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arXiv:2407.12242

Conditional Diffusion-based Parameter Generation for Quantum Approximate Optimization Algorithm

Fanxu Meng, Xiangzhen Zhou, Zhu, Pengcheng, Luo, Yu

Published Jul 17, 2024• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2407.12242

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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.
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.
Abstract The Quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm aiming to produce approximate solutions for combinatorial optimization problems. In the QAOA, the quantum part prepares a quantum parameterized state that encodes the solution, where the parameters are optimized by a classical optimizer. However, it is difficult to find optimal parameters when the quantum circuit becomes deeper. Hence, there is numerous active research on the performance and the optimization cost of QAOA. In this work, we build a convolutional neural network to predict parameters of depth p + 1 QAOA instance by the parameters from the depth p QAOA counterpart. We propose two strategies based on this model. First, we recurrently apply the model to generate a set of initial values for a certain depth QAOA. It successfully initiates depth 10 QAOA instances, whereas each model is only trained with the parameters from depths less than 6. Second, the model is applied repetitively until the maximum expected value is reached. An average approximation ratio of 0.9759 for Max-Cut over 264 Erdős–Rényi graphs is obtained, while the optimizer is only adopted for generating the first input of the model.
Variational Quantum Algorithms (VQAs) are widely used in the noisy intermediate-scale quantum (NISQ) era, but their trainability and performance depend critically on initialization parameters that shape the optimization landscape. Existing machine learning–based initializers achieve state-of-the-art results yet remain constrained to single-task domains and datasets of only hundreds of samples. We address these limitations by reformulating VQA parameter initialization as a generative modeling problem and introducing DiffQ, a parameter initializer based on the Denoising Diffusion Probabilistic Model (DDPM). To support robust training and evaluation, we construct a dataset of 15,085 instances spanning three domains and five representative tasks. Experiments demonstrate that DiffQ surpasses baselines, reducing initial loss by up to 8.95 and convergence steps by up to 23.4%. The code and data used in this paper are available at https://github.com/chizhang24/DiffQ.
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.
In recent years, quantum computing has emerged as a transformative force in the field of combinatorial optimization, offering novel approaches to tackling complex problems that have long challenged classical computational methods. Among these, the Quantum Approximate Optimization Algorithm (QAOA) stands out for its potential to efficiently solve the Max-Cut problem, a quintessential example of combinatorial optimization. However, practical application faces challenges due to current limitations on quantum computational resource. Our work optimizes QAOA initialization, using Graph Neural Networks (GNN) as a warm-start technique. This sacrifices affordable computational resource on classical computer to reduce quantum computational resource overhead, enhancing QAOA's effectiveness. Experiments with various GNN architectures demonstrate the adaptability and stability of our framework, highlighting the synergy between quantum algorithms and machine learning. Our findings show GNN's potential in improving QAOA performance, opening new avenues for hybrid quantum-classical approaches in quantum computing and contributing to practical applications.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading approach for solving combinatorial optimization problems on near-term quantum processors. However, finding good variational parameters remains a significant challenge due to the non-convex energy landscape, often resulting in slow convergence and poor solution quality. In this work, we propose a quantum meta-learning framework that trains advanced quantum sequence models to generate effective parameter initialization policies. We investigate four classical or quantum sequence models, including the Quantum Kernel-based Long Short-Term Memory (QK-LSTM), as learned optimizers in a "learning to learn" paradigm. Our numerical experiments on the Max-Cut problem demonstrate that the QK-LSTM optimizer achieves superior performance, obtaining the highest approximation ratios and exhibiting the fastest convergence rate across all tested problem sizes (n=10 to 13). Crucially, the QK-LSTM model achieves perfect parameter transferability by synthesizing a single, fixed set of near-optimal parameters, leading to a remarkable sustained acceleration of convergence even when generalizing to larger problems. This capability, enabled by the compact and expressive power of the quantum kernel architecture, underscores its effectiveness. The QK-LSTM, with only 43 trainable parameters, substantially outperforms the classical LSTM (56 parameters) and other quantum sequence models, establishing a robust pathway toward highly efficient parameter initialization for variational quantum algorithms in the NISQ era.