# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 5 Conclusions  
In this paper, we have proposed a new method for solving computationally complex problems in CFD applications with a substantial speedup and with a reduced development cost compared to standalone classical or quantum solvers. We have achieved this by combining quantum algorithms for solving systems of linear equations with suitable classical algorithms for data processing and reducing the complexity of the given problem (TPWL-POD). We carry this out by discretizing the physical model using FEM and solving the resulting system of linear equations for an initial training set of inputs. In the training stage, a suitable quantum algorithm is applied for solving the system of equations with an exponential speedup. In the second part, we use snapshot-based classical model order reduction techniques to reduce the complexity of the original problem, making it eligible to be solved classically with different excitations in future simulations, avoiding the need for a quantum solver. This set of linearized reduced-order models can be used for real-time calculations in embedded systems, model validation and optimization purposes.  
It is important to note that it is only possible to obtain accurate solutions if the parameter sets are in the range of the validity of the training set. Since it can be difficult to define the training trajectory in an optimal way, covering all the dynamics of interest of the system beforehand, this might require some experimenting at first. The positive side is that the training step only needs to be carried out once before the reduced models could be used for simulations, but any changes in the model, geometry and time-domain would require the retraining of the models. This would increase the number of times that the quantum solver is used, along with the development cost.  
After obtaining a set of linearized reduced-order models, they can be used for simulating the physical system with different excitation sets. Since the complexity and size of the original problem is greatly reduced at this point, it is clearly faster to solve the reduced models classically than to solve the original problem. We would also argue that solving the reduced models could be computationally faster on a classical computer than solving the initial problem with a quantum solver. If the system is large enough and has to be simulated with many different excitations or is used in embedded systems for real-time calculations, our hybrid approach could mean a faster computational time compared to both classical and quantum algorithms, and since a mostly reduced version of the original problem is solved on classical CPUs, the cost would be much less than a full quantum or classical solution each time with the full sized matrix.  
One of the other main advantages of our proposed method is that we only require the existence of the solutions, and do not have any prerequisites on how they are produced, meaning that it is general and flexible enough to replace the quantum solver and the reduction method for a more suitable one for a given application. This also makes it hard to estimate the limitations and constraints of our method, as it highly depends on the used algorithms. Therefore, one of the disadvantages of this method would be the difficulties of automatization, as there are no general rules for defining the training trajectories and critical points, as it can vary with each application.