S16 - Computational Fluid Dynamics VI

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.

The simulation of fluid flow using Computational Fluid Dynamics (CFD) is a central tool in many areas of science and engineering, including aerodynamics, energy systems, and climate modeling. For numerical simulations, the underlying partial differential equations, such as the Navier–Stokes equations, are discretized, leading to large and sparse linear systems of equations. Solving these systems efficiently plays an important role in the overall computational cost of CFD simulations.
Quantum algorithms are promising for advancing fluid simulations by leveraging an exponentially growing state space in the number of qubits [1]. In particular, Quantum Linear System Solvers (QLSS), such as the HHL algorithm, promise significant speedups under certain assumptions [2]. These approaches rely on matrix inversion. However, in classical CFD simulations, iterative schemes, ranging from classical methods such as the Jacobi method, Gauss–Seidel iteration, and Successive Over-Relaxation (SOR) to more advanced Krylov subspace methods such as the Conjugate Gradient method, are typically employed instead [3].
Recent work has therefore explored the implementation of iterative solvers within a quantum computing framework [4]. There, the implementation of the Jacobi scheme via block encoding results in a qubit cost that scales linearly with the number of Jacobi iterations.
In this work, we present an alternative implementation of the Jacobi method based on Quantum Singular Value Transformation (QSVT) [5]. Using this polynomial-based approach, the number of required qubits becomes independent of the number of Jacobi iterations. This enables an efficient quantum implementation of the Jacobi scheme in terms of circuit width and depth.

Solving partial differential equations (PDEs) is a cornerstone of computational physics and engineering. One important class of PDEs are parabolic PDEs that can be used to model transient diffusion processes. In fluid mechanics, diffusion processes describe the transport of a physical quantity (such as the concentration of a substance or heat) based on the random motion of molecules. In this talk, we present a quantum-based solver for a well-known parabolic model problem, which is referred to as heat equation. While quantum computers naturally excel at simulating conservative, reversible systems governed by the Schrödinger equation via Hamiltonian simulation, simulating dissipative systems like the heat equation is fundamentally more difficult. This is due to the fact that sequential time-stepping methods suffer from an exponentially decaying success probability. In particular, we discuss the design of an end-to-end implementation of a solver for the heat equation using the space-time formulation, which circumvents this decay by solving the entire time-evolution history simultaneously.