Abeynaya Gnanasekaran, Amit Surana, Hongyu Zhu

Author affiliations
  • Abeynaya Gnanasekaran1
  • Amit Surana2
  • Hongyu Zhu2
  1. SRI International, Menlo Park, CA, USA
  2. RTX Technology Research Center, 411 Silver Lane, East Hartford, CT, USA

pp. 260-289 · 10.2478/qic-2025-0014

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Abstract

We present a novel variational quantum framework for nonlinear partial differential equation (PDE) constrained optimization problems. The proposed work extends the recently introduced bi-level variational quantum PDE constrained optimization (BVQPCO) framework for linear PDE to a nonlinear setting by leveraging Carleman linearization (CL). CL framework allows one to transform a system of polynomial ordinary differential equations (ODE), i.e., ODE with polynomial vector field, into a system of infinite but linear ODE. For instance, such polynomial ODEs naturally arise when the PDE is semi-discretized in the spatial dimensions. By truncating the CL system to a finite order, one obtains a finite system of linear ODE to which the linear BVQPCO framework can be applied. In particular, the finite system of linear ODE is discretized in time and embedded as a system of linear equations. The variational quantum linear solver (VQLS) is used to solve the linear system for given optimization parameters and evaluate the design cost/objective function, and a classical black box optimizer is used to select the next set of parameter values based on this evaluated cost. We present detailed computational errors and complexity analysis and prove that under suitable assumptions, our proposed framework can provide potential advantages over classical techniques. We implement our framework using the PennyLane library and apply it to solve inverse Burgers’ problem. We also explore an alternative tensor product decomposition which exploits the sparsity/structure of linear system arising from PDE discretization to facilitate the computation of VQLS cost functions.
Open accessCC BY-NC-ND 4.0

Keywords

  • PDE constrained optimization
  • Carleman linearization
  • variational algorithms
  • quantum computing

Suggested citation

Abeynaya Gnanasekaran, Amit Surana, Hongyu Zhu (2025). Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization. Quantum Information & Computation, 25(3), 260-289. https://doi.org/10.2478/qic-2025-0014

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References (24)

  1. A. Surana and A. Gnanasekaran (2024). “Variational quantum framework for partial differential equation constrained optimization”. arXiv preprint arXiv:2405.16651.DOI: 10.1145/3762671Google Scholar
  2. H. Antil, D.P. Kouri, M.-D. Lacasse and D. Ridzal (2018). Frontiers in PDE-constrained Optimization, volume 163. Springer.DOI: 10.1007/978-1-4939-8636-1Google Scholar
  3. L.T. Biegler, O. Ghattas, M. Heinkenschloss, and B. van BloemenWaanders (2003) “Large-scale pde-constrained optimization: an introduction”, in Large-scale PDE-Constrained Optimization. Springer, pp. 3–13.DOI: 10.1007/978-3-642-55508-4_1Google Scholar
  4. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich (2008). Optimization with PDE Constraints, volume 23. Springer Science & Business Media.Google Scholar
  5. K. Kowalski and W.-H. Steeb (1991). Nonlinear Dynamical Systems and Carleman Linearization. Singapore: World Scientific.DOI: 10.1142/1347Google Scholar
  6. A. Amini, C. Zheng, Q. Sun, and N. Motee (2025). “Carleman linearization of nonlinear systems and its finite-section approximations”. Discrete and Continuous Dynamical Systems-B, 30: 2. 577–603.DOI: 10.3934/dcdsb.2024102Google Scholar
  7. M. Forets and A. Pouly (2017). “Explicit error bounds for Carleman linearization.” arXiv preprint arXiv:1711.02552.Google Scholar
  8. J.-P. Liu, H. Ø. Kolden, H.K. Krovi, N.F. Loureiro, K. Trivisa, and A.M. Childs. (2021). “Efficient quantum algorithm for dissipative nonlinear differential equations”. Proceedings of the National Academy of Sciences, 118: 35, e2026805118.DOI: 10.1073/pnas.2026805118Google Scholar
  9. A. Surana, A. Gnanasekaran, and T. Sahai (2024). “An efficient quantum algorithm for simulating polynomial dynamical systems”. Quantum Information Processing, 23: 3, 1–22.DOI: 10.1007/s11128-024-04311-2Google Scholar
  10. X. Li, X. Yin, N. Wiebe, J. Chun, G. K. Schenter, M. S. Cheung, and J. M”ulmenst”adt (2025). “Potential quantum advantage for simulation of fluid dynamics”. Physical Review Research 7, 013036.DOI: 10.1103/PhysRevResearch.7.013036Google Scholar
  11. J. Penuel, A. Katabarwa, P.D Johnson, C. Farquhar, Y. Cao, and M.C. Garrett (2024). “Feasibility of accelerating incompressible computational fluid dynamics simulations with fault-tolerant quantum computers”. arXiv preprint arXiv:2406.06323.Google Scholar
  12. R. Demirdjian, D. Gunlycke, C.A. Reynolds, J.D. Doyle, and S. Tafur. (2022). “Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics”. Quantum Information Processing, 21: 9, 322.DOI: 10.1007/s11128-022-03667-7Google Scholar
  13. C. B.-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles. (2023). “Variational quantum linear solver.” Quantum, 7, 1188.DOI: 10.22331/q-2023-11-22-1188Google Scholar
  14. A. Gnanasekaran and A. Surana. (2024). “Efficient variational quantum linear solver for structured sparse matrices.” In 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 1. IEEE, pp. 199–210.DOI: 10.1109/QCE60285.2024.00033Google Scholar
  15. M. A. Nielsen and I. L. Chuang (2010). Quantum Computation and Quantum Information. Cambridge University Press.Google Scholar
  16. J.C. D. Reyes (2015). Numerical PDE-Constrained Optimization. Springer.Google Scholar
  17. L. Hantzko, L. Binkowski, and S. Gupta. (2024). “Tensorized pauli decomposition algorithm.” Physica Scripta, 99: 8, 085128.DOI: 10.1088/1402-4896/ad6499Google Scholar
  18. H.-L. Liu, Y.-S. Wu, L.-C. Wan, S.-J. Pan, Su-Juan Qin, Fei Gao, and Qiao-Yan Wen (2021). “Variational quantum algorithm for the poisson equation.” Physical Review A, 104: 2, 022418.DOI: 10.1103/PhysRevA.104.022418Google Scholar
  19. P. Costa, P. Schleich, M.ES Morales, and D.W Berry (2023). “Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling.” arXiv preprint arXiv:2312.09518.Google Scholar
  20. S. Sim, P.D. Johnson, and A. Aspuru-Guzik (2019). “Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms.” Advanced Quantum Technologies, 2: 12, 1900070.DOI: 10.1002/qute.201900070Google Scholar
  21. A. Pellow-Jarman, I. Sinayskiy, A. Pillay, and F. Petruccione (2021). “A comparison of various classical optimizers for a variational quantum linear solver.” Quantum Information Processing, 20: 6, 202.DOI: 10.1007/s11128-021-03140-xGoogle Scholar
  22. P.C.S. Costa, D. An, Y.l.R. Sanders, Y. Su, R. Babbush, and D.W. Berry (2022). “Optimal scaling quantum linear-systems solver via discrete adiabatic theorem.” PRX quantum, 3: 4, 040303.DOI: 10.1103/PRXQuantum.3.040303Google Scholar
  23. D. Dervovic, M. Herbster, P. Mountney, S. Severini, N. Usher, and L. Wossnig (2018). “Quantum linear systems algorithms: a primer.” arXiv preprint arXiv:1802.08227.Google Scholar
  24. A.W. Harrow, A. Hassidim, and S. Lloyd (2009). “Quantum algorithm for linear systems of equations.” Physical Review Letters, 103: 15, 150502.DOI: 10.1103/PhysRevLett.103.150502Google Scholar