Due to the advances in the manufacturing of quantum hardware in the recent years, significant research efforts have been directed towards employing quantum methods to solving problems in various areas of interest. Thus a plethora of novel quantum methods have been developed in recent years. In this paper, we provide a survey of quantum sampling methods along-side needed theory and applications of those sampling methods as a starting point for research in this area. This work focuses in particular on Gaussian Boson sampling, quantum Monte Carlo methods, quantum variational Monte Carlo, quantum Boltzmann Machines and quantum Bayesian networks. We strive to provide a self-contained overview over the mathematical background, technical feasibility, applicability for other problems and point out potential areas of future research.

Many promising applications of quantum computing with a provable speedup center around the HHL algorithm. Due to restrictions on the hardware and its significant demand on qubits and gates in known implementations, its execution is prohibitive on near-term quantum computers. Aiming to facilitate such NISQ-implementations, we propose a novel circuit approximation technique that enhances the arithmetic subrou-tines in the HHL, which resemble a particularly resource-demanding component in small-scale settings. For this, we provide a description of the algorithmic implementation of space-efficient rotations of polynomial functions that do not demand explicit arithmetic calculations inside the quantum circuit. We show how these types of circuits can be reduced in depth by providing a simple and powerful approximation technique. Moreover, we provide an algorithm that converts lookup-tables for arbitrary function rotations into a structure that allows an application of the approximation technique. This allows implementing approximate rotation circuits for many polynomial and non-polynomial functions. Experimental results obtained for realistic early-application dimensions show significant improve-ments compared to the state-of-the-art, yielding small circuits while achieving good approximations.

To solve 3SAT instances on quantum annealers they need to be transformed to an instance of Quadratic Unconstrained Binary Optimization (QUBO). When there are multiple transformations available, the question arises whether different transformations lead to differences in the obtained solution quality. Thus, in this paper we conduct an empirical benchmark study, in which we compare four structurally different QUBO transformations for the 3SAT problem with regards to the solution quality on D-Wave’s Advantage_system4.1. We show that the choice of QUBO transformation can significantly impact the number of correct solutions the quantum annealer returns. Furthermore, we show that the size of a QUBO instance (i.e., the dimension of the QUBO matrix) is not a sufficient predictor for solution quality, as larger QUBO instances may produce better results than smaller QUBO instances for the same problem. We also empirically show that the number of different quadratic values of a QUBO instance, combined with their range, can significantly impact the solution quality.

The analysis of network structure is essential to many scientific areas ranging from biology to sociology. As the computational task of clustering these networks into partitions, i.e., solving the community detection problem, is generally NP-hard, heuristic solutions are indispensable. The exploration of expedient heuristics has led to the development of particularly promising approaches in the emerging technology of quantum computing. Motivated by the substantial hardware demands for all established quantum community detection approaches, we introduce a novel QUBO-based approach that only needs number-of-nodes qubits and is represented by a QUBO matrix as sparse as the input graph’s adjacency matrix. The substantial improvement in the sparsity of the QUBO matrix, which is typically very dense in related work, is achieved through the novel concept of separation nodes. Instead of assigning every node to a community directly, this approach relies on the identification of a separation-node set, which, upon its removal from the graph, yields a set of connected components, representing the core components of the communities. Employing a greedy heuristic to assign the nodes from the separation-node sets to the identified community cores, subsequent experimental results yield a proof of concept by achieving an up to 95% optimal solution quality on three established real-world benchmark datasets. This work hence displays a promising approach to NISQ-ready quantum community detection, catalyzing the application of quantum computers for the network structure analysis of large-scale, real-world problem instances.

As an application domain where the slightest qualitative improvements can yield immense value, finance is a
promising candidate for early quantum advantage. Focusing on the rapidly advancing field of Quantum Natural Language Processing (QNLP), we explore the practical applicability of the two central approaches DisCoCat and Quantum-Enhanced Long Short-Term Memory (QLSTM) to the problem of sentiment analysis in finance. Utilizing a novel ChatGPT-based data generation approach, we conduct a case study with more than 1000 realistic sentences and find that QLSTMs can be trained substantially faster than DisCoCat while also achieving close to classical results for their available software implementations.

