Graph computation is a core technique for solving realistic problems of graph representations. In solving the shortest path problem (SPP), the current classical methods are encountering a huge performance bottleneck. Attempting to solve this dilemma, we try to solve the SPP with a Quantum Approximate Optimal Algorithm (QAOA)-based quantum method. In this paper, we propose a QAOA-based shortest path algorithm (SPA) by constructing a suitable Hamiltonian quantity and using the idea of variational quantum computing, and verify the algorithm using a quantum simulator and an International Business Machines cloud quantum computer. The proposed algorithm is able to achieve a near-optimal solution with a correct rate that significantly exceeds the invalid solutions, reaching a good preliminary result. Furthermore, the proposed algorithm is expected to achieve a huge advantage over the classical algorithm and the SPA based on Grover’s algorithm with a suitable selection of parameters and number of steps. In addition, the proposed algorithm requires fewer quantum bits than other quantum algorithms, thus promising quantum computing superiority on current noisy intermediate-scale quantum (NISQ) quantum computing devices.
Graph computation is a core technique for solving realistic problems of graph representations. In solving the shortest path problem (SPP), the current classical methods are encountering a huge performance bottleneck. Attempting to solve this dilemma, we try to solve the SPP with a Quantum Approximate Optimal Algorithm (QAOA)-based quantum method. In this paper, we propose a QAOA-based shortest path algorithm (SPA) by constructing a suitable Hamiltonian quantity and using the idea of variational quantum computing, and verify the algorithm using a quantum simulator and an International Business Machines cloud quantum computer. The proposed algorithm is able to achieve a near-optimal solution with a correct rate that significantly exceeds the invalid solutions, reaching a good preliminary result. Furthermore, the proposed algorithm is expected to achieve a huge advantage over the classical algorithm and the SPA based on Grover’s algorithm with a suitable selection of parameters and number of steps. In addition, the proposed algorithm requires fewer quantum bits than other quantum algorithms, thus promising quantum computing superiority on current noisy intermediate-scale quantum (NISQ) quantum computing devices.
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 Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for solving NP hard combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) hardware. However, its performance is critically dependent on the selection of the circuit depth a parameter that must be specified a priori without clear guidance. In this paper, we introduce a variant of QAOA called dynamic depth Quantum Approximate Optimization Algorithm (DDQAOA) that resolves the challenge of pre selecting a fixed circuit depth. Our method adaptively expands circuit depth, starting from p = 1 and progressing up to p = 10, by transferring learned parameters to deeper circuits based on convergence criteria. We tested this approach on 100 instances of the Constrained Shortest Path Problem (CSPP) at 10 qubit and 16 qubit scales. Our DDQAOA achieved superior approximation ratios and success probabilities with fewer CNOT gate evaluations than the standard QAOA for p = 3, 5, 10, and 15. In particular, while standard QAOA at p = 15 achieved results close to our approach, it used 217% and 159.3% more CNOT gates for 10 qubit and 16 qubit instances, respectively. This demonstrates the performance and practical applicability of DDQAOA to solve combinatorial optimization problems on near term devices.
The Quantum Approximate Optimization Algorithm (QAOA), which is a variational quantum algorithm, aims to give sub-optimal solutions of combinatorial optimization problems. It is widely believed that QAOA has the potential to demonstrate application-level quantum advantages in the noisy intermediate-scale quantum(NISQ) processors with shallow circuit depth. Since the core of QAOA is the computation of expectation values of the problem Hamiltonian, an important practical question is whether we can find an efficient classical algorithm to solve quantum mean value in the case of general shallow quantum circuits. Here, we present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits in most optimization problems except for complete graph case. Numerical tests in Max-cut, graph coloring and Sherrington-Kirkpatrick model problems, compared to the state-of-the-art method, shows orders of magnitude performance improvement. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.
Quantum computing is no longer in its early stages. There already exists quantum computers with more qubits than a classical computer is capable of efficiently simulating. This current stage is considered intermediate and is therefore called the NISQ era (noisy intermediate-scale quantum). The main feature of this current stage is that there are still not enough qubits to perform quantum error correction, hence the noisy name. In this context of quantum computing without quantum error correction and with an intermediate number of qubits, variational algorithms gained prominence and, among them, there is one called QAOA (quantum approximate optimization algorithm). As the name suggests, this is a quantum algorithm that approximates the solution of optimization problems. The objective of this work was to apply this algorithm to solve an optimization problem in the finance area known as portfolio optimization. This application took place both in an ideal way (without noise) and in a way consistent with the current capacity of quantum computers (with noise). Both were simulated using IBM’s Python tool for simulation and access of quantum computers via cloud called Qiskit. The results suggest that the QAOA performance with noise was, as expected, worse than the ideal case, but still satisfactory within the limitations of the method.
Optimization problems are pervasive across various domains, including logistics, finance, machine learning, and operations research. Quantum computing has emerged as a promising frontier to address these challenges, offering potential speedups for certain classes of optimization tasks. This paper explores the development and application of quantum algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA), Variational Quantum Eigensolver (VQE), and Grover’s search algorithm, tailored for optimization. Recent advancements in hardware, hybrid quantum-classical approaches, and variational techniques have enabled practical implementations on Noisy Intermediate-Scale Quantum (NISQ) devices. Challenges such as noise, scalability, and performance limitations are also discussed. Through theoretical analysis and case studies, this work demonstrates how quantum computing can complement classical methods, paving the way for breakthroughs in solving complex optimization problems.
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.