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Improved approximation ratios for the Quantum Max-Cut problem on general, triangle-free and bipartite graphs

Sander Gribling, Lennart Sinjorgo, Renata Sotirov

TL;DR

This work advances classical approximation algorithms for Quantum Max-Cut by delivering improved, graph-class-specific ratios and a sharpened analysis framework. It refines SDP-based upper bounds and leverages a matching-polytope containment strategy to boost the general-graph ratio to $0.603$, while introducing two new algorithms for triangle-free ($0.61383$) and bipartite ($0.8162$) graphs. Central to the approach is the SDP-based relaxation at levels $k$ (with exactness at $k\ge n$) and the construction of real-valued Theta functions $\Theta\in\mathcal{A}$ that govern rounding and angle choices; for triangle-free graphs, the analysis shows $h^+$ lies in the matching polytope up to certain odd-set sizes, enabling stronger rounding. The results push the frontier of classical QMC approximations for specific graph classes and provide computational techniques and a repository for verification, offering a pathway toward tighter quantum-classical comparison bounds in these settings.

Abstract

We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph $G$ on n vertices, the QMC problem is to determine the largest eigenvalue of a particular $2^n \times 2^n$ matrix that corresponds to $G$. We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of $0.599$, which our analysis improves to $0.603$. Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of $0.61383$ and $0.8162$, respectively. These are the best-known approximation ratios for their respective graph classes.

Improved approximation ratios for the Quantum Max-Cut problem on general, triangle-free and bipartite graphs

TL;DR

This work advances classical approximation algorithms for Quantum Max-Cut by delivering improved, graph-class-specific ratios and a sharpened analysis framework. It refines SDP-based upper bounds and leverages a matching-polytope containment strategy to boost the general-graph ratio to , while introducing two new algorithms for triangle-free () and bipartite () graphs. Central to the approach is the SDP-based relaxation at levels (with exactness at ) and the construction of real-valued Theta functions that govern rounding and angle choices; for triangle-free graphs, the analysis shows lies in the matching polytope up to certain odd-set sizes, enabling stronger rounding. The results push the frontier of classical QMC approximations for specific graph classes and provide computational techniques and a repository for verification, offering a pathway toward tighter quantum-classical comparison bounds in these settings.

Abstract

We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph on n vertices, the QMC problem is to determine the largest eigenvalue of a particular matrix that corresponds to . We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of , which our analysis improves to . Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of and , respectively. These are the best-known approximation ratios for their respective graph classes.

Paper Structure

This paper contains 15 sections, 28 theorems, 101 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline.
  4. Notation
  5. Graph theory preliminaries
  6. SDP bounds
  7. Improved analysis of a QMC approximation algorithm
  8. New approximation algorithm on triangle-free graphs
  9. Properties of Theta
  10. Computing the approximation ratio of Algorithm 2
  11. Finding candidate functions Theta
  12. New approximation algorithm on bipartite graphs
  13. Conclusions and future work
  14. Additional lemmas and proofs
  15. Computational details of proof of Lemma 8

Key Result

Lemma 2.1

Let $G$ be a graph on $n$ vertices. The optimal value of $\textup{SDP}^n$ equals $\lambda_{\mathrm{max}}\mleft({H_G}\mright)$.

Figures (2)

  • Figure 4.1: Plot of $\zeta_{\Theta}(x,14/15,p^*)$ for $x \in [-0.1,0.8]$, $\Theta(x) = 1-e^{-x/20}{}$ and $p^*$ as in \ref{['eqn_bestBoundTriangleFree']}. We have $\min_{x \in (-1,1]} \zeta_{\Theta}(x,14/15,p^*) = \zeta_{\Theta}(0,14/15,p^*) = 0.61383{}$, marked by the dot.
  • Figure A.1: Plot of the function $g(p) = f(h(r,p),p)$, see \ref{['eqn_maxValNegative']}, for $p \in [\ell,1]$, where $\ell = (2r-1)/ \left( q(0) -1 \right) \approx 0.7961$.

Theorems & Definitions (53)

  • Definition 1.1: Graph matching
  • Definition 1.2: Edge-induced subgraph
  • Definition 1.3: Vertex cover number $\tau(G)$
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3: lee2024improved
  • Lemma 3.4
  • Lemma 3.5
  • ...and 43 more