We introduce a novel approach to translate arbitrary 3-sat instances to Quadratic Unconstrained Binary Optimization (qubo) as they are used by quantum annealing (QA) or the quantum approximate optimization algorithm (QAOA). Our approach requires fewer couplings and fewer physical qubits than the current state-of-the-art, which results in higher solution quality. We verified the practical applicability of the approach by testing it on a D-Wave quantum annealer.

Quantum Transfer Learning (QTL) recently gained popularity as a hybrid quantum-classical approach for image classification tasks by efficiently combining the feature extraction capabilities of large Convolutional Neural Networks with the potential benefits of Quantum Machine Learning (QML). Existing approaches, however, only utilize gate-based Variational Quantum Circuits for the quantum part of these procedures. In this work we present an approach to employ Quantum Annealing (QA) in QTL-based image classification. Specifically, we propose using annealing-based Quantum Boltzmann Machines as part of a hybrid quantum-classical pipeline to learn the classification of real-world, large-scale data such as medical images through supervised training. We demonstrate our approach by applying it to the three-class COVID-CT-MD dataset, a collection of lung Computed Tomography (CT) scan slices. Using Simulated Annealing as a stand-in for actual QA, we compare our method to classical transfer learning, using a neural network of the same order of magnitude, to display its improved classification performance. We find that our approach consistently outperforms its classical baseline in terms of test accuracy and AUC-ROC-Score and needs less training epochs to do this.

Black-box optimization (BBO) can be used to optimize functions whose analytic form is unknown. A common approach to realising BBO is to learn a surrogate model which approximates the target black-box function which can then be solved via white-box optimization methods. In this paper, we present our approach BOX-QUBO, where the surrogate model is a QUBO matrix. However, unlike in previous state-of-the-art approaches, this matrix is not trained entirely by regression, but mostly by classification between ‘good’ and ‘bad’ solutions. This better accounts for the low capacity of the QUBO matrix, resulting in significantly better solutions overall. We tested our approach against the state-of-the-art on four domains and in all of them BOX-QUBO showed better results. A second contribution of this paper is the idea to also solve white-box problems, i.e. problems which could be directly formulated as QUBO, by means of black-box optimization in order to reduce the size of the QUBOs to the information-theoretic minimum. Experiments show that this significantly improves the results for MAX-k-SAT.

One way of solving 3sat instances on a quantum computer is to transform the 3sat instances into instances of Quadratic Unconstrained Binary Optimizations (QUBOs), which can be used as an input for the QAOA algorithm on quantum gate systems or as an input for quantum annealers. This mapping is performed by a 3sat-to-QUBO transformation. Recently, it has been shown that the choice of the 3sat-to-QUBO transformation can significantly impact the solution quality of quantum annealing. It has been shown that the solution quality can vary up to an order of magnitude difference in the number of correct solutions received, depending solely on the 3sat-to-QUBO transformation. An open question is: what causes these differences in the solution quality when solving 3sat-instances with different 3sat-to-QUBO transformations? To be able to conduct meaningful studies that assess the reasons for the differences in the performance, a larger number of different 3sat-to-QUBO transformations would be needed. However, currently, there are only a few known 3sat-to-QUBO transformations, and all of them were created manually by experts, who used time and clever reasoning to create these transformations. In this paper, we will solve this problem by proposing an algorithmic method that is able to create thousands of new and different 3sat-to-QUBO transformations, and thus enables researchers to systematically study the reasons for the significant difference in the performance of different 3sat-to-QUBO transformations. Our algorithmic method is an exhaustive search procedure that exploits properties of 4×4 dimensional pattern QUBOs, a concept which has been used implicitly in the creation of 3sat-to-QUBO transformations before, but was never described explicitly. We will thus also formally and explicitly introduce the concept of pattern QUBOs in this paper.

Quantum computing provides powerful algorithmic tools that have been shown to outperform established classical solvers in specific optimization tasks. A core step in solving optimization problems with known quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is the problem formulation. While quantum optimization has historically centered around Quadratic Unconstrained Optimization (QUBO) problems, recent studies show, that many combinatorial problems such as the TSP can be solved more efficiently in their native Polynomial Unconstrained Optimization (PUBO) forms. As many optimization problems in practice also contain continuous variables, our contribution investigates the performance of the QAOA in solving continuous optimization problems when using PUBO and QUBO formulations. Our extensive evaluation on suitable benchmark functions, shows that PUBO formulations generally yield better results, while requiring less qubits. As the multi-qubit interactions needed for the PUBO variant have to be decomposed using the hardware gates available, i.e., currently single- and two-qubit gates, the circuit depth of the PUBO approach outscales its QUBO alternative roughly linearly in the order of the objective function. However, incorporating the planned addition of native multi-qubit gates such as the global Mølmer-Sørenson gate, our experiments indicate that PUBO outperforms QUBO for higher order continuous optimization problems in general.

