This document presents a quantum-inspired solver for 2D Euler equations, accepted at the final phase of the Airbus-BWM Group Quantum Computing Challenge (ABQCC) 2024. We tackle the case study of Quantum Solvers for Predictive Aeroacoustic and Aerodynamic modeling tasks. We propose a tensor network based solver that scales polylogarithmically with the mesh size, in both runtime and memory. This provides a promising avenue for tackling the curse of dimensionality that plagues the direct numerical simulations in the field of computational fluid dynamics.
This document presents a quantum-inspired solver for 2D Euler equations, accepted at the final phase of the Airbus-BWM Group Quantum Computing Challenge (ABQCC) 2024. We tackle the case study of Quantum Solvers for Predictive Aeroacoustic and Aerodynamic modeling tasks. We propose a tensor network based solver that scales polylogarithmically with the mesh size, in both runtime and memory. This provides a promising avenue for tackling the curse of dimensionality that plagues the direct numerical simulations in the field of computational fluid dynamics.
Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.
Abstract Computational fluid dynamics is both a thriving research field and a key tool for advanced industry applications. However, the simulation of turbulent flows in complex geometries is a compute-power intensive task due to the vast vector dimensions required by discretized meshes. We present a complete and self-consistent full-stack method to solve incompressible fluids with memory and run time scaling logarithmically in the mesh size. Our framework is based on matrix-product states, a compressed representation of quantum states. It is complete in that it solves for flows around immersed objects of arbitrary geometries, with non-trivial boundary conditions, and self-consistent in that it can retrieve the solution directly from the compressed encoding, i.e. without passing through the expensive dense-vector representation. This framework lays the foundation for a generation of more efficient solvers of real-life fluid problems.
Accurately simulating turbulent, compressible fluids remains a major computational challenge for the aviation industry and a key to achieving future climate goals. Despite advances in computational fluid dynamics (CFD), limitations in turbulence models and the immense cost of high-fidelity simulations make scale-resolving full-aircraft simulations unrealistic. Quantum computing could overcome these barriers, as even few qubits can represent vast degrees of freedom. However, due to numerous obstacles such as the restriction to unitary operators, the read-out problem, and the limited availability of actual quantum hardware, the development of efficient algorithms for solving fluid mechanics equations is a real scientific challenge. This presentation highlights current efforts within the DLR QCI project ToQuaFlics to explore the potential of quantum computing for aerodynamic simulations.
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it restricts the mesh size since the solution of the linear system becomes computationally intractable when the mesh resolution increases. The reliance on the ability of High-Performance Computers (HPC) to scale up and meet these requirements is myopic; such very high-fidelity simulations require a paradigm shift in computing. This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. Leveraging recent works in Quantum Linear Solver Algorithms (QLSA) and variational algorithms suitable for Quantum Simulation in HPC, we aspire to push the boundaries of CFD-relevant problems that can be solved on hybrid quantum-classical framework. To this end, we consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem and investigate the accuracy of our approach by comparing results with a classical linear system of equation solvers, such as the Generalized Minimal RESidual method (GMRES). Through rigorous testing, our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches. Additionally, we demonstrate the accuracy, scalability, and consistency of our quantum method. Lastly, we present an insightful estimation of the resources our quantum algorithm needs to solve systems with nearly 2 billion mesh points.
Computational Fluid Dynamics (CFD) plays a central role in mechanical system design by enabling the simulation of complex flow phenomena such as turbulence, phase transitions, and fluid–structure interactions. However, high-fidelity CFD simulations—particularly those involving turbulent and multiphase flows—are often limited by the computational cost of solving large-scale, nonlinear, and high-dimensional systems of equations. This study explores the potential of quantum-enhanced computational fluid dynamics, focusing on hybrid quantum–classical solvers as a next-generation approach to overcoming these limitations. By integrating quantum algorithms with classical numerical methods, hybrid frameworks aim to accelerate key CFD tasks such as linear system solving, optimization, and high-dimensional state exploration while preserving the stability and accuracy of established solvers. The paper examines how quantum principles such as superposition and entanglement can be leveraged to improve convergence rates and design-space exploration in turbulence modeling and multiphase flow simulations. Through a conceptual and methodological analysis, this work highlights the opportunities, current constraints, and future implications of quantum-enhanced CFD for mechanical system design, offering a forward-looking perspective on the convergence of quantum computing and fluid mechanics.
The accurate simulation of turbulent compressible fluids is a major computational challenge for the aviation industry and possibly a key technology for improving the efficiency of future aircraft so that the ambitious climate targets can be achieved. Even though great progress has been made in the field of CFD in recent decades, and simulation software is available to a wide range of users from both commercial providers and research institutes, it still has two major shortcomings: a) The accuracy of the results is usually limited by the quality of the turbulence models. Even very advanced turbulence models such as Reynolds-Stress models are not able to accurately reproduce a wide range of flows, in particular at the border of the flight envelope. For unseen test cases involving complex physical phenomena or unconventional aircraft configurations, selecting an appropriate turbulence model is often challenging and not straightforward. b) In order to overcome the shortcomings resulting from turbulence modeling, scale-resolving methods such as large eddy simulations or even direct numerical simulations are increasingly being carried out. However, the cost of these simulations for flight Reynolds numbers is so high that even under very optimistic assumptions about the development of future HPC resources and advances in algorithms, full-aircraft simulations seem unrealistic. Even though the possibilities for the use of quantum computers in CFD are still limited due to their current stage of development, they may offer the potential to accelerate calculations in such a way that scale-resolving methods for complete aircraft seem possible. Since algorithms that can be efficiently implemented on quantum computers for solving compressible nonlinear transport equations are also largely not yet available, the time is now to deal with them.