# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 2 Related Works  
The combination of quantum computing and finite element analysis holds great potential, especially in such cases where current classical computers are struggling to solve complex systems accurately in a relatively fast time, such as in computational fluid dynamics (CFD) applications or tightly coupled multiphysics systems. Nevertheless, this field has been mostly ignored before, but has been gaining more and more attention lately [28; 30; 31]. Since the output from FEM is often large-sized matrices that are produced algorithmically, it would be a great first step to find quantum algorithms that can solve a system of linear equations in the form \(Ax=b\) with exponential speedup. The FEM also leads to sparse system matrices, which is a useful and often required property to reach quantum speedup via the quantum solver algorithm. Lastly, as mentioned in the introduction, FEM has a wide range of industrial applications, such as solid mechanics, thermodynamics, computational fluid mechanics, chemistry and many more. Thus, any achievable speedup can mean a huge difference and may have a large impact on the industry.  
A quantum algorithm for solving large systems of linear equations was first introduced by [PERSON], [PERSON] and [PERSON] [32], and was later improved by [PERSON] [33]. In these papers, the quantum speedup in solving a linear system of equations is claimed to be exponential. However, there are challenges in efficiently preparing the classical input parameters and measuring the output states at each time step, as these two operations can cancel out the exponential speedup that the algorithm offers, as was later proven by [PERSON] and [PERSON] [28]. These problems need to be faced by any quantum algorithms that wish to have an exponential speedup compared to their classical counterparts. Nevertheless, there are new research activities mainly focused on solving non-linear systems of equations that claim to have found a workaround to the previously mentioned problems and have reached an exponential speedup in large scale systems [30; 31; 34; 35].  
In a recently published paper, [PERSON] et al. [30] proposed a method where the number of required resources scale quadratically and logarithmically in the integration time and in the dimension of space-time, respectively, thus preserving the exponential speedup of the solver. In a separate study by [PERSON] [33], a different approach was taken, where they used Carleman linearization to transform nonlinear problems into a system of linear equations. This system of linear equations consists of an infinite number of equations that have to be carefully truncated and reduced to a finite set to be applicable in real-world settings. In another recent paper by [PERSON] et al. [29], an alternative approach was proposed by taking advantage of quantum RAM (QRAM) in order to preserve the quantum speedup offered by the HHL algorithm. Although all of these results show great potential in the field of numerical simulation for a wide range of applications, such as in CFD, these methods will likely be refined, worked upon and used to develop new methods in the upcoming years, before a suitable quantum hardware is available.  
In this paper, we propose a method for applying the previously mentioned and other newly discovered quantum algorithms for real-life problems by taking advantage of their quantum speedup, but also taking the limited availability of quantum computers and their costs into account. We carry this out by minimizing the need to use quantum algorithms for solving PDEs by post-processing the solutions obtained from a few critical initial cases and creating ROMs, which can be efficiently solved on classical computers afterwards with different parameter sets. We solely focus on the FEM as our discretization method and will introduce it in Section 3. Nevertheless, the method we describe in this paper is generalizable in a sense that FEM can be replaced by other similar methods, such as FVM [36; 37].