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MLQAOA: Graph Learning Accelerated Hybrid Quantum-Classical Multilevel QAOA

Bao Bach, Jose Falla, Ilya Safro

TL;DR

MLQAOA introduces a scalable, multilevel hybrid quantum-classical framework for MAXCUT that fuses coarsening-uncoarsening with graph representation learning to accelerate QAOA parameterization. By embedding nodes on a sphere, performing spectral-based parameter transfer via a weighted Laplacian, and leveraging a quantum-informed recursive optimization (QIRO) for subproblem refinement, the method achieves fast runtimes and competitive quality on very large graphs. The approach demonstrates that spectral graph embeddings can outperform WL-based transfers for weighted graphs and that multilevel refinement can dramatically reduce the variational overhead, enabling practical exploration of QAOA on large instances. Empirical results on $G_{set}$, Karloff, and larger graphs show MLQAOA delivering high-quality solutions with runtimes far shorter than prior hybrid schemes, highlighting its potential for scalable quantum-assisted graph optimization in the NISQ era.

Abstract

Learning the problem structure at multiple levels of coarseness to inform the decomposition-based hybrid quantum-classical combinatorial optimization solvers is a promising approach to scaling up variational approaches. We introduce a multilevel algorithm reinforced with the spectral graph representation learning-based accelerator to tackle large-scale graph maximum cut instances and fused with several versions of the quantum approximate optimization algorithm (QAOA) and QAOA-inspired algorithms. The graph representation learning model utilizes the idea of QAOA variational parameters concentration and substantially improves the performance of QAOA. We demonstrate the potential of using multilevel QAOA and representation learning-based approaches on very large graphs by achieving high-quality solutions in a much faster time. Reproducibility: Our source code and results are available at https://github.com/bachbao/MLQAOA

MLQAOA: Graph Learning Accelerated Hybrid Quantum-Classical Multilevel QAOA

TL;DR

MLQAOA introduces a scalable, multilevel hybrid quantum-classical framework for MAXCUT that fuses coarsening-uncoarsening with graph representation learning to accelerate QAOA parameterization. By embedding nodes on a sphere, performing spectral-based parameter transfer via a weighted Laplacian, and leveraging a quantum-informed recursive optimization (QIRO) for subproblem refinement, the method achieves fast runtimes and competitive quality on very large graphs. The approach demonstrates that spectral graph embeddings can outperform WL-based transfers for weighted graphs and that multilevel refinement can dramatically reduce the variational overhead, enabling practical exploration of QAOA on large instances. Empirical results on , Karloff, and larger graphs show MLQAOA delivering high-quality solutions with runtimes far shorter than prior hybrid schemes, highlighting its potential for scalable quantum-assisted graph optimization in the NISQ era.

Abstract

Learning the problem structure at multiple levels of coarseness to inform the decomposition-based hybrid quantum-classical combinatorial optimization solvers is a promising approach to scaling up variational approaches. We introduce a multilevel algorithm reinforced with the spectral graph representation learning-based accelerator to tackle large-scale graph maximum cut instances and fused with several versions of the quantum approximate optimization algorithm (QAOA) and QAOA-inspired algorithms. The graph representation learning model utilizes the idea of QAOA variational parameters concentration and substantially improves the performance of QAOA. We demonstrate the potential of using multilevel QAOA and representation learning-based approaches on very large graphs by achieving high-quality solutions in a much faster time. Reproducibility: Our source code and results are available at https://github.com/bachbao/MLQAOA
Paper Structure (23 sections, 10 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 23 sections, 10 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. QAOA and MAXCUT
  4. Multilevel Methods
  5. Graph Representation Learning
  6. Related works
  7. QAOA-based Approaches
  8. Hybrid Quantum-Classical Algorithms For Large Graphs
  9. Graph Learning For Parameter Transferability
  10. Our methods
  11. Coarsening Phase
  12. Uncoarsening Phase
  13. Graph Representation Using Spectral Learning For Parameter Transferability On Weighted Graphs
  14. Quantum-Informed Recursive Optimization Algorithm
  15. Computational Results
  16. ...and 8 more sections

Figures (3)

  • Figure 1: MLQAOA scheme using $V$-cycle. The original graph is iteratively coarsened in the Coarsening Phase. The coarsest graph is solved using a classical solver. Building upon this solution, the Uncoarsening Phase leverages the previous level solution through interpolation and performs refinement by solving sub-problem graphs.
  • Figure 2: Approximation ratio of graph learning MLQAOA RQAOA-inspired QIRO MLQAOA on $G_{set}$ and Karloff graph. The whisker lines are drawn up (down) to the largest (lowest) observed data point from the dataset that falls within the $1.5$ interquartile range (IQR) value from the upper (lower) quartile. The upper (lower) notated approximation ratio of each box indicates the highest (lowest) approximation ratio over $20$ run
  • Figure 3: Average run time in seconds of graph learning MLQAOA and RQAOA-inspired QIRO MLQAOA on graphs from table \ref{['tab:total_table']} over 20 runs. The $x-$axis and $y-$axis are logarithmic scales with $x$-axis denoted the runtime and $y-$axis denoted the size of the graph calculated by $\abs{V} + \abs{E}$.