Through recent progress in hardware development, quantum computers have advanced to the point where benchmarking of (heuristic) quantum algorithms at scale is within reach. Particularly in combinatorial optimization - where most algorithms are heuristics - it is key to empirically analyze their performance on hardware and track progress towards quantum advantage. To this extent, we present ten optimization problem classes that are difficult for existing classical algorithms and can (mostly) be linked to practically relevant applications, with the goal to enable systematic, fair, and comparable benchmarks for quantum optimization methods. Further, we introduce the Quantum Optimization Benchmarking Library (QOBLIB) where the problem instances and solution track records can be found. The individual properties of the problem classes vary in terms of objective and variable type, coefficient ranges, and density. Crucially, they all become challenging for established classical methods already at system sizes ranging from less than 100 to, at most, an order of 100,000 decision variables, allowing to approach them with today's quantum computers. We reference the results from state-of-the-art solvers for instances from all problem classes and demonstrate exemplary baseline results obtained with quantum solvers for selected problems. The baseline results illustrate a standardized form to present benchmarking solutions, which has been designed to ensure comparability of the used methods, reproducibility of the respective results, and trackability of algorithmic and hardware improvements over time. We encourage the optimization community to explore the performance of available classical or quantum algorithms and hardware platforms with the benchmarking problem instances presented in this work toward demonstrating quantum advantage in optimization.

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

We study the problem of decomposing a graph into a weighted sum of a small number of graph matchings. This problem arises in network resource allocation problems such as peer-to-peer energy exchange, and it is challenging to solve with current classical algorithms even for small instances. To address this problem, we propose a hybrid quantum-classical algorithm, E-FCFW, based on the Fully-Corrective Frank-Wolfe (FCFW) algorithm. In particular, E-FCFW extends FCFW by incorporating a matching-sampling subroutine that can be carried out classically or with a quantum approach. We show how to design such a subroutine using QAOA, which aims at solving a constrained discrete optimisation problem approximately to obtain solution-variety. We benchmark our approach on complete, bipartite, and heavy-hex graphs, conducting experiments using the Qiskit Aer state-vector simulator (9-25 qubits), the Qiskit Aer MPS simulator (52-76 qubits) and on IBM Kingston (52-111 qubits), demonstrating performance at a utility-scale quantum hardware level. Our results show that E-FCFW with QAOA consistently yields sparser decompositions (mean and median) than the other methods (random sampling and simulated annealing) for small complete and bipartite graphs. For large heavy-hex graphs with 50 and 70 nodes, E-FCFW with QAOA also outperforms the other methods in terms of approximation error. Our findings highlight a promising role for quantum subroutines in classical algorithms.

Fidelity quantum kernels have shown promise in classification tasks, particularly when a group structure in the data can be identified and exploited through a covariant feature map. In fact, there exist classification problems on which covariant kernels provide a provable advantage, thus establishing a separation between quantum and classical learners. However, their practical application poses two challenges: on one side, the group structure may be unknown and approximate in real-world data, and on the other side, scaling to the `utility' regime (above 100 qubits) is affected by exponential concentration. In this work, we address said challenges by applying fidelity kernels to real-world data with unknown structure, related to the scheduling of a fleet of electric vehicles, and to synthetic data generated from the union of subspaces, which is then close to many relevant real-world datasets. Furthermore, we propose a novel error mitigation strategy specifically tailored for fidelity kernels, called Bit Flip Tolerance (BFT), to alleviate the exponential concentration in our utility-scale experiments. Our multiclass classification reaches accuracies comparable to classical SVCs up to 156 qubits, thus constituting the largest experimental demonstration of quantum machine learning on IBM devices to date. For the real-world data experiments, the effect of the proposed BFT becomes manifest on 40+ qubits, where mitigated accuracies reach 80%, in line with classical, compared to 33% without BFT. Through the union-of-subspace synthetic dataset with 156 qubits, we demonstrate a mitigated accuracy of 80%, compared to 83% of classical models, and 37% of unmitigated quantum, using a test set of limited size.

