Approximately solving complex nonlinear partial differential equations of industrial relevance remains a computational challenge. In the field of computational fluid dynamics (CFD), an established tool in academia, research and industry, nonlinear compressible transport equations arise, for example, for the simulation of aerodynamic effects in the optimization of transport aircraft. State-of-the-art methods for high Reynolds number flows typically solve the Reynolds averaged Navier-Stokes (RANS) equations in combination with a turbulence model. Years of experience show that the accuracy of predictions, especially at the border of an aircraft's flight envelope, is often not sufficient. Therefore scale-resolving methods such as large eddy simulations or even direct numerical simulations are considered. Unfortunately, even sophisticated algorithms on large supercomputers are generally not able to simulate a complete aircraft at flight Reynolds numbers in acceptable time. Furthermore, in many applications, such as optimizing the aerodynamic performance to reduce fuel consumption and costs for example, a large number of such simulations is required.[1, 2]
One way to harness the potential of quantum computers in the field of CFD is to identify the most computationally intensive operations in current algorithms and attempt to replace them with an algorithm suitable for quantum computers. Widely used algorithms for the temporal integration of fluid dynamics equations are implicit, multistage Runge-Kutta methods. A significant portion of the computational effort involved in these algorithms consists of approximately solving the linearized systems of equations that result from linearizing the given nonlinear equations [1, 2]. It therefore makes sense to replace precisely these components with a quantum algorithm in order to achieve a quantum advantage. Doing so introduces two challenges. First, the classical data needs to be prepared as a quantum state. Secondly, we need to access the quantum data and extract it back onto the classical system. To access the full quantum advantage, both steps need to be done efficiently.
In this talk, we will demonstrate how linearized systems of equations can be approximately solved on a quantum computer. Using a previously presented quantum linear system solver [3] the talk will focus on the retrieval of the solution, i.e. measurement of the quantum state to obtain the correction vector required for the outer nonlinear iteration. Limitations and realistic possibilities of hybrid methods will be discussed. The approach will be compared to possible CFD algorithms fully hosted on a quantum system.
