Quantum computing is a rapidly developing field that has seen a huge amount of interest in the last couple of decades due to its promise of revolutionizing several domains of business and science. It presents a new way of doing computations by making use of fundamental properties of quantum mechanics such as superposition and entanglement. Optimization, on the other hand, is a field that is omnipresent in the industry and where small improvements can have a significant impact. This thesis aims to tackle optimization problems using quantum algorithms.NP-hard optimization problems are not believed to be exactly solvable through general polynomial time algorithms. Variational quantum algorithms (VQAs) to address such combinatorial problems have been of great interest recently. Such algorithms are heuristic and aim to obtain an approximate solution. The hardware, however, is still in its infancy and the current Noisy Intermediate Scale Quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions.An issue with contemporary quantum optimization algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is that they scale linearly with problem size. To tackle this issue, we present the LogQ encoding, using which we can design quantum variational algorithms that scale logarithmically with problem size - opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. We show how this algorithm can be applied to several combinatorial optimization problems such as Maximum Cut, Minimum Partition, Maximum Clique and Maximum Weighted Independent Set (MWIS). Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.Next, we apply the LogQ encoding to two use-cases for large companies such as TotalEnergies. Fleet conversion is the process of transitioning a fleet of vehicles to more sustainable and environmentally friendly alternatives. It is modeled as a column generation scheme with the MWIS problem as the sub-problem or worker problem. We use the LogQ method to solve the MWIS Workers and demonstrate how quantum and classical solvers can be used together in a hybrid manner to approach an industrial-sized use-case. Mesh segmentation refers to the process of dividing a complex mesh (composed of vertices, edges, and faces) into meaningful and semantically coherent parts or regions. Mesh segmentation plays an important part in computer modeling, which is extensively used in the core domains of TotalEnergies' activities such as Earth imaging, physical modeling for reservoirs, and others. We define the problem as a graph optimization problem and use the LogQ encoding to solve it.
Quantum computing is a rapidly developing field that has seen a huge amount of interest in the last couple of decades due to its promise of revolutionizing several domains of business and science. It presents a new way of doing computations by making use of fundamental properties of quantum mechanics such as superposition and entanglement. Optimization, on the other hand, is a field that is omnipresent in the industry and where small improvements can have a significant impact. This thesis aims to tackle optimization problems using quantum algorithms.NP-hard optimization problems are not believed to be exactly solvable through general polynomial time algorithms. Variational quantum algorithms (VQAs) to address such combinatorial problems have been of great interest recently. Such algorithms are heuristic and aim to obtain an approximate solution. The hardware, however, is still in its infancy and the current Noisy Intermediate Scale Quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions.An issue with contemporary quantum optimization algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is that they scale linearly with problem size. To tackle this issue, we present the LogQ encoding, using which we can design quantum variational algorithms that scale logarithmically with problem size - opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. We show how this algorithm can be applied to several combinatorial optimization problems such as Maximum Cut, Minimum Partition, Maximum Clique and Maximum Weighted Independent Set (MWIS). Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.Next, we apply the LogQ encoding to two use-cases for large companies such as TotalEnergies. Fleet conversion is the process of transitioning a fleet of vehicles to more sustainable and environmentally friendly alternatives. It is modeled as a column generation scheme with the MWIS problem as the sub-problem or worker problem. We use the LogQ method to solve the MWIS Workers and demonstrate how quantum and classical solvers can be used together in a hybrid manner to approach an industrial-sized use-case. Mesh segmentation refers to the process of dividing a complex mesh (composed of vertices, edges, and faces) into meaningful and semantically coherent parts or regions. Mesh segmentation plays an important part in computer modeling, which is extensively used in the core domains of TotalEnergies' activities such as Earth imaging, physical modeling for reservoirs, and others. We define the problem as a graph optimization problem and use the LogQ encoding to solve it.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
Quantum computing is an emerging field on the multidisciplinary interface between physics, engineering, and computer science with the potential to make a large impact on computational intelligence (CI). The aim of this paper is to introduce quantum approximate optimization methods to the CI community because of direct relevance to solving combinatorial problems. We introduce quantum computing and variational quantum algorithms (VQAs). VQAs are an effective method for the near-term implementation of quantum solutions on noisy intermediate-scale quantum (NISQ) devices with less reliable qubits and early-stage error correction. Then, we explain Farhi et al.'s quantum approximate optimization algorithm (Farhi's QAOA, to prevent confusion). This VQA is generalized by Hadfield et al. to the quantum alternating operator ansatz (QAOA), which is a nature-inspired (particularly, adiabatic) quantum metaheuristic for approximately solving combinatorial optimization problems on gate-based quantum computers. We discuss connections of QAOA to relevant domains, such as computational learning theory and genetic algorithms, discussing current techniques and known results regarding hybrid quantum-classical intelligence systems. We present a schematic of how QAOA is constructed, and also discuss how CI techniques can be used to improve QAOA. We conclude with QAOA implementations for the well-known maximum cut, maximum bisection, and traveling salesperson problems, which can serve as templates for CI practitioners interested in using QAOA.
