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Comparative Analysis of Classical and Quantum-Inspired Solvers: A Preliminary Study on the Weighted Max-Cut Problem

Aitor Morais, Eneko Osaba, Iker Pastor, Izaskun Oregui

TL;DR

This work benchmarks three computational paradigms—classical metaheuristics (Genetic Algorithms), deep learning (Graph Neural Networks), and quantum-inspired tensor networks (DMRG with MPS/MPO)—on the Weighted Max-Cut problem for graphs with $N$ up to 250. It introduces a scalable MPO representation for the Max-Cut Hamiltonian and compares eight solver configurations, reporting that DMRG with small bond dimension achieves near-optimal solutions with favorable runtimes, GA-OC performs best among GAs and scales with problem size, and GNNs offer low memory usage with more variable performance on larger instances. The study provides a systematic, statistically supported view of strengths and trade-offs across paradigms, highlighting memory-time-accuracy regimes and guiding future hybridization and automation efforts. The findings suggest practical implications for selecting solvers under specific resource constraints and problem scales, and point to avenues for automating MPO construction and expanding to other combinatorial problems.

Abstract

Combinatorial optimization is essential across numerous disciplines. Traditional metaheuristics excel at exploring complex solution spaces efficiently, yet they often struggle with scalability. Deep learning has become a viable alternative for quickly generating high-quality solutions, particularly when metaheuristics underperform. In recent years, quantum-inspired approaches such as tensor networks have shown promise in addressing these challenges. Despite these advancements, a thorough comparison of the different paradigms is missing. This study evaluates eight algorithms on Weighted Max-Cut graphs ranging from 10 to 250 nodes. Specifically, we compare a Genetic Algorithm representing metaheuristics, a Graph Neural Network for deep learning, and the Density Matrix Renormalization Group as a tensor network approach. Our analysis focuses on solution quality and computational efficiency (i.e., time and memory usage). Numerical results show that the Genetic Algorithm achieves near-optimal results for small graphs, although its computation time grows significantly with problem size. The Graph Neural Network offers a balanced solution for medium-sized instances with low memory demands and rapid inference, yet it exhibits more significant variability on larger graphs. Meanwhile, the Tensor Network approach consistently yields high approximation ratios and efficient execution on larger graphs, albeit with increased memory consumption.

Comparative Analysis of Classical and Quantum-Inspired Solvers: A Preliminary Study on the Weighted Max-Cut Problem

TL;DR

This work benchmarks three computational paradigms—classical metaheuristics (Genetic Algorithms), deep learning (Graph Neural Networks), and quantum-inspired tensor networks (DMRG with MPS/MPO)—on the Weighted Max-Cut problem for graphs with up to 250. It introduces a scalable MPO representation for the Max-Cut Hamiltonian and compares eight solver configurations, reporting that DMRG with small bond dimension achieves near-optimal solutions with favorable runtimes, GA-OC performs best among GAs and scales with problem size, and GNNs offer low memory usage with more variable performance on larger instances. The study provides a systematic, statistically supported view of strengths and trade-offs across paradigms, highlighting memory-time-accuracy regimes and guiding future hybridization and automation efforts. The findings suggest practical implications for selecting solvers under specific resource constraints and problem scales, and point to avenues for automating MPO construction and expanding to other combinatorial problems.

Abstract

Combinatorial optimization is essential across numerous disciplines. Traditional metaheuristics excel at exploring complex solution spaces efficiently, yet they often struggle with scalability. Deep learning has become a viable alternative for quickly generating high-quality solutions, particularly when metaheuristics underperform. In recent years, quantum-inspired approaches such as tensor networks have shown promise in addressing these challenges. Despite these advancements, a thorough comparison of the different paradigms is missing. This study evaluates eight algorithms on Weighted Max-Cut graphs ranging from 10 to 250 nodes. Specifically, we compare a Genetic Algorithm representing metaheuristics, a Graph Neural Network for deep learning, and the Density Matrix Renormalization Group as a tensor network approach. Our analysis focuses on solution quality and computational efficiency (i.e., time and memory usage). Numerical results show that the Genetic Algorithm achieves near-optimal results for small graphs, although its computation time grows significantly with problem size. The Graph Neural Network offers a balanced solution for medium-sized instances with low memory demands and rapid inference, yet it exhibits more significant variability on larger graphs. Meanwhile, the Tensor Network approach consistently yields high approximation ratios and efficient execution on larger graphs, albeit with increased memory consumption.

Paper Structure

This paper contains 18 sections, 13 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Metaheuristics
  4. Deep Learning
  5. Tensor Networks
  6. Motivation
  7. Tensor Network Basics
  8. Matrix Product States
  9. Matrix Product Operators
  10. Densitity Matrix Renormalization Group
  11. Description of Compared Algorithms
  12. The Weighted Max-Cut Problem
  13. Algorithms
  14. Experimental Setup
  15. Max-Cut Benchmark Description
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Diagrammatic representation of (a) the $c_{q_1,q_2,...,q_N}$ tensor and (b) its MPS representation for a system of $N=4$ qubits. Here, filled shapes are used to represent tensors, while lines extending from these shapes denote their indices.
  • Figure 2: Diagrammatic representation of (a) the $c_{q_1,...,q_N}^{q'_1,...,q'_N}$ tensor, and (b) its MPO representation for a system of $N=4$ qubits.
  • Figure 3: Critical Difference Diagram, with Friedman test $p\text{-value}<0.05$ and critical distance $CD=1.753$. The plot shows the average ranks (in brackets) achieved by each approach, calculated based on $\bar{\text{AR}}$. In this case, lower average ranks indicate better performance. The algorithms are indexed from 0 to 7 as follows: 0DMRG 2, 1 DMRG $0.10N$, 2 DMRG $0.20N$, 3 GNN, 4 GA-OC, 5 cGA 500, 6 cGA 1000, 7 cGA 2000.