This paper establishes a theoretical framework for constraint-preserving quantum optimization that maintains problem structure throughout quantum algorithm execution, addressing a fundamental challenge in translating real-world optimization problems to quantum hardware. We develop mathematical formulations that embed hard constraints directly into quantum circuit design rather than relying solely on penalty methods, proving that certain constraint classes can be exactly enforced through quantum state space restriction. Our theory characterizes the complexity of constraint preservation across different problem classes including facility location, network optimization, and resource allocation, deriving conditions under which quantum algorithms can explore only feasible solution spaces. We introduce automated repair mechanisms with theoretical guarantees on convergence to valid solutions and establish bounds on the additional quantum resources required for constraint maintenance. The framework provides rigorous analysis of trade-offs between constraint flexibility, quantum circuit depth, and solution quality, identifying problem structures amenable to efficient constraint-preserving quantum formulations. We extend the theory to socioeconomic weighting and multi-objective scenarios, proving that preference structures can be incorporated without compromising feasibility guarantees. This theoretical foundation advances practical quantum optimization by ensuring quantum algorithms produce implementable solutions while maintaining provable performance characteristics across diverse constrained optimization domains including logistics, energy systems, and infrastructure planning.
This paper establishes a theoretical framework for constraint-preserving quantum optimization that maintains problem structure throughout quantum algorithm execution, addressing a fundamental challenge in translating real-world optimization problems to quantum hardware. We develop mathematical formulations that embed hard constraints directly into quantum circuit design rather than relying solely on penalty methods, proving that certain constraint classes can be exactly enforced through quantum state space restriction. Our theory characterizes the complexity of constraint preservation across different problem classes including facility location, network optimization, and resource allocation, deriving conditions under which quantum algorithms can explore only feasible solution spaces. We introduce automated repair mechanisms with theoretical guarantees on convergence to valid solutions and establish bounds on the additional quantum resources required for constraint maintenance. The framework provides rigorous analysis of trade-offs between constraint flexibility, quantum circuit depth, and solution quality, identifying problem structures amenable to efficient constraint-preserving quantum formulations. We extend the theory to socioeconomic weighting and multi-objective scenarios, proving that preference structures can be incorporated without compromising feasibility guarantees. This theoretical foundation advances practical quantum optimization by ensuring quantum algorithms produce implementable solutions while maintaining provable performance characteristics across diverse constrained optimization domains including logistics, energy systems, and infrastructure planning.
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.
Constrained combinatorial optimization with strict linear constraints underpins applications in drug discovery, power grids, logistics, and finance, yet remains computationally demanding for classical algorithms, especially at large scales. The Quantum Approximate Optimization Algorithm (QAOA) offers a promising quantum framework, but conventional penalty-based formulations distort optimization landscapes and demand deep circuits, undermining scalability on near-term hardware. In this work, we introduce Hamming Weight Operators, a new class of constraint-aware operators that confine quantum evolution strictly within the feasible subspace. Building on this idea, we develop Adaptive Hamming Weight Operator QAOA, which dynamically selects the most effective operators to construct shallow, problem-tailored circuits. We validate our approach on benchmark tasks from both finance and high-energy physics, specifically portfolio optimization and two-jet clustering with energy balance. Across these problems, our method inherently satisfies all constraints by construction, converges faster, and achieves higher Approximation Ratios than penalty-based QAOA, while requiring roughly half as many gates. By embedding constraint-aware operators into an adaptive variational framework, our approach establishes a scalable and hardware-efficient pathway for solving practical constrained optimization problems on near-term quantum devices.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.
Quantum Optimization (QO) is emerging as a promising approach to tackle hard combinatorial optimization problems by leveraging the unique computational properties of quantum hardware. In recent years, QO has gained traction in Machine Learning (ML), where problems such as clustering and vector quantization can be naturally expressed as Quadratic Unconstrained Binary Optimization (QUBO) problems. These ML applications often involve complex, high-dimensional data, which directly influences both the structure and scale of the resulting QUBO formulations. However, current quantum devices fall under the category of Noisy Intermediate-Scale Quantum (NISQ) hardware, which imposes significant limitations on qubit counts and small error rates. This dissertation investigates the impact of data complexity and data scale—as encountered in ML settings—on the effectiveness and feasibility of QO on NISQ devices. We propose novel, hardware-aware methods to overcome these challenges. The first part of this thesis focuses on the effect of data complexity for QO. We analyze how problem structure—particularly when QUBO formulations are derived from ML tasks—affects quantum solvability. We identify the spectral gap as a key factor in the success of adiabatic quantum algorithms and provide a theoretical and empirical analysis linking data-induced complexity to optimization performance.To address noise sensitivity and limited precision, we introduce a preprocessing framework based on a principled Branch-and-Bound algorithm that reduces dynamic range while preserving global optima. This improves robustness on both quantum annealers and classical solvers. In the second part, the thesis addresses challenges introduced by data scale. First, we propose a recursive Divide-and-Conquer strategy that breaks down large QUBO problems into tractable subproblems, demonstrated through an application to Bundle Adjustment in 3D scene reconstruction. Second, we introduce a variable and constraint generation method inspired by column generation for Integer Linear Programming, enabling dynamic QUBO size growth. This technique is successfully applied to Multi-Agent Pathfinding, where it scales well on NISQ devices while maintaining compatibility with classical solvers. Finally, we explore QUBO reformulations that reduce problem size through compact encodings and cyclic expansion, enabling the use of quantum hardware for tasks such as FPGA placement in chip design. Across all contributions, the developed algorithms are evaluated both theoretically and empirically, with experiments conducted on real quantum and quantum-inspired hardware. These methods significantly improve solution quality, scalability, and hardware compatibility, paving the way for more effective use of QO in practical settings. While tailored for NISQ era constraints, the results remain relevant for fault-tolerant quantum computing, offering insights into the long-term viability of QO across hardware generations.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the p-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.
Optimizing objective functions stands to benefit significantly from leveraging quantum computers, promising enhanced solution quality across various application domains in the future. However, harnessing the potential of quantum solvers necessitates formulating problems according to the Quadratic Unconstrained Binary Optimization (QUBO) model, demanding significant expertise in quantum computation and QUBO formulations. This expertise barrier limits access to quantum solutions. Fortunately, automating the conversion of conventional optimization problems into QUBO formulations presents a solution for promoting accessibility to quantum solvers. This article addresses the unmet need for a comprehensive automatic framework to assist users in utilizing quantum solvers for optimization tasks while preserving interfaces that closely resemble conventional optimization practices. The framework prompts users to specify variables, optimization criteria, as well as validity constraints and, afterwards, allows them to choose the desired solver. Subsequently, it automatically transforms the problem description into a format compatible with the chosen solver and provides the resulting solution. Additionally, the framework offers instruments for analyzing solution validity and quality. Comparative analysis against existing libraries and tools in the literature highlights the comprehensive nature of the proposed framework. Two use cases (the knapsack problem and linear regression) are considered to show the completeness and efficiency of the framework in real-world applications. Finally, the proposed framework represents a significant advancement towards automating quantum computing solutions and widening access to quantum optimization for a broader range of users. The framework is publicly available on GitHub (https://github.com/cda-tum/mqt-qao) as part of the Munich Quantum Toolkit (MQT).