Over the past few years, many researchers around the world have been keen to know the potential and efficiency of quantum computers. The researchers have focused on specific issues that classical computers cannot solve or issues that quantum computers can handle in a better way. Among these various attractive research topics in quantum computers, this paper introduces the Quantum Approximate Optimization Algorithm (QAOA) which guarantees relatively considerable performances in many combinatorial optimization problems. For the comprehensive understanding of QAOA, this paper also describes the approximate optimization, the Quantum Alternating Operator Ansatz, and applications. Besides the theories of QAOA and Quantum Alternating Operator Ansatz, this paper explains the applications of QAOA to major combinatorial optimization problems such as maximum cut (MaxCut) problem and the max-independent set (MIS) problem.
Over the past few years, many researchers around the world have been keen to know the potential and efficiency of quantum computers. The researchers have focused on specific issues that classical computers cannot solve or issues that quantum computers can handle in a better way. Among these various attractive research topics in quantum computers, this paper introduces the Quantum Approximate Optimization Algorithm (QAOA) which guarantees relatively considerable performances in many combinatorial optimization problems. For the comprehensive understanding of QAOA, this paper also describes the approximate optimization, the Quantum Alternating Operator Ansatz, and applications. Besides the theories of QAOA and Quantum Alternating Operator Ansatz, this paper explains the applications of QAOA to major combinatorial optimization problems such as maximum cut (MaxCut) problem and the max-independent set (MIS) problem.
In the past few years, quantum computers and quantum algorithms have shown advantages compared to their classical counterparts. In combinatorial optimization problems, quantum approximate optimization algorithm(QAOA) has good performance. This paper focuses on the application of QAOA to one combinational problem called maximum independent set(MIS). Two different Hamiltonian construction methods are employed in this algorithm using IBM Qiskit. It is shown that the key is to design proper Hamiltonians to map the solution to the feasible states of the Hilbert space.
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, while simultaneously reducing the required number of CNOT gates and optimization parameters. We provide evidence that this speedup is connected to the concept of shortcuts to adiabaticity.
Quantum computing has emerged as a promising paradigm to tackle computationally intensive problems that classical computers struggle with. In this study, we explore the application of the Edge-based quantum approximate optimization algorithm (QAOA) to the MAX-CUT problem, a well-known combinatorial optimization challenge. MAX-CUT aims to partition the vertices of a graph into two subsets such that the number of edges between the subsets is maximized. We define the edge-based MAX-CUT problem and propose a method for applying QAOA specifically tailored to this formulation. We conduct simulations using IBM’s Qiskit framework, examining both vertex-based and edge-based QAOA implementations across various graph structures. Our results highlight the comparative performance of these approaches in terms of solution quality and computational efficiency. Specifically, we analyze the impact of different graph sizes and edge densities on the complexity and CNOT gate counts of the proposed algorithms. This analysis provides insights into leveraging quantum computing for combinatorial optimization tasks, particularly focusing on the implications for practical applications and future research directions.
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA Ansatz is not optimal, there is no systematic approach for finding better Ans\"atze. We address this problem by developing an iterative version of QAOA that is problem tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, while simultaneously reducing the required number of CNOT gates and optimization parameters. We provide evidence that this speedup is connected to the concept of shortcuts to adiabaticity.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.
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.