The quantum approximate optimization algorithm (QAOA) holds promise for combinatorial optimization but is constrained by limited qubits. While divide-and-conquer frameworks like QAOA$^{2}$ address scalability by partitioning graphs into subgraphs, existing methods suffer from two fundamental limitations: i) misalignment between heuristic partitioning metrics and quantum optimization goals, and ii) topology-blind parameter initialization that leads to optimization cold starts. To bridge these gaps, we propose Neural QAOA$^{2}$, an end-to-end differentiable framework that jointly generates graph partitions and initial parameters. By integrating a generative evaluative network (GEN), our method utilizes a differentiable quantum evaluator as a high-fidelity performance surrogate to provide direct gradient guidance, enabling the joint generator to learn the intrinsic mapping from graph topology to high-quality partition and parameter configurations. Extensive experiments on 183 QUBO, Ising, and MaxCut instances (21 to 1000 variables) demonstrate that our gradient-driven approach broadly outperforms heuristic baselines, ranking first on 101 instances. It exhibits zero-shot generalization across out-of-distribution graph topologies and scales.
Neural QAOA2: Differentiable Joint Graph Partitioning and Parameter Initialization for Quantum Combinatorial Optimization
The quantum approximate optimization algorithm (QAOA) holds promise for combinatorial optimization but is constrained by limited qubits. While divide-and-conquer frameworks like QAOA2 address scalability by partitioning graphs into subgraphs, existing methods suffer from two fundamental limitations: i) misalignment between heuristic partitioning metrics and quantum optimization goals, and ii) topology-blind parameter initialization that leads to optimization cold starts. To bridge these gaps, we propose Neural QAOA2, an end-to-end differentiable framework that jointly generates graph partitions and initial parameters. By integrating a generative evaluative network (GEN), our method utilizes a differentiable quantum evaluator as a high-fidelity performance surrogate to provide direct gradient guidance, enabling the joint generator to learn the intrinsic mapping from graph topology to high-quality partition and parameter configurations. Extensive experiments on 183 QUBO, Ising, and MaxCut instances (21 to 1000 variables) demonstrate that our gradient-driven approach broadly outperforms heuristic baselines, ranking first on 101 instances. It exhibits zero-shot generalization across out-of-distribution graph topologies and scales.
Approximate combinatorial optimisation has emerged as one of the most promising application areas for quantum computers, particularly those in the near term. In this work, we focus on the quantum approximate optimisation algorithm (QAOA) for solving the MaxCut problem. Specifically, we address two problems in the QAOA, how to initialise the algorithm, and how to subsequently train the parameters to find an optimal solution. For the former, we propose graph neural networks (GNNs) as a warm-starting technique for QAOA. We demonstrate that merging GNNs with QAOA can outperform both approaches individually. Furthermore, we demonstrate how graph neural networks enables warm-start generalisation across not only graph instances, but also to increasing graph sizes, a feature not straightforwardly available to other warm-starting methods. For training the QAOA, we test several optimisers for the MaxCut problem up to 16 qubits and benchmark against vanilla gradient descent. These include quantum aware/agnostic and machine learning based/neural optimisers. Examples of the latter include reinforcement and meta-learning. With the incorporation of these initialisation and optimisation toolkits, we demonstrate how the optimisation problems can be solved using QAOA in an end-to-end differentiable pipeline.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm that shows promise in efficiently solving the Max-Cut problem, a representative example of combinatorial optimization. However, its effectiveness heavily depends on the parameter optimization pipeline, where the parameter initialization strategy is nontrivial due to the non-convex and complex optimization landscapes characterized by issues with low-quality local minima. Recent inspiration comes from the diffusion of classical neural network parameters, which has demonstrated that neural network training can benefit from generating good initial parameters through diffusion models. However, whether the diffusion model can enhance the parameter optimization and performance of QAOA by generating well-performing initial parameters is still an open topic. Therefore, in this work, we formulate the problem of finding good initial parameters as a generative task and propose the initial parameter generation scheme through dataset-conditioned pre-trained parameter sampling. Concretely, the generative machine learning model, specifically the denoising diffusion probabilistic model (DDPM), is trained to learn the distribution of pre-trained parameters conditioned on the graph dataset. Intuitively, the proposed framework aims to effectively distill knowledge from pre-trained parameters to generate well-performing initial parameters for QAOA. To benchmark our framework, we adopt trotterized quantum annealing (TQA)-based and graph neural network (GNN) prediction-based initialization protocols as baselines. Through numerical experiments on Max-Cut problem instances of various sizes, we show that conditional DDPM can consistently generate high-quality initial parameters, improve convergence to the approximation ratio, and exhibit greater robustness against local minima over baselines. Additionally, the experimental results also indicate that the conditional DDPM trained on small problem instances can be extrapolated to larger ones, thus demonstrating the extrapolation capacity of our framework in terms of the qubit number.