Applying new computing paradigms like quantum computing to the field of machine learning has recently gained attention. However, as high-dimensional real-world applications are not yet feasible to be solved using purely quantum hardware, hybrid methods using both classical and quantum machine learning paradigms have been proposed. For instance, transfer learning methods have been shown to be successfully applicable to hybrid image classification tasks. Nevertheless, beneficial circuit architectures still need to be explored. Therefore, tracing the impact of the chosen circuit architecture and parameterization is crucial for the development of beneficially applicable hybrid methods. However, current methods include processes where both parts are trained concurrently, therefore not allowing for a strict separability of classical and quantum impact. Thus, those architectures might produce models that yield a superior prediction accuracy whilst employing the least possible quantum impact . To tackle this issue, we propose Sequential Quantum Enhanced Training (SEQUENT) an improved architecture and training process for the traceable application of quantum computing methods to hybrid machine learning. Furthermore, we provide formal evidence for the disadvantage of current methods and preliminary experimental results as a proof-of-concept for the applicability of SEQUENT.

In recent years, quantum machine learning has seen a substantial increase in the use of variational quantum circuits (VQCs). VQCs are inspired by artificial neural networks, which achieve extraordinary performance in a wide range of AI tasks as massively parameterized function approximators. VQCs have already demonstrated promising results, for example, in generalization and the requirement for fewer parameters to train, by utilizing the more robust algorithmic toolbox available in quantum computing. A VQCs’ trainable parameters or weights are usually used as angles in rotational gates and current gradient-based training methods do not account for that. We introduce weight re-mapping for VQCs, to unambiguously map the weights to an interval of length 2π, drawing inspiration from traditional ML, where data rescaling, or normalization techniques have demonstrated tremendous benefits in many circumstances. We employ a set of five functions and evaluate them on the Iris and Wine datasets using variational classifiers as an example. Our experiments show that weight re-mapping can improve convergence in all tested settings. Additionally, we were able to demonstrate that weight re-mapping increased test accuracy for the Wine dataset by 10% over using unmodified weights.

Quadratic unconstrained binary optimization (QUBO) can be seen as a generic language for optimization problems. QUBOs attract particular attention since they can be solved with quantum hardware, like quantum annealers or quantum gate computers running QAOA. In this paper, we present two novel QUBO formulations for 𝑘-SAT and Hamiltonian Cycles that scale significantly better than existing approaches. For 𝑘-SAT we reduce the growth of the QUBO matrix from 𝑂(𝑘)to 𝑂(𝑙𝑜𝑔(𝑘)). For Hamiltonian Cycles the matrix no longer grows quadratically in the number of nodes, as currently, but linearly in the number of edges and logarithmically in the number of nodes. We present these two formulations not as mathematical expressions, as most QUBO formulations are, but as meta-algorithms that facilitate the design of more complex QUBO formulations and allow easy reuse in larger and more complex QUBO formulations.

Quadratic unconstrained binary optimization (QUBO) has become the standard format for optimization using quantum computers, i.e., for both the quantum approximate optimization algorithm (QAOA) and quantum annealing (QA). We present a toolkit of methods to transform almost arbitrary problems to QUBO by (i) approximating them as a polynomial and then (ii) translating any polynomial to QUBO. We showcase the usage of our approaches on two example problems (ratio cut and logistic regression).

Power grid partitioning is an important requirement for resilient distribution grids. Since electricity production is progressively shifted to the distribution side, dynamic identification of self-reliant grid subsets becomes crucial for operation. This problem can be represented as a modification to the well-known NP-hard Community Detection (CD) problem. We formulate it as a Quadratic Unconstrained Binary Optimization (QUBO) problem suitable for solving using quantum computation, which is expected to find better-quality partitions faster. The formulation aims to find communities with maximal self-sufficiency and minimal power flowing between them. To assess quantum optimization for sizeable problems, we apply a hierarchical divisive method that solves sub-problem QUBOs to perform grid bisections. Furthermore, we propose a customization of the Louvain heuristic that includes self-reliance. In the evaluation, we first demonstrate that this problem examines exponential runtime scaling classically. Then, using different IEEE power system test cases, we benchmark the solution quality for multiple approaches: D-Wave’s hybrid quantum-classical solvers, classical heuristics, and a branch-and-bound solver. As a result, we observe that the hybrid solvers provide very promising results, both with and without the divisive algorithm, regarding solution quality achieved within a given time frame. Directly utilizing D-Wave’s Quantum Annealing (QA) hardware shows inferior partitioning.

