# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 1 Introduction  
The aim of numerical simulations is to approximate the behavior of physical systems on a computer with high precision. It is now a well established methodology that is widely used in many fields, such as solid mechanics, thermodynamics, computational fluid dynamics and chemistry, for a wide range of industries that require accurate solutions. It plays a crucial part in the designing phase of product development, where numerical simulations are carried out in advance for virtual prototyping, prior testing and optimization for time and cost efficiency. Numerical simulations can give us insights into understanding component and system behaviors for large-scale complex applications under extreme conditions and use cases that could be impossible to test physically beforehand. The range of application areas is growing constantly, with some examples being stress-strain simulations estimating the crash worthiness of automobile parts [1], stress analyses of high speed helical gears to reduce failures and for optimal design [2] and accurately predicting and confirming vehicle flight characteristics for Boeing commercial airplanes [3]. Among all the available numerical methods, the most commonly used ones are the finite difference method (FDM) [4; 5], the finite element method (FEM) [6; 7; 8] and the finite volume method (FVM) [9; 10; 11]. These methods approximately solve differential equations that describe the underlying physics behind the mathematical models in the simulation. It works by discretizing the domain and converting the differential equations into a (typically large) set of linear equations. Although solving these large systems of linear equations with high performance computing (HPC) seems to be acceptable in many use cases, there are still many real-life processes that are difficult to simulate accurately due to their complexity and large size. Turbulent flows are examples of such complex systems that require more advanced methods to approach and to predict their behaviors [12; 13; 14]. Model order reduction (MOR) is a method that aims to reduce the computational complexity of large-scale dynamical and control systems. The main motivation behind MOR techniques is to remove the features and information of the full-order model (FOM) that are less important in terms of a given application while still maintaining a relatively high accuracy. By reducing the state-space dimension of the model (or degrees of freedom), an approximation to the original model is computed, which is commonly referred to as a reduced-order model (ROM) [15; 16; 17; 18]. ROMs are small in complexity and cheap in terms of computational time; thus, they can be effectively applied in the early stages of product development, as in conceptual design, virtual prototyping and optimization (such as in naval shape design [19] and wind-driven ocean flows [20]), where highly accurate results are not yet desired. This allows for experimenting with different geometries and parameters under certain scenarios in a fast, iterative process before a high-order simulation is carried out for more accurate results and validation purposes. Reduced-order models are also extremely useful in settings where it is often unfeasible to perform numerical simulations using the complete full-order model due to limitations in computational resources or the requirements of the simulations setting. This is often the case in real-time simulation settings where a large number of simulations need to be performed [21; 22]. For more applications of MOR in fields such as engineering, electromagnetics and neuroscience, refer to [23]. Most MOR methods can be classified as projection- or truncation-based techniques. Projection-based MOR techniques aim to lower the computational complexity by finding an appropriate space of reduced dimension where the high dimensional state-space can be projected onto and approximated by.