A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.
A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.
Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.
While variational quantum algorithms (VQAs) have demonstrated considerable success in unconstrained optimization, their application to constrained combinatorial problems face a trade-off. Penalty-based methods, despite their circuit simplicity, suffer from a fundamental limitation: inefficient sampling in vast infeasible regions. This often results in suboptimal solutions that violate constraints and impede convergence to high-quality results. In contrast, ansatz-based approaches enforce solution feasibility by design but require complex, problem-specific circuits that are challenging to implement on current noisy intermediate-scale quantum devices. To overcome these limitations, we introduce an alternative VQA whose core innovation lies in a strategically designed loss function. This function offers a dual advantage. First, it is provably guaranteed that its global minimum corresponds uniquely to the optimal feasible solution, as this is achieved by ensuring universally higher loss values for all infeasible solutions. Second, it furnishes distinct computational pathways for feasible versus infeasible regions, thus creating clear and non competing guidance for the optimizer. As a result of these combined features, the algorithm's overall performance is significantly enhanced. Regarding hardware overhead, our design requires adding only an efficient validation oracle module to the penalty-based circuit, resulting in a circuit complexity significantly lower than that of ansatz-based approaches with their custom mixers. To validate the practical efficiency of our method, we empirically demonstrate its effectiveness by solving minimum vertex cover and maximum independent set problems on random graphs of varying small-scale sizes.
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.
We propose a post-processing variationally scheduled quantum algorithm (pVSQA) for solving constrained combinatorial optimization problems (COPs). COPs are typically transformed into ground-state search problems of the Ising model on a quantum annealer or gate-based quantum device. Variational methods are used to find an optimal schedule function that leads to high-quality solutions in a short amount of time. Post-processing techniques convert the output solutions of the quantum devices to satisfy the constraints of the COPs. pVSQA combines the variational methods and the post-processing technique. We obtain a sufficient condition for constrained COPs to apply pVSQA based on a greedy post-processing algorithm. We apply the proposed method to two constrained NP-hard COPs: the graph partitioning problem and the quadratic knapsack problem. pVSQA on a simulator shows that a small number of variational parameters is sufficient to achieve a (near-)optimal performance within a predetermined operation time. Then building upon the simulator results, we implement pVSQA on a quantum annealer and a gate-based quantum device. The experimental results demonstrate the effectiveness of our proposed method.
Understanding the benefits of quantum computing for solving combinatorial optimization problems (COPs) remains an open research question. In this work, we extend and analyze algorithms that solve COPs by recursively shrinking them. The algorithms leverage correlations between variables extracted from quantum or classical subroutines to recursively simplify the problem. We compare the performance of the algorithms equipped with correlations from the quantum approximate optimization algorithm (QAOA) as well as the classical linear programming (LP) and semi-definite programming (SDP) relaxations. This allows us to benchmark the utility of QAOA correlations against established classical relaxation algorithms. We apply the recursive algorithm to MaxCut problem instances with up to a hundred vertices at different graph densities. Our results indicate that LP outperforms all other approaches for low-density instances, while SDP excels for high-density problems. Moreover, the shrinking algorithm proves to be a viable alternative to established methods of rounding LP and SDP relaxations. In addition, the recursive shrinking algorithm outperforms its bare counterparts for all three types of correlations, i.e., LP with spanning tree rounding, the Goemans-Williamson algorithm, and conventional QAOA. While the lowest depth QAOA consistently yields worse results than the SDP, our tensor network experiments show that the performance increases significantly for deeper QAOA circuits.
This dissertation addresses fundamental challenges in solving constrained combinatorial optimization problems using hybrid quantum-classical computing approaches. Quantum computing shows promise for tackling computationally intractable problems, current approaches face significant limitations. The path forward in computing involves combining quantum and classical computing approaches. Quantum processing units (QPUs) will play a key role in accelerating algorithms in optimization, simulation, and machine learning applications beyond what classical computers can achieve on their own, similar to how graphical processing units (GPUs) have become integral to general-purpose computing. The current generation of quantum devices is limited by noise and circuit depth constraints, making it challenging to gain practical utility. Hybrid quantum-classical approaches, which combine quantum computing resources with classical computing resources, are essential for addressing these limitations on near-term devices. In the area of combinatorial optimization, current practical state-of-the-art hybrid approaches are variational in nature, relying on parameterized quantum circuits to encode solutions and classical optimization techniques to find optimal parameters. While the ultimate goal of solving computationally intractable problems is to find provably optimal solutions, practical constraints of real-world scenarios often necessitate focusing on efficiently obtaining high-quality, near-optimal solutions. The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical approach for tackling these challenging problems that are encoded using quadratic and higher-order unconstrained binary optimization problems (QUBO and HUBO). QAOA alternates between two operators: a problem-specific cost Hamiltonian that encodes the optimization objective, and a mixer Hamiltonian that explores the solution space. The algorithm's parameters are iteratively optimized using classical methods to maximize the probability of sampling high-quality solutions. However, these methods often struggle with solution feasibility, especially for problems where the solutions must satisfy specific constraints. Penalty-based encodings require careful parameter tuning and risk producing infeasible solutions, while feasibility-preserving methods demand deeper quantum circuits that are challenging to implement on near-term quantum devices. In this dissertation, we present novel strategies that reduce or omit dependency on penalty parameters while maintaining solution quality, effectively capturing both optimal and near-optimal solutions, and balancing the trade-offs between circuit depth and solution feasibility. We present SCOOP, a novel QAOA-based framework for solving constrained optimization problems. SCOOP transforms a constrained problem into an unconstrained counterpart, forming SCOOP problem twins. QAOA operates on the unconstrained twin to identify potential optimal and near-optimal solutions. Effective classical post-processing reduces the solution set to the constrained problem space. Our SCOOP approach is solution-enhanced, objective-function-compatible, and scalable. We demonstrate the effectiveness of the SCOOP framework on selected NP-hard problems, some of which can be encoded quadratically (such as Minimum Vertex Cover and Maximum Independent Set), and some that can be encoded using higher-order terms (such as Minimum Dominating Set, Minimum Maximal Matching, and Minimum Set Cover). We also apply our framework to constrained combinatorial optimization problems that can be solved in polynomial time, such as Minimum Edge Cover and Maximum Matching. Experimental results on full statevector simulators and tensor network simulators show that SCOOP significantly improves the probability of finding high-quality feasible solutions across various problem classes, on problems that can be encoded quadratically as well as problems that can be encoded using higher-order terms. We also demonstrate the effectiveness of SCOOP on real quantum hardware, specifically IBM Quantum Heron R2 processors with 156 qubits, showing that SCOOP can be applied to practical problems with current quantum hardware limitations. This work makes significant contributions toward achieving quantum utility by enabling systematic investigation of constrained optimization problems on quantum devices. While current quantum computing research primarily focuses on unconstrained problems like MaxCut, real-world applications typically involve constraints. SCOOP provides a scalable framework for encoding constrained problems, extending quantum computing's practical utility beyond simple unconstrained cases. By eliminating dependency on penalty parameters and providing natural problem formulations, SCOOP makes it feasible to explore quantum solutions for constrained problems where classical exact methods become intractable. This capability is crucial for identifying promising avenues for demonstrating quantum utility across diverse constrained optimization problems, advancing the practical applicability of quantum computing approaches.