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arXiv:2512.01120 · Physical Review Applied

Variational quantum algorithm for anion exchange across electrolyzer membrane

Timur Gubaev, Philipp Pfeffer, Christian Dreßler, Jörg Schumacher

physics.flu-dynquant-phSubmitted Nov 30, 2025• Announced Dec 2, 2025• 0 citations• Revised May 29, 2026 (v2)
v1 · Nov 30, 2025v2 · May 29, 2026v3 · Sep 21, 2026
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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 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.

Journal reference
Phys. Rev. Applied 26, 014051 (2026)
DOI
10.1103/149q-4z6s
Comments
24 pages, 16 figures, 4 tables

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The mixing of scalar substances in fluid flows by stirring and diffusion is ubiquitous in natural flows, chemical engineering, and microfluidic drug delivery. Here, we present a spectral quantum algorithm for scalar mixing by solving the advection-diffusion equation in a quantum computational fluid dynamics framework. The exact gate decompositions of the advection and diffusion operators in spectral space are derived. For all but the simplest one-dimensional flows, these operators do not commute. Therefore, we use operator splitting to construct quantum circuits capable of simulating arbitrary polynomial velocity profiles in multiple dimensions, such as the Blasius profile of a laminar boundary layer. Periodic, Neumann, and Dirichlet boundary conditions can be imposed with the appropriate quantum spectral transform. We evaluate the approach in statevector simulations of a Couette flow, plane Poiseuille flow, and a polynomial Blasius profile approximation. For an advection-diffusion problem in one dimension, we compare the time evolution of an ideal quantum simulation with those of real quantum computers with superconducting and trapped-ion qubits. The required number of two-qubit gates grows with the logarithm of the number of grid points raised to one higher power than the order of the polynomial velocity profile.
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