The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for addressing combinatorial optimization problems on near-term quantum hardware. In this work, we conduct an empirical evaluation of QAOA on the Max-Cut problem, using the Goemans-Williamson (GW) algorithm as a classical baseline for comparison. Unlike many prior studies, our methodology treats QAOA implementations as black-box optimizers, relying solely on default parameter settings without manual fine-tuning. We evaluate specific off-the-shelf QAOA implementations under default settings, not the algorithmic potential of QAOA with optimized parameters. This reflects a more realistic use case for end users who may lack the resources or expertise for instance-specific optimization. To facilitate fair and informative evaluation, we construct benchmark instances using well-known graph generation models that emulate practical graph structures, avoiding synthetic constructions tailored to either quantum or classical algorithms. A central component of our analysis is a per-shot statistical framework, which tracks the quality of QAOA outputs as a function of the number of circuit executions. This enables probabilistic comparisons with the GW algorithm by examining when and how frequently QAOA surpasses classical performance baselines such as the GW expectation and lower bound. Our results provide insight into the practical applicability of QAOA for Max-Cut and highlight its current limitations, offering a framework that can guide the assessment and development of future QAOA implementations.
Enhanced Distributed Variational Quantum Eigensolver for Large-Scale MaxCut Problem
Shenzhen SpinQ Technology Co., Ltd（深圳 spinq 科技有限公司）；College of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University（四川师范大学物理与电子工程学院计算科学中心）；State Key Laboratory of Genome and Multi-omics Technologies, BGI Research（华大基因研究院基因组与多组学技术国家重点实验室）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）
The quantum approximate optimization algorithm (QAOA) is a near-term combinatorial optimization algorithm suitable for noisy quantum devices. However, little is known about performance guarantees for $p>2$. A recent work \cite{Wurtz_guarantee} computing MaxCut performance guarantees for 3-regular graphs conjectures that any $d$-regular graph evaluated at particular fixed angles has an approximation ratio greater than some worst-case guarantee. In this work, we provide numerical evidence for this fixed angle conjecture for $p<12$. We compute and provide these angles via numerical optimization a…
The Quantum Approximate Optimization Algorithm (QAOA) is a promising approach for programming a near-term gate-based hybrid quantum computer to find good approximate solutions of hard combinatorial problems. However, little is currently know about the capabilities of QAOA, or of the difficulty of the requisite parameters optimization. Here, we study the performance of QAOA on the MaxCut combinatorial optimization problem, optimizing the quantum circuits on a classical computer using automatic differentiation and stochastic gradient descent, using QuantumFlow, a quantum circuit simulator implem…
The quantum approximate optimization algorithm (QAOA) is a variational method for noisy, intermediate-scale quantum computers to solve combinatorial optimization problems. Quantifying performance bounds with respect to specific problem instances provides insight into when QAOA may be viable for solving real-world applications. Here, we solve every instance of MaxCut on non-isomorphic unweighted graphs with nine or fewer vertices by numerically simulating the pure-state dynamics of QAOA. Testing up to three layers of QAOA depth, we find that distributions of the approximation ratio narrow with …
The Quantum Approximate Optimization Algorithm (QAOA) constitutes one of the often mentioned candidates expected to yield a quantum boost in the era of near-term quantum computing. In practice, quantum optimization will have to compete with cheaper classical heuristic methods, which have the advantage of decades of empirical domain-specific enhancements. Consequently, to achieve optimal performance we will face the issue of algorithm selection, well-studied in practical computing. Here we introduce this problem to the quantum optimization domain. Specifically, we study the problem of detecting…
Local Hamiltonians with topological quantum order exhibit highly entangled ground states that cannot be prepared by shallow quantum circuits. Here, we show that this property may extend to all low-energy states in the presence of an on-site $\mathbb{Z}_2$ symmetry. This proves a version of the No Low-Energy Trivial States (NLTS) conjecture for a family of local Hamiltonians with symmetry protected topological order. A surprising consequence of this result is that the Goemans-Williamson algorithm outperforms the Quantum Approximate Optimization Algorithm (QAOA) for certain instances of MaxCut, …
Quantum technology provides a ground-breaking methodology to tackle challenging computational issues in power systems, especially for Distributed Energy Resources (DERs) dominant cyber-physical systems that have been widely developed to promote energy sustainability. The systems' maximum power or data sections are essential for monitoring, operation, and control, while high computational effort is required. Quantum Approximate Optimization Algorithm (QAOA) provides a promising means to search for these sections by leveraging quantum resources. However, its performance highly relies on the crit…
There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combi…
Combinatorial optimization problems are computationally hard in general, but they are ubiquitous in our modern life. A coherent Ising machine (CIM) based on a multiple-pulse degenerate optical parametric oscillator (DOPO) is an alternative approach to solve these problems by a specialized physical computing system. To evaluate its potential performance, computational experiments are performed on maximum cut (MAX-CUT) problems against traditional algorithms such as semidefinite programming relaxation of Goemans-Williamson and simulated annealing by Kirkpatrick, et al. The numerical results empi…
We introduce a method to solve the MaxCut problem efficiently based on quantum imaginary time evolution (QITE). We employ a linear Ansatz for unitary updates and an initial state involving no entanglement, as well as an imaginary-time-dependent Hamiltonian interpolating between a given graph and a subgraph with two edges excised. We apply the method to thousands of randomly selected graphs with up to fifty vertices. We show that our algorithm exhibits a 93% and above performance converging to the maximum solution of the MaxCut problem for all considered graphs. Our results compare favorably wi…
Many computational problems are intractable through classical computing and, as Moore's law is drawing to a halt, demand for finding alternative methods in tackling these problems is growing. Here, we realize a liquid light machine for the NP-hard max-3-cut problem based on a network of synchronized exciton-polariton condensates. We overcome the binary limitation of the decision variables in Ising machines using the continuous-phase degrees of freedom of a coherent network of polariton condensates. The condensate network dynamical transients provide optically-fast annealing of the XY Hamiltoni…
Imaginary Hamiltonian variational ansatz for combinatorial optimization problems
Peking University（北京大学）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学项目（iTHEMS））；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Computation-Based Science and Technology Research Center, The Cyprus Institute（塞浦路斯研究所计算科学与技术研究中心）；Institut für Physik, Humboldt-Universität zu Berlin（柏林洪堡大学物理研究所）
We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problem's approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this p…
Optimizing parameterized quantum circuits promises efficient use of near-term quantum computers to achieve the potential quantum advantage. However, there is a notorious tradeoff between the expressibility and trainability of the parameter ansatz. We find that in combinatorial optimization problems, since the solutions are described by bit strings, one can trade the expressiveness of the ansatz for high trainability. To be specific, by focusing on the max-cut problem we introduce a simple yet efficient algorithm named Quantum Qubit Rotation Algorithm (QQRA). The quantum circuits are comprised …
Sampling (noisy) quantum circuits through randomized rounding
IBM France（法国国际商业机器公司）；Univ Lyon, ENS Lyon, UCBL, Inria, LIP（里昂大学、巴黎高等师范学校里昂分校、里昂第一大学、法国国家信息与自动化研究所、计算与图像实验室）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
Evolving Hard Maximum Cut Instances for Quantum Approximate Optimization Algorithms
LIACS, Leiden University（莱顿大学信息科学研究中心）；Optimisation and Logistics, School of Computer and Mathematical Sciences, The University of Adelaide（计算机与数学科学学院优化与物流系，阿德莱德大学）；Mathematical Sciences, The University of Adelaide（数学科学，阿德莱德大学）
