Combinatorial optimisation constitutes a central class of computational problems with broad relevance across science and industry, combining foundational theoretical interest with significant real-world impact. Many problems of practical importance admit concise mathematical formulations, yet resist efficient classical solution methods due to their NP-hard structure. Quantum computing offers several algorithmic paradigms that promise structural speedups for such problems. Most quantum and hybrid frameworks fall into one of the following three categories: algorithms based on continuous adiabatic evolution, such as quantum annealing; circuit-based variational quantum algorithms; circuit-based approaches based on Grover's algorithm. These three algorithmic families are conceptualised for different kinds and generations of quantum architectures, often studied in isolation from each other and from preexisting sophisticated classical methods. This thesis develops a unified, predominantly circuit-based treatment of quantum algorithms for combinatorial optimisation, with a particular focus on principled design transfers between near-term parametric algorithms and far-term fault-tolerant approaches. Starting from the Boolean circuit model, reversible and quantum circuit formulations are derived that allow for precise size and depth analysis of classical control logic, reversible embeddings, and quantum realisations of Boolean and pseudo-Boolean functions. These constructions form a common backbone for optimisation algorithms based on adiabatic evolution, variational principles, and amplitude amplification. Adiabatic quantum computation is analysed within this circuit framework, leading to explicit convergence conditions and resource bounds for discretised adiabatic evolutions driven by pseudo-Boolean cost functions of bounded degree. Variational quantum algorithms are then revisited with an emphasis on feasibility-preserving operations and mixing families. In this context, a novel class of exhaustive variational quantum algorithms is introduced, characterised by finite parameterisations that provably reach optimal solutions. These constructions naturally induce state-preparation routines that guarantee to generate superpositions of all feasible solutions. Finally, Grover’s algorithm and its generalisations are studied as optimisation methods, highlighting their structural dependence on feasible-state preparation and their conceptual proximity to variational and adiabatic approaches. By establishing explicit design transfer principles between these paradigms, this thesis provides a unified perspective on quantum optimisation algorithms, accompanied by concrete circuit constructions and quantitative resource analyses.
Combinatorial optimisation constitutes a central class of computational problems with broad relevance across science and industry, combining foundational theoretical interest with significant real-world impact. Many problems of practical importance admit concise mathematical formulations, yet resist efficient classical solution methods due to their NP-hard structure. Quantum computing offers several algorithmic paradigms that promise structural speedups for such problems. Most quantum and hybrid frameworks fall into one of the following three categories: algorithms based on continuous adiabatic evolution, such as quantum annealing; circuit-based variational quantum algorithms; circuit-based approaches based on Grover's algorithm. These three algorithmic families are conceptualised for different kinds and generations of quantum architectures, often studied in isolation from each other and from preexisting sophisticated classical methods. This thesis develops a unified, predominantly circuit-based treatment of quantum algorithms for combinatorial optimisation, with a particular focus on principled design transfers between near-term parametric algorithms and far-term fault-tolerant approaches. Starting from the Boolean circuit model, reversible and quantum circuit formulations are derived that allow for precise size and depth analysis of classical control logic, reversible embeddings, and quantum realisations of Boolean and pseudo-Boolean functions. These constructions form a common backbone for optimisation algorithms based on adiabatic evolution, variational principles, and amplitude amplification. Adiabatic quantum computation is analysed within this circuit framework, leading to explicit convergence conditions and resource bounds for discretised adiabatic evolutions driven by pseudo-Boolean cost functions of bounded degree. Variational quantum algorithms are then revisited with an emphasis on feasibility-preserving operations and mixing families. In this context, a novel class of exhaustive variational quantum algorithms is introduced, characterised by finite parameterisations that provably reach optimal solutions. These constructions naturally induce state-preparation routines that guarantee to generate superpositions of all feasible solutions. Finally, Grover’s algorithm and its generalisations are studied as optimisation methods, highlighting their structural dependence on feasible-state preparation and their conceptual proximity to variational and adiabatic approaches. By establishing explicit design transfer principles between these paradigms, this thesis provides a unified perspective on quantum optimisation algorithms, accompanied by concrete circuit constructions and quantitative resource analyses.
This paper presents a comprehensive theoretical framework that unifies various hybrid quantumclassical optimization approaches for combinatorial problems. Building upon recent advances in Quantum Approximate Optimization Algorithm (QAOA) and quantum-enhanced heuristics, we establish a mathematical foundation that connects amplitude amplification, counterdiabatic annealing, and classical relaxation techniques. We derive general convergence criteria and performance bounds for hybrid algorithms, demonstrating how quantum subroutines can be systematically integrated with classical optimization methods. The framework encompasses both unconstrained and constraint-preserving formulations, providing theoretical guarantees for problem classes including graph optimization, scheduling, and resource allocation. We analyze the trade-offs between quantum circuit depth, classical preprocessing overhead, and solution quality, establishing conditions under which exponential or polynomial speedups can be achieved. Our theoretical analysis reveals fundamental limitations and opportunities in hybrid algorithm design, offering guidelines for selecting appropriate quantum-classical decompositions based on problem structure. This work contributes to the growing understanding of quantum advantage in optimization and provides a roadmap for developing provably efficient hybrid algorithms across diverse application domains.