NP-hard problems are not believed to be exactly solvable through general polynomial time algorithms. Hybrid quantum-classical algorithms to address such combinatorial problems have been of great interest in the past few years. Such algorithms are heuristic in nature and aim to obtain an approximate solution. Significant improvements in computational time and/or the ability to treat large problems are some of the principal promises of quantum computing in this regard. The hardware, however, is still in its infancy and the current noisy intermediate-scale quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions. An issue with quantum optimization algorithms such as the quantum approximate optimization algorithm is that they scale linearly with problem size. In this paper, we build upon a proprietary methodology which scales logarithmically with problem size---opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. To test the performance of the algorithm, we first find a way to apply it to a handful of NP-hard problems: Maximum Cut, Minimum Partition, Maximum Clique, Maximum Weighted Independent Set. Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
Méthodes d'optimisation variationnelles quantiques et leurs applications L'informatique quantique est un domaine en plein essor qui a suscité un intérêt considérable au cours des deux dernières décennies en raison de sa promesse de révolutionner plusieurs domaines des affaires et de la science. Il s'agit d'une nouvelle façon d'effectuer des calculs en utilisant les propriétés fondamentales de la mécanique quantique telles que la superposition et l'intrication. L'optimisation, quant à elle, est un domaine omniprésent dans l'industrie et où de petites améliorations peuvent avoir un impact significatif. Cette thèse vise à résoudre des problèmes d'optimisation à l'aide d'algorithmes quantiques.Les problèmes d'optimisation NP-difficiles ne sont pas considérés comme pouvant être résolus exactement par des algorithmes généraux en temps polynomial. Les algorithmes quantiques variationnels (VQA en anglais) destinés à résoudre ces problèmes combinatoires ont récemment fait l'objet d'un grand intérêt. Ces algorithmes sont heuristiques et visent à obtenir une solution approximative. Cependant, le matériel en est encore à ses débuts et les ordinateurs quantiques bruyants à échelle intermédiaire (NISQ en anglais) actuels ne sont pas en mesure d'optimiser les problèmes d'intérêt industriel. De plus, le stockage des qubits et l'introduction de l'intrication nécessitent des conditions physiques extrêmes. Les algorithmes d'optimisation quantique contemporains, tels que le algorithme d'optimisation approximative quantique, Quantum Approximate Optimization Algorithm (QAOA) en anglais, posent un problème : leur échelle est linéaire en fonction de la taille du problème. Pour résoudre ce problème, nous présentons l'encodage LogQ, qui permet de concevoir des algorithmes variationnels quantiques dont l'échelle est logarithmique avec la taille du problème, ce qui ouvre la voie au traitement de problèmes d'optimisation d'une ampleur sans précédent sur des ordinateurs quantiques basés sur des portes. Nous montrons comment cet algorithme peut être appliqué à plusieurs problèmes d'optimisation combinatoire tels que Maximum Cut, Minimum Partition, Maximum Clique and Maximum Weighted Independent Set (MWIS). Ensuite, ces algorithmes sont testés sur un simulateur quantique avec des graphes de plus d'une centaine de nœuds et sur un véritable ordinateur quantique jusqu'à des graphes de taille 256. À notre connaissance, il s'agit des plus grands problèmes réalistes d'optimisation combinatoire jamais exécutés sur une machine NISQ, dépassant de près de dix fois la taille des problèmes résolus précédemment.Ensuite, nous appliquons le codage LogQ à deux cas d'utilisation pour de grandes entreprises telles que TotalEnergies. La conversion de la flotte est le processus de transition d'une flotte de véhicules vers des alternatives plus durables et plus respectueuses de l'environnement. Il est modélisé comme un schéma de génération de colonnes avec le problème MWIS comme sous-problème ou problème de travailleur. Nous utilisons la méthode LogQ pour résoudre le problème de travailleur MWIS et démontrons comment les solveurs quantiques et classiques peuvent être utilisés ensemble pour aborder un cas d'utilisation de taille industrielle. La segmentation de maillage fait référence au processus de division d'un maillage complexe (composé de sommets, d'arêtes et de faces) en parties ou régions significatives et sémantiquement cohérentes. La segmentation du maillage joue un rôle important dans la modélisation informatique, qui est largement utilisée dans les domaines clés des activités de TotalEnergies, tels que l'imagerie terrestre, la modélisation physique des réservoirs, etc. Nous définissons le problème comme un problème d'optimisation de graphe et utilisons l'encodage LogQ pour le résoudre.