The formation of energy communities is pivotal for advancing decentralized and sustainable energy management. Within this context, Coalition Structure Generation (CSG) emerges as a promising framework. The complexity of CSG grows rapidly with the number of agents, making classical solvers impractical for even moderate sizes. This suggests CSG as an ideal candidate for benchmarking quantum algorithms against classical ones. Facing ongoing challenges in attaining computational quantum advantage for exact optimization, we pivot our focus to benchmarking quantum and classical solvers for approximate optimization. Approximate optimization is particularly critical for industrial use cases requiring real-time optimization, where finding high-quality solutions quickly is often more valuable than achieving exact solutions more slowly. Our findings indicate that quantum annealing (QA) on DWave can achieve solutions of comparable quality to our best classical solver, but with more favorable runtime scaling, showcasing an advantage. This advantage is observed when compared to solvers, such as Tabu search, simulated annealing, and the state-of-the-art solver Gurobi, in finding approximate solutions for energy community formation involving over 100 agents. DWave also surpasses 1-round QAOA on IBM hardware. Our findings represent the largest benchmark of quantum approximate optimizations for a real-world dense model beyond the hardware's native topology, where D-Wave demonstrates a scaling advantage.

In this work, we aim to solve a practical use-case of unsupervised clustering which has applications in predictive maintenance in the energy operations sector using quantum computers. Using only cloud access to quantum computers, we complete a thorough performance analysis of what some current quantum computing systems are capable of for practical applications involving non-trivial mid-to-high dimensional datasets. We first benchmark how well distance estimation can be performed using two different metrics based on the swap-test, using angle and amplitude data embedding. Next, for the clustering performance analysis, we generate sets of synthetic data with varying cluster variance and compare simulation to physical hardware results using the two metrics. From the results of this performance analysis, we propose a general, competitive, and parallelized version of quantum $k$-means clustering to avoid some pitfalls discovered due to noisy hardware and apply the approach to a real energy grid clustering scenario. Using real-world German electricity grid data, we show that the new approach improves the balanced accuracy of the standard quantum $k$-means clustering by $67.8\%$ with respect to the labeling of the classical algorithm.

We propose a Hybrid classical-quantum Autoencoder (HAE) model, which is a synergy of a classical autoencoder (AE) and a parametrized quantum circuit (PQC) that is inserted into its bottleneck. The PQC augments the latent space, on which a standard outlier detection method is applied to search for anomalous data points within a classical dataset. Using this model and applying it to both standard benchmarking datasets, and a specific use-case dataset which relates to predictive maintenance of gas power plants, we show that the addition of the PQC leads to a performance enhancement in terms of precision, recall, and F1 score. Furthermore, we probe different PQC Ansätze and analyse which PQC features make them effective for this task.

In this project summary paper, we summarize the key results and use-cases explored in the German Federal Ministry of Education and Research (BMBF) funded project "Q-GRID" which aims to assess potential quantum utility optimization applications in the electrical grid. The project focuses on two layers of optimization problems relevant to decentralized energy generation and transmission as well as novel energy transportation/exchange methods such as Peer-2-Peer energy trading and microgrid formation. For select energy grid optimization problems, we demonstrate exponential classical optimizer runtime scaling even for small problem instances, and present initial findings that variational quantum algorithms such as QAOA and hybrid quantum annealing solvers may provide more favourable runtime scaling to obtain similar solution quality. These initial results suggest that quantum computing may be a key enabling technology in the future energy transition insofar that they may be able to solve business problems which are already challenging at small problem instance sizes.

Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields. Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging. In this article, we develop a quantum version of multi-output Gaussian Process (QGP) by implementing a well-known quantum algorithm called HHL, to perform the Kernel matrix inversion within the Gaussian Process. To reduce the large circuit depth of HHL a circuit optimization technique called Approximate Quantum Compiling (AQC) has been implemented. We further showcase the application of QGP for a real-world problem to estimate line parameters of an electrical grid. Using AQC, up to 13-qubit HHL circuit has been implemented for a 32x32 kernel matrix inversion on IBM Quantum hardware for demonstrating QGP based line parameter estimation experimentally. Finally, we compare its performance against noise-less quantum simulators and classical computation results.

Demand Side Response (DSR) is a strategy that enables consumers to actively participate in managing electricity demand. It aims to alleviate strain on the grid during high demand and promote a more balanced and efficient use of (renewable) electricity resources. We implement DSR through discount scheduling, which involves offering discrete price incentives to consumers to adjust their electricity consumption patterns to times when their local energy mix consists of more renewable energy. Since we tailor the discounts to individual customers' consumption, the Discount Scheduling Problem (DSP) becomes a large combinatorial optimization task. Consequently, we adopt a hybrid quantum computing approach, using D-Wave's Leap Hybrid Cloud. We benchmark Leap against Gurobi, a classical Mixed Integer optimizer in terms of solution quality at fixed runtime and fairness in terms of discount allocation. Furthermore, we propose a large-scale decomposition algorithm/heuristic for the DSP, applied with either quantum or classical computers running the subroutines, which significantly reduces the problem size while maintaining solution quality. Using synthetic data generated from real-world data, we observe that the classical decomposition method obtains the best overall \newp{solution quality for problem sizes up to 3200 consumers, however, the hybrid quantum approach provides more evenly distributed discounts across consumers.

Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing are prominent approaches for solving combinatorial optimization problems, such as those formulated as Quadratic Unconstrained Binary Optimization (QUBO). These algorithms aim to minimize the objective function $x^T Q x$, where $Q$ is a QUBO matrix. However, the number of two-qubit CNOT gates in QAOA circuits and the complexity of problem embeddings in Quantum Annealing scale linearly with the number of non-zero couplings in $Q$, contributing to significant computational and error-related challenges. To address this, we introduce the concept of \textit{semi-symmetries} in QUBO matrices and propose an algorithm for identifying and factoring these symmetries into ancilla qubits. \textit{Semi-symmetries} frequently arise in optimization problems such as \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, and \textit{Graph Isomorphism}. We theoretically demonstrate that the modified QUBO matrix $Q_{\text{mod}}$ retains the same energy spectrum as the original $Q$. Experimental evaluations on the aforementioned problems show that our algorithm reduces the number of couplings and QAOA circuit depth by up to $45\%$. For Quantum Annealing, these reductions also lead to sparser problem embeddings, shorter qubit chains and better performance. This work highlights the utility of exploiting QUBO matrix structure to optimize quantum algorithms, advancing their scalability and practical applicability to real-world combinatorial problems.

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.

In this article we demonstrate the applications of classical and quantum machine learning in quantum transport and spintronics. With the help of a two-terminal device with magnetic impurity we show how machine learning algorithms can predict the highly non-linear nature of conductance as well as the non-equilibrium spin response function for any random magnetic configuration. By mapping this quantum mechanical problem onto a classification problem, we are able to obtain much higher accuracy beyond the linear response regime compared to the prediction obtained with conventional regression methods. We finally describe the applicability of quantum machine learning which has the capability to handle a significantly large configuration space. Our approach is applicable for solid state devices as well as for molecular systems. These outcomes are crucial in predicting the behavior of large-scale systems where a quantum mechanical calculation is computationally challenging and therefore would play a crucial role in designing nano devices.

Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems.

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{\color{blue}, 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.

In this article we present the application of classical and quantum-classical hybrid anomaly detection schemes to explore exotic configuration with anomalous features. We consider the Anderson model as a prototype where we define two types of anomalies - a high conductance in presence of strong impurity and low conductance in presence of weak impurity - as a function of random impurity distribution. Such anomalous outcome constitutes an imperceptible fraction of the data set and is not a part of the training process. These exotic configurations, which can be a source of rich new physics, usually remain elusive to conventional classification or regression methods and can be tracked only with a suitable anomaly detection scheme. We also present a systematic study of the performance of the classical and the quantum-classical hybrid anomaly detection method and show that the inclusion of a quantum circuit significantly enhances the performance of anomaly detection which we quantify with suitable performance metrics. Our approach is quite generic in nature and can be used for any system that relies on a large number of parameters to find their new configurations which can hold exotic new features.

