Variational quantum algorithm for anion exchange across electrolyzer membrane

Variational quantum algorithm for anion exchange across electrolyzer membrane 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 ions 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 VQA method. We demonstrate the applicability of the quantum algorithm to a problem with non-trivial 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. Dataset description and structure The provided dataset contains Python scripts and data generated thereof that can be used to reproduce figures 2-4, 7, 9-16 of [1]. It is organized as a set of .zip files, each corresponding to its purpose and figures as indicated in the table below. Each .zip file contains .ipynb interactive Python notebooks with the corresponding data. File Purpose Related figures 1dPWCDE_analytical_solution.zip Solves the problem analytically. Figure 2 1dPWCDE_relaxation_rate_analysis.zip Computes the relaxation rate of the problem for various values of $D_2$. Figure 3 1dPWCDE_gradients.zip Computes the steady-state solution of the problem for various values of $r_D$, $c_A$ and $\tilde x_1$. Figure 4 rpqc_expressibility_simulations.zip Computes expressibility of the real-valued parameterized quantum circuit for various numbers of qubits $n$ and layers $d$. Figure 7 rpqc_expressibility_analysis.zip Plots expressibility of the real-valued parameterized quantum circuit. Figure 7 1dPWCDE_456_4_5_6_sv_bfgs.zip Performs state-vector simulations of the VQA algorithm for $n = 4, 5, 6$ qubits (with $d = 4, 5, 6$ layers, respectively) using the BFGS algorithm for the classical optimization. Figures 9-11, 13 and 16 1dPWCDE_45_345_456_sv_bfgs.zip Performs state-vector simulations of the VQA algorithm for $n = 4, 5$ qubits with $d = 3, 4, 5$ in 4-qubit experiments and $d = 4, 5, 6$ in 6-qubit experiments. All experiments use the BFGS algorithm for the classical optimization. Figure 12 1dPWCDE_4_4_sv_bfgs_nm_sbo_cma.zip Performs state-vector simulations of the VQA for $n = 4$ qubits with $d = 4$ using Nelder-Mead, BFGS, surrogate-based optimization (SBO) [2], and covariance matrix adaptation evolution strategy (CMA-ES) [3] algorithms for the classical optimization. Figure 14 1dPWCDE_5_5_sv_bfgs_nm_sbo.zip Performs state-vector simulations of the VQA for $n = 5$ qubits with $d = 5$ using Nelder-Mead, BFGS, SBO algorithms for the classical optimization. No associated figures 1dPWCDE_4_4_sb_bfgs_nm_cma.zip Performs shot-based simulations of the VQA for $n = 4$ qubits with $d = 4$ using Nelder-Mead, BFGS, and CMA-ES algorithms for the classical optimization. Figure 15 Python requirements All simulations were conducted using the following Python packages with the specified versions: NumPy 2.0.2 SciPy 1.15.1 Matplotlib 3.10.0 Qiskit 1.4.2 Moreover, two additional packages are required: Custom parameterized quantum circuit expressibility estimator: https://github.com/tgubaev/pqc-expressibility-estimator A slightly modified version of the SBO [2] implementation by R. Shaffer: https://github.com/tgubaev/sbovqaopt-nq Funding The work of T. Gubaev and P. Pfeffer is funded by the European Union (ERC, MesoComp, 101052786). References T. Gubaev, P. Pfeffer, C. Dreßler, and J. Schumacher, Variational quantum algorithm for anion exchange across electrolyzer membrane, preprint arxiv.2512.01120 arXiv. https://doi.org/10.48550/arxiv.2512.01120 (2025) R. Shaffer, L. Kocia, and M. Sarovar, Surrogate-based optimization for variational quantum algorithms, Phys. Rev. A 107, 032415 (2023). H. Nikolaus, The CMA evolution strategy: A tutorial, preprint arXiv:1604.00772 10.48550/arXiv.1604.00772 (2023).