Variational quantum algorithm for anion exchange across electrolyzer membrane

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 multi-layer membrane in an alkaline electrolyzer. We use $16$ to $64$ grid points across the membrane, resulting from $n=4$ to $6$ data qubits for the ideal quantum simulations that are based on the Qiskit software. 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.

Figures (12)

• Figure 1: Sketch of the anion-exchange membrane (AEM). (a) The 4-layer AEM electrolysis system. (b) The simplified configuration, which will be used for the actual quantum computation approximates the two electrolyte reservoirs by Dirichlet conditions for the anion concentration at the corresponding outer membrane interfaces. The layer $\text{Zr}\text{O}_2$ refers to Zirfon, a composite material consisting of $\text{Zr}\text{O}_2$ embedded in a polysulfone matrix.
• Figure 2: Analytical solution compared at different time instants together with the finite-difference solution. Panel (a) shows the initial conditions, panels (b) and (c) depict time evolution at different time instants, and panel (d) demonstrates how the analytical and numerical finite-difference solutions tend to the steady-state solution.
• Figure 3: Relaxation rate analysis of the problem. (a) The function $\delta(t)$ for various values of $D_2$. (b) The time constant $\tau_s$ and smallest positive eigenvalue reciprocals for various values of $D_2$.
• Figure 4: Quantum circuits for computing individual terms of the cost function. Circuit (a) evaluates the terms $S_\text{PER}$ and $S^\pm$, where the unitary $P$ incorporates the respective coefficients. Circuit (b) evaluates the term $S_\text{LIN}$.
• Figure 5: Sketch of the principle of the state preparation algorithm shown in panels (a)-(d) and a general quantum circuit produced by the state preparation algorithm shown in panel (e). The arrows in panels (a)-(d) indicate the direction in which each corresponding segment is shifted — either upward or downward. Gray rectangles with dashed borders depict the areas under the various segments. In panels (e), blocks $B_1$ and $B_2$ produce the second and the third approximations of the desired state, respectively. Block $B_{n - 1}$ produces the final approximation.