Quantum technology enters in the new era of Noisy Intermediate-Scale Quantum (NISQ) with 50-100 qubits quantum computers already available. It is now important to compare quantum methods with the classical approach to solve optimization problems in order to determine if there is a potential speed-up and what quantum technologies will be able to perform. The Quantum Approximate optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. We provide the method to apply QAOA on a shortest path problem and a study of the performance of this quantum method by using heuristic strategies based on observed patterns in optimal parameters. First, we will present the state of the art of quantum hybrid algorithms and technologies. Then we will discuss the effect of the shape of the graph, the number of valid solutions, and the number of nodes of the graph we consider on the quality of the QAOA’s results and on the patterns in optimal variational parameters.
Quantum technology enters in the new era of Noisy Intermediate-Scale Quantum (NISQ) with 50-100 qubits quantum computers already available. It is now important to compare quantum methods with the classical approach to solve optimization problems in order to determine if there is a potential speed-up and what quantum technologies will be able to perform. The Quantum Approximate optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. We provide the method to apply QAOA on a shortest path problem and a study of the performance of this quantum method by using heuristic strategies based on observed patterns in optimal parameters. First, we will present the state of the art of quantum hybrid algorithms and technologies. Then we will discuss the effect of the shape of the graph, the number of valid solutions, and the number of nodes of the graph we consider on the quality of the QAOA’s results and on the patterns in optimal variational parameters.
The transition from theoretical quantum computing to practical real-world applications heavily relies onhybrid quantum-classical architectures designed for NoisyIntermediate-Scale Quantum (NISQ) devices. Among these approaches, the Quantum Approximate Optimization Algorithm(QAOA) is a prominent strategy for addressing complex combinatorial optimization challenges, such as the NP-hard MaxCut problem. Instead of performing the entire computationon a quantum processor, QAOA utilizes a quantum circuit toevaluate a parameterized cost function, whereas a classical optimizer refines the parameters iteratively. This study presentsa rigorous empirical evaluation comparing the performanceof QAOA against established classical solvers on small-scaleMax-Cut instances. All experiments were implemented andexecuted using Python (Qiskit Aer) on a state-vector simulator.Using Erdős–Rényi random graphs and regular graphs with4 to 20 nodes, we benchmark QAOA at depths p = 2 andp = 3 against simulated annealing, the Goemans–WilliamsonSDP-based algorithm, and a linear-time greedy heuristic. Ourexperimental results demonstrate that low-depth QAOA implementations exhibit severe performance degradation, prohibitive computational overhead (O(2^n) scaling), and low convergencerates compared to mature classical algorithms. At 20 nodes,QAOA (p = 2) achieves only a 0.467 approximation ratio versus0.856 for Goemans–Williamson, with a 210× runtime slowdownover greedy heuristics
Path planning in complex and obstacle-constrained environments remains a challenging combinatorial optimization problem due to the exponential growth of the solution space and the limitations of classical algorithms in dynamic decision-making scenarios. Traditional approaches such as Dijkstra or A* may experience high computational costs and susceptibility to local optima when dealing with large-scale directed graphs containing obstacles. This study proposes a hybrid quantum computing approach that integrates the Grover Search Algorithm and the Quantum Approximate Optimization Algorithm (QAOA) to address optimal path planning on obstacle-oriented directed graphs. The Grover algorithm is first employed as an oracle-based filtering mechanism to detect and eliminate obstacle-containing edges, thereby generating a valid subgraph of feasible paths. Following this preprocessing step, QAOA is applied to the filtered graph within a nonlinear optimization framework to determine the minimum-cost path between source and destination nodes. The optimization model represents the graph as G=(V,E), where binary decision variables indicate whether an edge is included in the selected route while satisfying flow conservation constraints. By narrowing the solution space through Grover-based obstacle detection and performing optimization only on valid edges, the proposed method significantly reduces computational complexity compared to conventional approaches. Experimental analysis demonstrates that the hybrid quantum model effectively identifies optimal routes and adapts to different obstacle configurations while maintaining efficient convergence behavior. The results highlight the potential of integrating quantum search and quantum optimization techniques for solving combinatorial graph problems. This work contributes to the growing body of research on quantum-assisted optimization and demonstrates the feasibility of applying hybrid quantum algorithms to directed path planning problems relevant to autonomous navigation systems.
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.
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.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.
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.