The integration of quantum algorithms into machine learning pipelines offers a promising direction for addressing complex combinatorial optimization problems inherent in graph-structured data. This study investigates the use of the Quantum Approximate Optimization Algorithm (QAOA) as a quantum subroutine within a hybrid quantum–classical framework to enhance machine learning solutions for graph-based tasks. QAOA is employed to solve NP-hard graph optimization problems such as Max-Cut, graph coloring, and community detection, which are reformulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. The solutions provided by QAOA are then incorporated into machine learning models, such as Graph Neural Networks (GNNs) or clustering algorithms, as enriched features or optimized graph partitions. The hybrid approach is evaluated on benchmark datasets, demonstrating competitive accuracy and improved scalability in certain problem instances compared to purely classical methods. The findings highlight the potential of leveraging near-term quantum hardware for augmenting classical learning systems in domains such as logistics, social network analysis, and computational biology.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm that shows promise in efficiently solving the MaxCut problem, a representative example of combinatorial optimization. However, its effectiveness heavily depends on the parameter optimization pipeline, where the parameter initialization strategy is nontrivial due to the non-convex and complex optimization landscapes characterized by issues with low-quality local minima. Recent inspiration comes from the diffusion of classical neural network parameters, which has demonstrated that neural network training can benefit from generating good initial parameters through diffusion models. Therefore, in this work, we formulate the problem of finding good initial parameters as a generative task and propose the initial parameter generation scheme through dataset-conditioned pre-trained parameter sampling. Concretely, the generative machine learning model, specifically the denoising diffusion probabilistic model (DDPM), is trained to learn the distribution of pretrained parameters conditioned on the graph dataset. Intuitively, our proposed framework aims at effectively distilling knowledge from pre-trained parameters to generate well-performing initial parameters for QAOA. Compared to random parameter initialization, experiments on various-sized Max-Cut problem instances consistently show that our conditional DDPM is capable of improving the approximation ratio by as much as 14.4%, 11.0%, 11.4% and 7.49%, 8.31%, 6.08% on average for random, regular, and Watts-Strogatz graphs, respectively. Additionally, the experimental results also indicate that the conditional DDPM trained on small problem instances can be extrapolated to larger ones, improving the approximation ratio by up to 28.4% and 12.1% on average.
Abstract The quantum approximate optimization algorithm (QAOA) is a leading iterative variational quantum algorithm for heuristically solving combinatorial optimization problems. A large portion of the computational effort in QAOA is spent by the optimization steps, which require many executions of the quantum circuit. Therefore, there is active research focusing on finding better initial circuit parameters, which would reduce the number of required iterations and hence the overall execution time. While existing methods for parameter initialization have shown great success, they often offer a single set of parameters for all problem instances. We propose a practical method that uses a simple, fully connected neural network that leverages previous executions of QAOA to find better initialization parameters tailored to a new given problem instance. We benchmark state-of-the-art initialization methods for solving the MaxCut problem of Erdős-Rényi graphs using QAOA and show that our method is consistently the fastest to converge while also yielding the best final result. Furthermore, the parameters predicted by the neural network are shown to match very well with the fully optimized parameters, to the extent that no iterative steps are required, thereby effectively realizing an iteration-free QAOA scheme.
The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Because of the Z2 symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. In view of this, we propose QAOA-in-QAOA (QAOA2) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of QAOA2 is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, QAOA2 attains a competitive or even better performance over classical algorithms when the node count is around 2000. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems.