- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Variational quantum algorithm for anion exchange across an electrolyzer membrane

Timur Gubaev1, Philipp Pfeffer1, Christian Dreßler2, and Jörg Schumacher1

PDF

Phys. Rev. Applied 26, 014051 – Published 17 July, 2026

DOI: https://doi.org/10.1103/149q-4z6s

Abstract

We present a variational quantum algorithm that solves the one-dimensional diffusion problem with a space-dependent diffusion constant D(x). This problem is relevant for the exchange of hydroxide ion across a two-layer membrane in an alkaline electrolyzer, where the concentration of OH− ion determines the chemical stability for longer time periods. We use 16 to 64 grid points across the membrane, resulting from n=4 to 6 data qubits for the ideal statevector and shot-based quantum simulations implemented using Qiskit. For these qubit numbers, the depth of the parametric quantum circuit has been chosen to ensure sufficient expressibility. The state preparation requires particular attention since the diffusivity D is piecewise constant in the different layers with discontinuities at the interface. Furthermore, we compare different classical optimization schemes with respect to their convergence in the variational quantum algorithms (VQA) method. We demonstrate the applicability of the quantum algorithm to a problem with nontrivial boundary conditions and jump conditions of the diffusion constant and outline possible extensions of the proof-of-concept application case of quantum computing. Our simulations show that pronounced hydroxide-ion concentration gradients, and thus chemical instabilities, can occur only when the ratio of diffusivity in both layers of the membrane exceeds approximately 50.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. Z. Yan, J. L. Hitt, J. A. Turner, and T. E. Mallouk, Renewable electricity storage using electrolysis, Proc. Natl. Acad. Sci. U.S.A. 117, 12558 (2020).
  2. M. El-Shafie, Hydrogen production by water electrolysis technologies: A review, Results Eng. 20, 101426 (2023).
  3. H. I. Lee, M. Mehdi, S. K. Kim, H. S. Cho, M. J. Kim, W. C. Cho, Y. W. Rhee, and C. H. Kim, Advanced Zirfon-type porous separator for a high-rate alkaline electrolyser operating in a dynamic mode, J. Membr. Sci. 616, 118541 (2020).
  4. M. Schalenbach, W. Lueke, and D. Stolten, Hydrogen diffusivity and electrolyte permeability of the Zirfon perl separator for alkaline water electrolysis, J. Electrochem. Soc. 163, F1480 (2016).
  5. J. Brauns, J. Schönebeck, M. R. Kraglund, D. Aili, J. Hnát, J. Žitka, W. Mues, J. O. Jensen, K. Bouzek, and T. Turek, Evaluation of diaphragms and membranes as separators for alkaline water electrolysis, J. Electrochem. Soc. 168, 014510 (2021).
  6. E. Delp, R. Mishra, and E. Wagner, Evaluation of complex multi-physics phenomena at gas diffusion electrodes during high-pressure water electrolysis with AEMs, Sci. Rep. 15, 24866 (2025).
  7. N. Du, C. Roy, R. Peach, M. Turnbull, S. Thiele, and C. Bock, Anion-exchange membrane water electrolyzers, Chem. Rev. 122, 11830 (2022).
  8. D. Henkensmeier, W.-C. Cho, P. Jannasch, J. Stojadinovic, Q. Li, D. Aili, and J. O. Jensen, Separators and membranes for advanced alkaline water electrolysis, Chem. Rev. 124, 6393 (2024).
  9. Y. Sugawara, S. Sankar, S. Miyanishi, R. Illathvalappil, P. K. Gangadharan, H. Kuroki, G. M. Anilkumar, and T. Yamaguchi, Anion exchange membrane water electrolyzers: An overview, J. Chem. Eng. Jpn. 56, 2210195 (2023).
  10. K. Yassin and D. R. Dekel, Grand challenges in anion exchange membrane energy applications, Front. Membr. Sci. Technol. 4, 1691069 (2025).
  11. W. E. Mustain, M. Chatenet, M. Page, and Y. S. Kim, Durability challenges of anion exchange membrane fuel cells, Energy Environ. Sci. 13, 2805 (2020).
