# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 4 Quantum Reduced Finite Element Method  
In this section, we will describe the different steps and details of our proposed method, combining all the preliminaries that we have seen in the previous sections. Our main motivation was to develop an approach where challenging problems in CFD applications can be accurately simulated with a minimization in computational time and development costs. The approach we have found combines quantum and classical algorithms that output a reusable set of linearized reduced-order models that can be efficiently simulated with different excitation sets on classical computers. The steps in our method are briefly summarized in Figure 5, starting with the modeling of the physically relevant system that we plan to simulate numerically. This can be carried out in many commercially available FEA softwares, such as COMSOL, Abaqus, Ansys, OpenFOAM and Siemens NX, just to name a few of the most widely used ones. After obtaining an accurate model of the physical system, along with the governing equations and properties, we have to solve the originally large system for some initial states, which are called trajectories (this step is also called the offline training). The initial trajectories have to be carefully selected in order to obtain accurate results for the different excitation sets that we would like to simulate the physical system with using the reduced state later on. The quantum solver in this context is taken as a black box that is assumed to solve the system of linear equations (exponentially) faster than existing classical algorithms in every step of the time integration. This is the part where we can achieve the most speedup, but it can also be the most expensive step if we are not careful enough and use the quantum hardware for more time than necessary, assuming that the price of a QCPU is much higher  
Figure 5: Visualization of the proposed quantum-aided model order reduction method highlighting the different steps in the workflow. This method can be used to generate linearized reduced-order models for efficiently solving complex simulations approximately with a high accuracy on classical computers. It is a combination of quantum and classical algorithms, where a quantum algorithm is used for solving the originally large problem in the offline training phase, with the results being post-processed by classical MOR techniques. than that of a regular CPU. The solutions that we obtain from the solver need to be post-processed classically in order to acquire a reusable system of linearized reduced-order models, as seen in Section 3.3. For this, we first need to find the critical points in our system that hold the most nonlinearity and to linearize the system around these points. After the local linearization, we can start the reduction procedure at the given points to gather a set of reduced linear models. In the last step, a weight has to be assigned to each model. The weighting function typically assigns a large weight to models close to the given point in the simulation and assigns a small weight otherwise, as seen in Equation (41). The result is a weighted linearized set of ROMs that can be used for approximating the solutions of the original system. This approximation can be carried out with different parameter sets, besides what was used at the offline training part (but within its range of validity) on classical computers with much efficiency. The disadvantage of the TPWL is that it is only applicable for excitations that are in the validation range of the offline training part. If different excitations need to be used that are outside the validation range, a new offline training part needs to be carried out, and the quantum solver has to be applied again in that case. Additional steps could include an evaluation of the results and validation by comparing the gathered solutions to the ones obtained by solving the original full system.