ASML and QAL develop quantum-assisted Finite Element Method for plasma modelling
For modelling plasmas in their machines, ASML engineers rely on the Finite Element Method (FEM). FEM discretises physical space into a mesh to calculate approximate results. This process generates a large linear system, Ax=b, describing the physics of the entire problem.
There are many ways to solve these systems, some of which can involve decomposing the matrix A. During this process, “fill-in” can occur, which leads to values in previously empty matrix positions, wasting memory and power. Luckily, the matrix can be re-ordered and each re-ordering yields a different amount of fill-in. Needless to say, one would ideally want to find the re-ordering that minimises the amount of fill-in generated.
ASML and QAL developed a quantum-assisted method to find such good re-orderings. First off, we identified nested dissection as a fruitful way to re-order the matrix. It re-orders the matrix into an "arrow-head" shape, which minimises fill-in during decomposition. Mathematically, this task is similar to finding a graph bisection; a well-known problem in graph theory, which is NP-complete. Then we showed how this graph bisection problem could be translated into a Quadratic Unconstrained Binary Optimisation (QUBO) problem, which is then solved by the D-Wave quantum annealer. The solution is then fed back into a classical computer to finalise the matrix re-ordering.
The method shows significant potential when the same matrix structure can be reused for multiple consecutive calculations. While current quantum hardware is not (yet) better than state-of-the-art classical solvers, this method marks a step toward integrating quantum power into standard engineering workflows.
Stay tuned for more insight: an explainer video is currently being developed on this project.