The simulation of many industrially relevant physical processes can be executed up to exponentially faster using quantum algorithms. However, this speedup can only be leveraged if the data input and output of the simulation can be implemented efficiently. While we show that recent advancements for optimal state preparation can effectively solve the problem of data input at a moderate cost of ancillary qubits in many cases, the output problem can provably not be solved efficiently in general. By acknowledging that many simulation problems arise only as a subproblem of a larger optimization problem in many practical applications however, we identify and define a class of practically relevant problems that does not suffer from the output problem: Quantum Simulation-based Optimization (QuSO). QuSO represents optimization problems whose objective function and/or constraints depend on summary statistic information on the result of a simulation, i.e., information that can be efficiently extracted from a quantum state vector. In this article, we focus on the LinQuSO subclass of QuSO, which is characterized by the linearity of the simulation problem, i.e., the simulation problem can be formulated as a system of linear equations. By cleverly combining the quantum singular value transformation (QSVT) with the quantum approximate optimization algorithm (QAOA), we prove that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component. Finally, we present two practically relevant use cases that fall within this subgroup of QuSO problems.

This work is a benchmark study for quantum-classical computing method with a real-world optimization problem from industry. The problem involves scheduling and balancing jobs on different machines, with a non-linear objective function. We first present the motivation and the problem description, along with different modeling techniques for classical and quantum computing. The modeling for classical solvers has been done as a mixed-integer convex program, while for the quantum-classical solver we model the problem as a binary quadratic program, which is best suited to the D-Wave Leap’s Hybrid Solver. This ensures that all the solvers we use are fetched with dedicated and most suitable model(s). Henceforth, we carry out benchmarking and comparisons between classical and quantum-classical methods, on problem sizes ranging till approximately 150000 variables. We utilize an industry grade classical solver and compare its results with D-Wave Leap’s Hybrid Solver. The results we obtain from D-Wave are highly competitive and sometimes offer speedups, compared to the classical solver.

This paper presents a novel optimization approach for allocating grid operation costs in Peer-to-Peer (P2P) electricity markets using Quantum Computing (QC). We develop a Quadratic Unconstrained Binary Optimization (QUBO) model that matches logical power flows between producer-consumer pairs with the physical power flow to distribute grid usage costs fairly. The model is evaluated on IEEE test cases with up to 57 nodes, comparing Quantum Annealing (QA), hybrid quantum-classical algorithms, and classical optimization approaches. Our results show that while the model effectively allocates grid operation costs, QA performs poorly in comparison despite extensive hyperparameter optimization. The classical branch-and-cut method outperforms all solvers, including classical heuristics, and shows the most advantageous scaling behavior. The findings may suggest that binary least-squares optimization problems may not be suitable candidates for near-term quantum utility.

Quantum Simulation-based Optimization (QuSO) is a recently proposed class of optimization problems that entails industrially relevant problems characterized by cost functions or constraints that depend on summary statistic information about the simulation of a physical system or process. This work extends initial theoretical results that proved an up-to-exponential speedup for the simulation component of the QAOA-based QuSO solver proposed by Stein et al. for the unit commitment problem by an empirical evaluation of the optimization component using a standard benchmark dataset, the IEEE 57-bus system. Exploiting clever classical pre-computation, we develop a very efficient classical quantum circuit simulation that bypasses costly ancillary qubit requirements by the original algorithm, allowing for large-scale experiments. Utilizing more than 1000 QAOA layers and up to 20 qubits, our experiments complete a proof of concept implementation for the proposed QuSO solver, showing that it can achieve both highly competitive performance and efficiency in its optimization component compared to a standard classical baseline, i.e., simulated annealing.

