S14 - Computational Fluid Dynamics V

The simulations of fluid dynamics play an important role in many areas of research and engineering, including climate prediction [1], biomedical applications [2], and the optimization of aircraft designs [3]. Among the available numerical methods, direct numerical simulation (DNS) provides the highest level of fidelity because it resolves all relevant flow scales; however, this comes at the cost of extremely fine computational meshes [4]. Quantum computing may offer a promising way to represent and process such large-scale solutions more efficiently, by encoding classical information into the amplitudes of a quantum state. However, the operations on quantum computers are restricted to linear and unitary operations, while fluid flows are governed by non-unitary and non-linear dynamics.
Both, non-unitary differential operators, as well as non-linear operations can be implemented probabilistically using quantum circuits that include mid-circuit measurements [5,6].  The scaling of the success probability of this operations play a crucial role in the overall scaling of the algorithm. While it can be constant with system size for specific use cases, especially the success probability of the non-linear operation can decay steeply in the context of fluid dynamical problems. This results in high or even unfeasible number of required measurements when implementing existing approaches for large simulations. Alternative linearization strategies, like the Carleman linearization, are widely studied but are predicted to scale unfavorably for increasingly turbulent flows [7].
We address this challenge by introducing a hybrid-quantum-classical tensor network scheme, which uses tensor-network strategies to realize the non-linearity more efficiently. Based on our analysis of large-scale fluid simulations, we predict significant savings over classical tensor-network routines as well as over prior quantum implementations. 

We employ a variational quantum algorithm (VQA) to numerically simulate hydroxide-ion transport across a two-layer anion exchange membrane (AEM). An AEM electrolyzer, depicted in Fig. 1(a), uses the AEM to efficiently convert electrical energy into chemical energy stored in hydrogen, while safely separating hydrogen and oxygen production. Developing efficient hydrogen production methods like this is crucial for enabling sustainable energy systems and supporting the transition to a low-carbon economy.


(a) An AEM electrolyzer (Fraunhofer IKTS Arnstadt)(b) Extended model with three layers
Figure 1: The AEM electrolyzer (left) and the extended model (right).

The mathematical model of the process is governed by a one-dimensional diffusion equation with a piecewise-constant diffusion coefficient and Dirichlet boundary conditions,


(1)
where c(x, t) is a time-dependent hydroxide-ion concentration, D(x) is a space-dependent piecewise-constant diffusion coefficient, and ΩT is a space-time domain.

In [1], we simulated the model (1) both classically and using the VQA, assuming simple spatial dependence, a piecewise constant diffusion coefficient D(x). The currently ongoing work investigates extensions of this model depicted in Fig. 1(b). In this extension, three layers are involved in the hydroxide-ion transport, whereas the water concentration cw(x, t) is also considered, such that the membrane's diffusivity D becomes a nonlinear function of the water concentration D = D(cw). Moreover, instead of Dirichlet boundary conditions, Neumann boundary conditions are imposed at the sides.

Electrochemical energy storage systems are a cornerstone of the global transition towards renewable energies. Lithium-ion batteries are used in portable electronical devices, electric vehicles and grid-scale renewable energy systems. Their internal dynamics are described by complex physical processes which pose in general a multiscale problem. Classical numerical methods operate on the limits of existing hardware as the demand for finer resolution and higher complexity of computational results increases, especially in three-dimensional models. Quantum methods with their inherently exponential latent space emerges as a promising candidate to capture high-resolution descriptions of multi-scale effects [1].
In our work we focus on the continuum model for the transport in an electrochemical cell and its transport dynamics. These processes are described by a system of coupled, nonlinear partial differential equations (PDEs), including the Nernst-Planck equation for ion diffusion, Poisson's equation for electrostatic potential, and non-linear reaction kinetics models for the electrochemical reactions at the electrode-electrolyte interface. Classical numerical methods, such as finite element or finite differences schemes struggle with these problems, especially when applied to 3d microstructures or multiscale effects in the electrodes.
Concurrent to numerical methods, Physics Informed Neural Networks (PINN) [2] provide a new approach to this kind of problem as they avoid discretization on a grid and have gathered the interest of the research community in recent years [3]. Transferring the idea of physics informed learning to quantum computing offers the potential to harness the exponential Hilbert-space for function descriptions together with additional advantages in terms of trainability, the required number of parameters in such a model.
The possible real-world impact is of high significance. Accelerated and more accurate battery simulation can improve the performance and longevity of lithium-ion batteries if used for battery management systems. Simulations on a larger scale and with more dimensions than currently possible would enable us to gain a better understanding and develop better batteries. The described method can be easily extended to new generations of batteries with alternative chemistries such as lithium-sulfur batteries which are a promising technology for the electrification of aviation. The battery simulation and modelling software market is projected to grow to 4 billion USD by 2030 [4], and sits upstream of a multi billion USD market [5].