Electric Vehicles (EVs) are emerging as battery energy storage systems (BESSs) of increasing importance for different power grid services. However, the unique characteristics of EVs makes them more difficult to operate than dedicated BESSs. In this work, we apply a data-driven learning approach to leverage EVs as a BESS to provide capacity-related services to the grid. The approach uses machine learning to predict how to charge and discharge EVs while satisfying their operational constraints. As a paradigm application, we use flexible energy commercialization in the wholesale markets, but the approach can be applied to a broader range of capacity-related grid services. We evaluate the proposed approach numerically and show that when the number of EVs is large, we can obtain comparable objective values to CPLEX and approximate dynamic programming, but with shorter run times. These reduced run times are important because they allow us to (re)optimize frequently to adapt to the time-varying system conditions.

Quantum computing is emerging as a promising technology for solving complex optimization problems that arise in various engineering fields, and therefore has the potential to also significantly impact power electronics applications. This paper offers a concise tutorial on fundamental concepts in quantum computing, serving as both an introduction to the field and a bridge to its potential applications in power electronics. As a first step in this direction, the use of quantum computing for solving offline mixed-integer optimization problems commonly encountered in power electronics is examined. To this end, a simplified power electronics design problem is reformulated as a quadratic unconstrained binary optimization (QUBO) problem and executed on quantum hardware, despite current limitations such as low qubit counts and hardware noise. This demonstration marks a pioneering step towards leveraging quantum methods in power electronics. Moreover, the implications of ongoing advancements in quantum algorithms and hardware are discussed, highlighting their potential to enable the efficient solution of large-scale, multiobjective design and control problems. The presented findings suggest that early adoption and exploration of quantum computing could significantly expand the capabilities and performance of power electronic systems in the near future.

Quantum machine learning (QML) presents potential for early industrial adoption, yet limited access to quantum hardware remains a significant bottleneck for deployment of QML solutions. This work explores the use of classical surrogates to bypass this restriction, which is a technique that allows to build a lightweight classical representation of a (trained) quantum model, enabling to perform inference on entirely classical devices. We reveal prohibiting high computational demand associated with previously proposed methods for generating classical surrogates from quantum models, and propose an alternative pipeline enabling generation of classical surrogates at a larger scale than was previously possible. Previous methods required at least a high-performance computing (HPC) system for quantum models of below industrial scale (ca. 20 qubits), which raises questions about its practicality. We greatly minimize the redundancies of the previous approach, utilizing only a minute fraction of the resources previously needed. We demonstrate the effectiveness of our method on a real-world energy demand forecasting problem, conducting rigorous testing of performance and computation demand in both simulations and on quantum hardware. Our results indicate that our method achieves high accuracy on the testing dataset while its computational resource requirements scale linearly rather than exponentially. This work presents a lightweight approach to transform quantum solutions into classically deployable versions, facilitating faster integration of quantum technology in industrial settings. Furthermore, it can serve as a powerful research tool in search practical quantum advantage in an empirical setup.

With the increase of intermittent renewable generation resources feeding into the electrical grid, Distribution System Operators (DSOs) must find ways to incorporate these new actors and adapt the grid to ensure stability and enable flexibility. Dividing the grid into logical clusters entails several organization and technical benefits, helping overcome these this http URL, finding the optimal grid partitioning remains a challenging task due to its complexity. At the same time, a new technology has gained traction in the last decades for its promising speed-up potential in solving non-trivial combinatorial optimization problems: quantum computing. This work explores its application in Graph Partitioning using electrical modularity. We benchmarked several quantum annealing and hybrid methods on IEEE well-known test cases. The results obtained for the IEEE 14-bus test case show that quantum annealing DWaveSampler brings equal solutions or, for the optimal number partitions, a 1% improvement. For the more significant test cases, hybrid quantum annealing shows a relative error of less than 0.02% compared to the classical benchmark and for IEEE 118-bus test case shows time performance speed-up. The increment in performance would enable real time planning and operations of electrical grids in real time. This work intends to be the first step to showcase the potentials of quantum computing towards the modernization and adaption of electrical grids to the decentralized future of energy systems.