#### Automatic Feature Map Generation

The demonstrator shows the generation of a quantum feature map for a simple classification or regression problem. Techniques from reinforcement learning are used here and the decision-making process of the AI agent is shown by means of a simple visualization. The results show a feature map design tailored to the problem.

In this demonstration, we introduce a quantum Neural Network (QNN) designed to tackle a 2D partial differential equation (PDE) problem. The demonstrator consists of two parts. In the first part of the notebook, we elucidate the motivation of using QNN to solve PDEs and delve into the theoretical aspect of the QNN. The notebook provides a step-by-step guide on employing variational quantum circuits (VQC) to construct the QNN and the associated training process. The concept of Physics Informed Quantum Neural Network (PIQNN) is also introduced in this section to expedite convergence. The second part focuses on applying the QNN and PIQNN to solve the Poisson equation as the target PDE. Execution takes place on both a quantum simulator and the IBM quantum system at Ehningen. Results and insights derived from these experiments are presented and discussed to provide an understanding of the QNN's performance and the differences between QNN and PIQNN.

#### Error Mitigation by Zero Noise Extrapolation

Strategies for error mitigation are of crucial importance for the further development of quantum computing. Zero-noise extrapolation (ZNE) is a widely used method. In this demonstrator, we introduce "Inverted-Circuit Zero-Noise Extrapolation (IC-ZNE)", which provides a new approach to error estimation and mitigation. The code can also be used to easily adapt ZNE to an algorithm.

#### Zero-Noise Extrapolation

The idea of zero-noise extrapolation (ZNE for short) is based on the assumption that it is possible to increase the strength of the noise in a quantum circuit, e.g. by inserting additional gates. If the strength of the noise is changed several times (single, triple, quintuple error strength), a fit can then be performed through the measured points. This extrapolates to the error-free case. Since the main errors are caused by faulty CNOT gates, the simplest method is to replace each CNOT gate with 3 (or even 5) CNOT gates and thus amplify the error by a factor of 3 (or 5). In the error-free case, this would not change the circuit. The Python library developed in the project is used for all algorithms in which expected values are calculated. In the Jupyter Notebook example, we demonstrate the application of the ZNE for the HHL algorithm with 4 qubits. This algorithm solves a two-dimensional, linear system of equations. The function 𝐹 we are interested in is the norm of the corresponding solution.

#### Error Mitigation Service

The Error Mitigation Service was developed as part of the SEQUOIA project and can be used to reduce the effects of errors in noisy measurement results of a quantum computer. It is available as an open source project on GitHub. This service enables the creation and management of calibration and mitigation data for various QPU vendors. It also allows users to improve their execution results based on newly generated or existing mitigation data. The Error Mitigation Service currently implements several methods, such as Mthree or TPNM for IBMQ and IonQ. It also supports error mitigation for results obtained on quantum simulators with emulated noise. Users can choose between full noise models and noise models containing only readout errors.

#### ADMM-Surrogate

Mixed-integer linear programs or mixed-binary linear programs are an important optimization problem and interesting object for quantum computing. This notebook shows how a mixed-binary problem can be optimized with the help of a quantum computer. Two different optimization strategies are implemented; a strategy inspired by classical ADMM algorithms and another one using a Kriging surrogate model on top. Both of them use the VQE algorithm to optimize the binary problem. They can be used and tested with different ansatz functions and optimizers for VQE. Currently mixed-binary equality constraints and intervals for the continuous variables are supported.

#### 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.