Time series forecasting is a valuable tool for many applications, such as stock price predictions, demand forecasting or logistical optimization. There are many well-established statistical and machine learning models that are used for this purpose. Recently in the field of quantum machine learning many candidate models for forecasting have been proposed, however in the absence of theoretical grounds for advantage thorough benchmarking is essential for scientific evaluation.
To this end, we performed a benchmarking study using real data of various quantum models, both gate-based and annealing-based, comparing them to the state-of-the-art classical approaches, including extensive hyperparameter optimization. Overall we found that the best classical models outperformed the best quantum models. Most of the quantum models were able to achieve comparable results and for one data set two quantum models outperformed the classical ARIMA model. These results serve as a useful point of comparison for the field of forecasting with quantum machine learning.

To increase efficiency in automotive manufacturing, newly produced vehicles can move autonomously from the production line to the distribution area. This requires an optimal placement of sensors to ensure full coverage while minimizing the number of sensors used. The underlying optimization problem poses a computational challenge due to its large-scale nature. Currently, classical solvers rely on heuristics, often yielding non-optimal solutions for large instances, resulting in suboptimal sensor distributions and increased operational costs.
We explore quantum computing methods that may outperform classical heuristics in the future. We implemented quantum annealing with D-Wave, transforming the problem into a quadratic unconstrained binary optimization formulation with one-hot and binary encoding. Hyperparameters like the penalty terms and the annealing time are optimized and the results are compared with default parameter settings.
Our results demonstrate that quantum annealing is capable of solving instances derived from real-world scenarios. Through the use of decomposition techniques, we are able to scale the problem size further, bringing it closer to practical, industrial applicability. Through this work, we provide key insights into the importance of quantum annealing parametrization, demonstrating how quantum computing could contribute to cost-efficient, large-scale optimization problems once the hardware matures.

This study evaluates the performance of a quantum-classical metaheuristic and a traditional classical mathematical programming solver, applied to two mathematical optimization models for an industry-relevant scheduling problem with autonomous guided vehicles (AGVs). The two models are: (1) a time-indexed mixed-integer linear program, and (2) a novel binary optimization problem with linear and quadratic constraints and a linear objective. Our experiments indicate that optimization methods are very susceptible to modeling techniques and different solvers require dedicated methods. We show in this work that quantum-classical metaheuristics can benefit from a new way of modeling mathematical optimization problems. Additionally, we present a detailed performance comparison of the two solution methods for each optimization model.

Exploiting the fact that samples drawn from a quantum annealer inherently follow a Boltzmann-like distribution, annealing-based Quantum Boltzmann Machines (QBMs) have gained increasing popularity in the quantum research community. While they harbor great promises for quantum speed-up, their usage currently stays a costly endeavor, as large amounts of QPU time are required to train them. This limits their applicability in the NISQ era. Following the idea of Noè et al. (2024), who tried to alleviate this cost by incorporating parallel quantum annealing into their unsupervised training of QBMs, this paper presents an improved version of parallel quantum annealing that we employ to train QBMs in a supervised setting. Saving qubits to encode the inputs, the latter setting allows us to test our approach on medical images from the MedMNIST data set (Yang et al., 2023), thereby moving closer to real-world applicability of the technology. Our experiments show that QBMs using our approach already achieve reasonable results, comparable to those of similarly-sized Convolutional Neural Networks (CNNs), with markedly smaller numbers of epochs than these classical models. Our parallel annealing technique leads to a speed-up of almost 70 % compared to regular annealing-based BM executions.

Optimizing the architecture of variational quantum circuits (VQCs) is crucial for advancing quantum computing (QC) towards practical applications. Current methods range from static ansatz design and evolutionary methods to machine learned VQC optimization, but are either slow, sample inefficient or require infeasible circuit depth to realize advantages. Quality diversity (QD) search methods combine diversity-driven optimization with user-specified features that offer insight into the optimization quality of circuit solution candidates. However, the choice of quality measures and the representational modeling of the circuits to allow for optimization with the current state-of-the-art QD methods like covariance matrix adaptation (CMA), is currently still an open problem. In this work we introduce a directly matrix-based circuit engineering, that can be readily optimized with QD-CMA methods and evaluate heuristic circuit quality properties like expressivity and gate-diversity as quality measures. We empirically show superior circuit optimization of our QD optimization w.r.t. speed and solution score against a set of robust benchmark algorithms from the literature on a selection of NP-hard combinatorial optimization problems.