# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 6 Future Works  
Regarding the next steps, we would like to scale up our approach to other applications besides CFD. This would include both linear and nonlinear use cases, such as thermodynamics, structural mechanics, electromagnetics and coupled multiphysics models, just to name a few. This would allow for the generalization of our method in order to use it for a wide range of industrial problems in the field of numerical simulations.  
The majority of quantum solvers promise exponential speedup compared to classical algorithms for problems where the problem domain is large enough. Thus, we can make our method more efficient by making a prior estimation of whether it is actually worth using a quantum solver as opposed to a classical one in terms of the computational time depending on the problem's complexity. A similar estimation could also be given for the reduced models, as their size could still be large enough to gain quantum speedup. Thisestimation is visualized as a decision tree in Figure 8. If the size is close to the boundaries of gaining any speedup from the quantum algorithm, an additional trade-off could be made. If the size of the ROM is further reduced, it can be solved faster and cheaper classically, but with less accuracy (as when we reduced the model size, we lost additional information). On the other hand, if the size of the ROM is kept at a larger size, it can be solved faster and more accurately with a quantum algorithm, but with a higher cost.  
Simulation models are often parameterized to allow for different variations in the geometry, shape, material, loading, boundary and initial conditions during their design. The described MOR methods in this paper are not robust enough with respect to parametric variations, and a new set of ROMs needs to be generated each time a parameter is changed in the system. This limits the applicability of our method for product design and optimization problems, as the whole workflow needs to be reset and started from sketch. Thus, another important work would be the development and integration of parametric reduced-order models (PROMs) into our workflow, which could reflect the parameter-dependent behavior in the reduced-order systems. Parametric model order reduction (PMOR) methods are intended to solve this exact problem, and could benefit greatly in product design and optimization problems, where exploration cycles can be achieved by carrying out the parametric simulations in the reduced space [65; 66; 67] and with parametrized geometries [68; 69].  
Conceptualization, S.J., Z.Z., A.K., T.M.; methodology, S.J., Z.Z., A.K., T.M.; software, S.J.; validation, S.J., Z.Z., A.K., T.M.; investigation, S.J., Z.Z., A.K., T.M.; writing--original draft preparation, S.J., Z.Z., A.K., T.M.; writing--review and editing, S.J., Z.Z., A.K., T.M.; supervision, A.K., Z.Z.; project administration, A.K. All authors have read and agreed to the published version of the manuscript.  
This research was supported by grants of the \"Application Domain Specific Highly Reliable IT Solutions\" project, which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding scheme. This research was also supported by the Ministry of Innovation and Technology and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary (NKFIH-873-1/2020).  
The authors declare no conflict of interest.  
Figure 8: Decision tree for whether it is worth using quantum solver over classical ones. The same question can be applied for both the original large system and its corresponding reduced model.