NP-hard problems are one of the most computationally intensive challenges in computer science, appearing in many domains such as logistics, cybersecurity, drug design, and financial modelling. Classical algorithms become impractical as problem size increases due to combinatorial explosion. This paper presents a structured framework for analysing quantum algorithms targeting NP-hard optimization problems, like the Traveling Salesman Problem (TSP), Max-Cut, and Boolean Satisfiability (SAT). We have developed a taxonomy which categorizes quantum approaches—quantum annealing, variational algorithms (QAOA/VQE), and Grover-based search—based on problem encoding strategies, resource requirements, and theoretical guarantees. From the comparative analysis of existing literature, we: 1) Map algorithm classes to problem domains (e.g., QAOA for Max-Cut, annealing for TSP) 2) Identify performance trade-offs between solution quality, qubit counts, and circuit depth 3) Analyse hardware compatibility constraints across NISQ platforms 4) Reveal fundamental scalability barriers including noise susceptibility and embedding overhead The framework establishes implementation guidelines for near- term quantum hardware and proposes hybrid quantum-classical pathways to bridge theoretical potential and practical deployment. This systematic analysis prioritizes research directions for achieving quantum advantage in combinatorial optimization.
The semiconductor supply chain gives rise to many complex, large and heavily constrained combinatorial optimization problems. We derive simpler problems from these complex formulations and analyze quantum algorithms for their solution. We study two main regimes: variational algorithms derived from the Quantum Approximate Optimization Algorithm (QAOA) and quantum search algorithms based on Grover’s search algorithm and Amplitude Amplification. In the first case, the main challenge lies in integrating problem constraints into the QAOA framework, which was originally devised for unconstrained problems such as MaxCut. The canonical method of adding penalty terms to the objective leads to an undesirable inflation of the energy spectrum. Instead, we examine alternative problem encodings as well as constraint-preserving mixing operators, which are designed to limit the quantum algorithm to the subspace of feasible states. We make explicit suggestions for a wide selection of problems. In the second case, we operate under the premise that any quantum search protocol for combinatorial optimization must use tailored initial states– otherwise the square root speedup of quantum search is not enough to offset the exponential size of the search space in a combinatorial optimization problem. Consequently, we examine state preparation routines for a wide range or problems. The main devices lie in preparing superpositions of only feasible states and superpositions which are centered around a reference solution. We present a benchmarking technique, in an exact and an approximate version, which allows us to generate performance metrics for our algorithms based on quantum search. We use the benchmarking tools to carry out numerical studies for a selection of problems. We conclude that our variational methods tailored to constrained problems while technically sound– lead to deep quantum circuits which are not suitable for the NISQ machines which QAOA was originally developed for. Our conclusion for optimization algorithms based on quantum search is more optimisitic. Their compatability with classical optimization principles, such as neighborhood search, gives rise to promising further research directions. Advanced benchmarking techniques allow us to test these algorithms independently of quantum hardware developments.
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.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Many important physical and mathematical problems are difficult to solve on classical computers, resulting in difficulty in finding new advanced materials, simulating quantum field theories and solving large combinatorial optimisation tasks. Quantum computers have shown great promise in being able to solve these (and many other) problems faster than their classical counterparts, and therefore should be thoroughly investigated as a significant tool for enabling the acceleration of scientific progress. This work develops methods to tackle several problems across the spectrum of quantum algorithms, providing methods suitable for the near-term hardware, to the future where large fault-tolerant quantum computers exist. Therefore, the works in this thesis range from efficiently making use of limited quantum hardware to taking advantage of hardware benefits (e.g. excess qubits) as quantum computation advances. Covered topics include: an alternative paradigm for the optimisation of variational circuits and quantum neural networks; techniques for improving energy estimation of quantum states through the collection and processing of large amounts of randomised measurement data; the use of entanglement in guiding the automatic discovery of quantum circuits; investigating the role of symmetries in adiabatic state preparation of molecules and the large scale parallelisation of a common fault-tolerant subroutine, potentially producing large runtime speed-ups for the simulation of chemistry on quantum computers. Although there are many challenges remaining from both hardware and algorithms perspectives before the large-scale utility of quantum computers can be demonstrated, the contributions of this work make steps towards the efficient utilisation of quantum hardware that will be available in the coming years.