This paper presents a novel approach to quantum architecture search by integrating the techniques of ZX-calculus with Genetic Programming (GP) to optimize the structure of parameterized quantum circuits employed in quantum machine learning (QML). Recognizing the challenges in designing efficient quantum circuits for QML, we propose a GP framework that utilizes mutations defined via ZX-calculus, a graphical language that can simplify visualizing and working with quantum circuits. Our methodology focuses on evolving quantum circuits with the aim of enhancing their capability to approximate functions relevant in various machine learning tasks. We introduce several mutation operators inspired by the transformation rules of ZX-calculus and investigate their impact on the learning efficiency and accuracy of quantum circuits. The empirical analysis involves a comparative study where these mutations are applied to a diverse set of quantum regression problems, measuring performance metrics such as the percentage of valid circuits after the mutation, improvement of the objective, and circuit depth and width. Our results indicate that certain ZX-calculus-based mutations perform significantly better than others for quantum architecture search (QAS) in all metrics considered. They suggest that ZX-diagram-based QAS results in shallower circuits and more uniformly allocated gates than crude genetic optimization based on the circuit model. The code used for the numerical experiments is open source and can be found at TODO https://gitlab.cc-asp.fraunhofer.de/itwm-fm-qc-public/cvqa.

In this paper, we consider the inclusion of the solvency capital requirement (SCR) into portfolio optimization by the use of a quadratic proxy model. The Solvency II directive requires insurance companies to calculate their SCR based on the complete loss distribution for the upcoming year. Since this task is, in general, computationally challenging for insurance companies (and therefore, not taken into account during portfolio optimization), employing more feasible proxy models provides a potential solution to this computational difficulty. Here, we present an approach that is also suitable for future applications in quantum computing. We analyze the approximability of the solvency capital ratio in a quadratic form using machine learning techniques. This allows for an easier consideration of the SCR in the classical mean-variance analysis. In addition, it allows the problem to be formulated as a quadratic unconstrained binary optimization (QUBO), which benefits from the potential speedup of quantum computing. We provide a detailed description of our model and the translation into a QUBO. Furthermore, we investigate the performance of our approach through experimental studies.

Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning. We analyze the frequency spectrum using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area $$A:=R\cdot L$$ that preserves the frequency spectrum, where R denotes the number of qubits and L the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in R and L in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area $$A=RL$$ and not on the individual values of R and L . Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area A , thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on $$A = RL$$ , but also on the choice of ( R , L ), so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability. While the maximality analysis limits the space of attainable frequency spectra and characterizing these spectra helps to understand the effect of the ansatz choice, it does not by itself provide sufficient criteria for ansatz selection, and further considerations are required.

Risk measures are important key figures to measure the adequacy of the reserves of a company. The most common risk measures in practice are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Recently, quantum-based algorithms are introduced to calculate them. These procedures are based on the so-called quantum amplitude estimation algorithm which lead to a quadratic speed up compared to classical Monte-Carlo based methods. Based on these ideas, we construct quantum-based algorithms to calculate alternatives for VaR and CVaR, namely the Expectile Value-at-Risk (EVaR) and the Range Value-at-Risk (RVaR). We construct quantum algorithms to calculate them. These algorithms are based on quantum amplitude estimation. In a case study, we compare their performance with the quantum-based algorithms for VaR and CVaR. We find that all of the algorithms perform sufficiently well on a quantum simulator. Further, the calculations of EVaR and VaR are robust against noise on a real quantum device. This is not the case for CVaR and RVaR.

Classical Monte Carlo algorithms can theoretically be sped up on a quantum computer by employing amplitude estimation (AE). To realize this, an efficient implementation of state-dependent functions is crucial. We develop a straightforward approach based on pretraining parameterized quantum circuits, and show how they can be transformed into their conditional variant, making them usable as a subroutine in an AE algorithm. To identify a suitable circuit, we propose a genetic optimization approach that combines variable ansatzes and data encoding. We apply our algorithm to the problem of pricing financial derivatives. At the expense of a costly pretraining process, this results in a quantum circuit implementing the derivatives' payoff function more efficiently than previously existing quantum algorithms. In particular, we compare the performance for European vanilla and basket options.

This paper presents methodological improvements to variational quantum algorithms (VQAs) for solving multicriteria optimization problems. We introduce two key contributions. First, we reformulate the parameter optimization task of VQAs as a multicriteria problem, enabling the direct use of classical algorithms from various multicriteria metaheuristics. This hybrid framework outperforms the corresponding single-criteria VQAs in both average and worst-case performance across diverse benchmark problems. Second, we propose a method that augments the hypervolume-based cost function with coverage-oriented indicators, allowing explicit control over the diversity of the resulting Pareto front approximations. Experimental results show that our method can improve coverage by up to 40\% with minimal loss in hypervolume. Our findings highlight the potential of combining quantum variational methods with classical population-based search to advance practical quantum optimization.