#### Quantum-based Computational Fluid Dynamics

Current quantum computers are characterized by small and error-prone systems. The search for algorithms that can be used advantageously under these conditions has attracted a lot of attention in recent years. Promising candidates in this context are variational quantum algorithms, where only part of the algorithm is executed on the quantum computer. Therefore, they usually require fewer qubits and qubit gates and can deal with the errors of a real quantum computer. An important example of variational quantum algorithms is the so-called quantum circuit learning algorithm (QCL), which can approximate arbitrary functions. It is particularly interesting because it can be combined with the parameter shift rule to solve non-linear differential equations. This demonstrator aims to explain the basics of QCL and uses an example to show how different functions can be approximated. It also shows how the multi-qubit character of QCL can be utilized in a targeted manner. Thus, several functions can be approximated simultaneously with the same QCL circuit by measuring multiple qubits. Finally, it is described how arbitrary differential equations can be solved in combination with the parameter shift rule.

#### Computational Fluid Dynamics (VQLS)

In this demonstrator, we show how partial differential equations can be solved on a quantum computer with linear systems of equations. The demonstrator consists of three Jupyter notebooks. The first notebook presents the theory for converting partial differential equations into linear systems of equations using finite difference methods together with an interactive code for solving the equations. The quantum algorithm VQLS is the focus of the second and third notebooks. The second notebook discusses the theory and implementation of the global cost function, explains the code and illustrates the results using a simple example. In the third notebook, the theory and implementation of local cost functions are discussed using another illustrative example.

#### Disclaimer

The interactive demonstrator notebooks have been licensed under the Apache licence (version 2.0). The files may only be used in accordance with the licence. A copy of the licence can be downloaded from http://www.apache.org/licenses/LICENSE-2.0 Except as required by applicable law or agreed to in writing, software distributed under this licence is distributed on an "AS IS" basis, without warranties or conditions of any kind, either express or implied. See the licence for the specific rights and restrictions associated with it.

This is a research prototype. Liability for loss of profit, loss of production, business interruption, loss of use, loss of data and information, financing costs and other financial and consequential damage is excluded, except in cases of gross negligence, intent and personal injury.