# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 4 Quantum Reduced Finite Element Method
### Study: Flow Past a Cylinder  
As a demonstration of the POD reduction method, we show an example of an unsteady, incompressible flow past a cylinder placed in a channel at a right angle to the oncoming fluid. The cylinder is offset from the center of the flow to make the steady-state symmetrical flow unstable. Vortex shedding is an interesting and well-known phenomenon in fluid dynamics that has been studied in relation with MOR techniques for reduction purposes [47; 54; 55]. For Reynolds numbers below 100, the flow can be treated as a steady laminar flow, and for values above 100, the flow starts developing turbulent behavior. In this simulation, the Reynolds number is equal to 161, which is already enough to show signs of the Karman vortex street, as seen in Figure 6.  
The geometry definition and the simulation were carried out using OpenFOAM 2012 following the work of [PERSON] [56]. The full-order model has 85.720 degrees of freedom. The simulation was run for 3 s, with a fixed time step of \(dt=0.00001\) on eight cores in parallel on an Intel(R) Xeon(R) Gold 6248R, 3.00 GHz CPU for a total time of 10.309 s (\(\approx\)171 min). For the model-order reduction step, Illinois Rocstar LLC's AccelerateCFD was used, which is an open-source software that reduces a full-order CFD flow (defined in OpenFOAM) using the global POD reduction method. The POD basis was calculated using a total number of 250 snapshots evenly collected from the solutions of OpenFOAM. In Figure 7, we can see the cumulative flow energy in terms of how many basis vectors are selected. This can be used to decide the number of POD basis vectors used for computing the ROM.  
The ROM simulations were carried out using different values of the POD velocity space dimension \(N_{r}=14,33,54,75,95\). In these cases, the cumulative energies are 99.1%,  
Figure 6: Incompressible flow past a cylinder placed in a channel at a right angle to the oncoming fluid. The simulation was carried out in OpenFOAM, with Reynolds number set to 161. Comparison of velocity U between the full-order model (**a**) and the reduced-order model solutions (**b**).  
Figure 7: Cumulative energies of the obtained POD base vectors. The first 95 POD bases vectors have a cumulative energy above 99.9%, which means that the rest of the base vectors can be dismissed while still maintaining a high accuracy.  
99.91%, 99.99%, 99.99% and 99.9999%, respectively. The simulations were run in series on the same processor that was used for the FOM simulation. The ROM simulation results can be seen in Table 1.  
Naturally, the proposed method is optimal in terms of computational time and expenses compared to using either classical or quantum solvers alone. Since, for the offline training phase, a quantum solver is used, in this phase, the computations could be exponentially faster compared to a classical solver. On the other hand, if we consider the online simulation phase after the model-order reduction, the system's small complexity allows for a much faster simulation time compared to both quantum and classical solvers, as the reduction and speedup factor can be substantial, as seen in Table 1. In this paper, \(Re=161\) was used for simplicity, but for simulations of flow past a cylinder at high Reynolds numbers, refer to [57; 58]. For POD reduction methods applied to turbulent flows with high Reynolds numbers, refer to [59; 60; 61].