  12. C. E. Diesendruck and D. R. Dekel, Water – A key parameter in the stability of anion exchange membrane fuel cells, Curr. Opin. Electrochem. 9, 173 (2018).
  13. S. Koch, J. Disch, S. K. Kilian, Y. Han, L. Metzler, A. Tengattini, L. Helfen, M. Schulz, M. Breitwieser, and S. Vierrath, Water management in anion-exchange membrane water electrolyzers under dry cathode operation, RSC Adv. 12, 20778 (2022).
  14. M. G. Marino, J. P. Melchior, A. Wohlfarth, and K. D. Kreuer, Hydroxide, halide and water transport in a model anion exchange membrane, J. Membr. Sci. 464, 61 (2014).
  15. T. Zelovich, L. Vogt-Maranto, M. A. Hickner, S. J. Paddison, C. Bae, D. R. Dekel, and M. E. Tuckerman, Hydroxide ion diffusion in anion-exchange membranes at low hydration: Insights from ab initio molecular dynamics, Chem. Mater. 31, 5778 (2019).
  16. C. Chen, Y.-L. S. Tse, G. E. Lindberg, C. Knight, and G. A. Voth, Hydroxide solvation and transport in anion exchange membranes, J. Am. Chem. Soc. 138, 991 (2016).
  17. M. Al Otmi, P. Lin, W. Schertzer, C. M. Colina, R. Ramprasad, and J. Sampath, Investigating correlations in hydroxide ion transport in anion exchange membranes from atomistic molecular dynamics simulations, ACS Appl. Polym. Mater. 6, 11270 (2024).
  18. A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications (John Wiley & Sons, New York, 2001).
  19. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2010).
  20. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  21. S. S. Bharadwaj and K. R. Sreenivasan, Quantum computation of fluid dynamics, Indian Acad. Sci. Conf. Ser. 3, 77 (2020).
  22. S. Succi, W. Itani, K. Sreenivasan, and R. Steijl, Quantum computing for fluids: Where do we stand?, Europhys. Lett. 144, 10001 (2023).
  23. F. Tennie, S. Laizet, S. Lloyd, and L. Magri, Quantum computing for nonlinear differential equations and turbulence, Nat. Rev. Phys. 7, 1 (2025).
  24. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  25. R. Demirdjian, D. Gunlycke, C. A. Reynolds, J. D. Doyle, and S. Tafur, Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics, Quantum Inf. Process. 21, 322 (2022).
  26. F. Y. Leong, W.-B. Ewe, and D. E. Koh, Variational quantum evolution equation solver, Sci. Rep. 12, 10817 (2022).
  27. F. Y. Leong, D. E. Koh, W.-B. Ewe, and J. F. Kong, Variational quantum simulation of partial differential equations: Applications in colloidal transport, Int. J. Numer. Methods Heat Fluid Flow 33, 3669 (2023).
  28. D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quantum algorithms for computational fluid dynamics, AIAA J. 61, 1885 (2023).
  29. J. Ingelmann, S. S. Bharadwaj, P. Pfeffer, K. R. Sreenivasan, and Jörg Schumacher, Two quantum algorithms for solving the one-dimensional advection-diffusion equation, Comput. Fluids 281, 106369 (2024).
  30. S. Bengoechea, P. Over, D. Jaksch, and T. Rung, Toward variational quantum algorithms for generalized linear and nonlinear transport phenomena, AIAA J. 64, 585 (2025).
  31. P. Over, S. Bengoechea, T. Rung, F. Clerici, L. Scandurra, E. de Villiers, and D. Jaksch, Boundary treatment for variational quantum simulations of partial differential equations on quantum computers, Comput. Fluids 288, 106508 (2025).
  32. N. M. Guseynov, A. A. Zhukov, W. V. Pogosov, and A. V. Lebedev, Depth analysis of variational quantum algorithms for the heat equation, Phys. Rev. A 107, 052422 (2023).
  33. Y. Y. Liu, Z. Chen, C. Shu, S. C. Chew, B. C. Khoo, X. Zhao, and Y. D. Cui, Application of a variational hybrid quantum-classical algorithm to heat conduction equation and analysis of time complexity, Phys. Fluids 34, 117121 (2022).
  34. M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101, 010301(R) (2020).
  35. N. Köcher, H. Rose, S. S. Bharadwaj, J. Schumacher, and S. Schumacher, Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm, Sci. Rep. 15, 23478 (2025).
  36. A. J. Pool, A. D. Somoza, M. Lubasch, and B. Horstmann, Solving partial differential equations using a quantum computer, in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), Broomfield, CO, USA (2022), pp. 864–866, 10.1109/QCE53715.2022.00146.
  37. A. J. Pool, A. D. Somoza, C. Mc Keever, M. Lubasch, and B. Horstmann, Nonlinear dynamics as a ground-state solution on quantum computers, Phys. Rev. Res. 6, 033257 (2024).
  38. P. Pfeffer, P. Brearley, S. Laizet, and J. Schumacher, Spectral quantum algorithm for passive scalar transport in shear flows, Sci. Rep. 15, 41172 (2025).
  39. R. Shaffer, L. Kocia, and M. Sarovar, Surrogate-based optimization for variational quantum algorithms, Phys. Rev. A 107, 032415 (2023).
  40. Z. B. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  41. S. Sim, P. D. Johnson, and Aán Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technol. 2, 1900070 (2019).
  42. P. C. Bressloff, Accumulation time of diffusion in a two-dimensional singularly perturbed domain, Proc. R. Soc. A 478, 20210847 (2022).
  43. A. G. Neelan, F. S. Don Bosco, N. S. Jarugumalli, and S. B. Vedarethinam, Physics-informed neural networks: Bridging the divide between conservative and non-conservative equations, arXiv:2506.22413.
  44. J. Lee, Stability of finite difference schemes on the diffusion equation with discontinuous coefficients, Master’s thesis, Department of Mathematics, Massachusetts Institute of Technology, 2017.
  45. B. Zindorf and S. Bose, Efficient implementation of multicontrolled quantum gates, Phys. Rev. Appl. 24, 044030 (2025).
  46. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with Qiskit, arXiv:2405.08810v3.
  47. K. Życzkowski and H.-J. Sommers, Average fidelity between random quantum states, Phys. Rev. A 71, 032313 (2005).
  48. A. J. da Silva and D. K. Park, Linear-depth quantum circuits for multiqubit controlled gates, Phys. Rev. A 106, 042602 (2022).
  49. J. A. Nelder and R. Mead, A simplex method for function minimization, Comput. J. 7, 308 (1965).
  50. H. Nikolaus, The CMA evolution strategy: A tutorial, arXiv:1604.00772.
  51. S. Liu, S.-X. Zhang, S.-K. Jian, and H. Yao, Training variational quantum algorithms with random gate activation, Phys. Rev. Res. 5, L032040 (2023).
  52. A. Arrasmith, M. Cerezo, P. Czarnik, L. Cincio, and P. J. Coles, Effect of barren plateaus on gradient-free optimization, Quantum 5, 558 (2021).
  53. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  54. J. Gacon, C. Zoufal, G. Carleo, and S. Woerner, Simultaneous perturbation stochastic approximation of the quantum Fisher information, Quantum 5, 567 (2021).
  55. D. Fitzek, R. S. Jonsson, W. Dobrautz, and C. Schäfer, Optimizing variational quantum algorithms with qBang: Efficiently interweaving metric and momentum to navigate flat energy landscapes, Quantum 8, 1313 (2024).
  56. V. Novák, I. Zelinka, and V. Snášel, Optimization strategies for variational quantum algorithms in noisy landscapes, arXiv:2506.01715.
  57. K. Yassin, I. G. Rasin, S. Brandon, and D. R. Dekel, Quantifying the critical effect of water diffusivity in anion exchange membranes for fuel cell applications, J. Membr. Sci. 608, 118206 (2020).
  58. D. R. Dekel, I. G. Rasin, M. Page, and S. Brandon, Steady state and transient simulation of anion exchange membrane fuel cells, J. Power Sources 375, 191 (2018).
  59. T. Gubaev, P. Pfeffer, C. Dreßler, and J. Schumacher, Variational quantum algorithm for anion exchange across an electrolyzer membrane, Zenodo, https://zenodo.org/records/20444655 (2026).

Outline

Information

Phys. Rev. Applied 26, 014051– Published 17 July, 2026
  • Received 30 November 2025
  • Revised 30 March 2026
  • Accepted 18 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation