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Quantum Approximate Optimization Algorithms for Maximum Cut on Low-Girth Graphs

Tongyang Li, Yuexin Su, Ziyi Yang, Shengyu Zhang

TL;DR

The paper investigates QAOA for MaxCut on low-girth expander graphs, focusing on additive product (X-Ramanujan) graphs, and introduces ma-QAOA to leverage their structured composition. It derives an iterative formula to compute the expected MaxCut fraction at fixed depth $p$ for these graphs and extends the framework to quantum MaxCut, enabling systematic analysis beyond high-girth assumptions. Empirically, QAOA and ma-QAOA outperform leading classical local algorithms by notable margins on several additive-product graphs and planar tilings, with ma-QAOA often providing additional gains and in some cases outperforming classical methods while plain QAOA does not. The work provides a practical analytical tool and demonstrates the potential of quantum approaches on low-girth expander graphs, while highlighting open questions about broader graph families and comparisons to classical approximation bounds.

Abstract

Maximum cut (MaxCut) on graphs is a classic NP-hard problem. In quantum computing, Farhi, Gutmann, and Goldstone proposed the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Its guarantee on cut fraction (the fraction of edges in the output cut over all edges) was mainly studied for high-girth graphs, i.e., graphs with only long cycles. On the other hand, low-girth graphs are ubiquitous in theoretical computer science, including expander graphs being outstanding examples with wide applications in theory and beyond. In this paper, we apply QAOA to MaxCut on a set of expander graphs proposed by Mohanty and O'Donnell known as additive product graphs. Additionally, we apply multi-angle QAOA (ma-QAOA) to better utilize the graph structure of additive product graphs in ansatz design. In theory, we derive an iterative formula to calculate the expected cut fraction of such graphs. This formula also extends to the quantum MaxCut problem. On the other hand, we conduct numerical experiments to compare between best-known classical local algorithms and QAOA with constant depth. Our results demonstrate that QAOA outperforms the best-known classical algorithms by 0.3% to 5.2% on several additive product graphs, while ma-QAOA further enhances this advantage by an additional 0.6% to 2.5%. In particular, we observe cases that ma-QAOA exhibits superiority over best-known classical algorithms but QAOA does not. Furthermore, we extend our experiments to planar graphs such as tiling grid graphs, where QAOA also demonstrates an advantage.

Quantum Approximate Optimization Algorithms for Maximum Cut on Low-Girth Graphs

TL;DR

The paper investigates QAOA for MaxCut on low-girth expander graphs, focusing on additive product (X-Ramanujan) graphs, and introduces ma-QAOA to leverage their structured composition. It derives an iterative formula to compute the expected MaxCut fraction at fixed depth for these graphs and extends the framework to quantum MaxCut, enabling systematic analysis beyond high-girth assumptions. Empirically, QAOA and ma-QAOA outperform leading classical local algorithms by notable margins on several additive-product graphs and planar tilings, with ma-QAOA often providing additional gains and in some cases outperforming classical methods while plain QAOA does not. The work provides a practical analytical tool and demonstrates the potential of quantum approaches on low-girth expander graphs, while highlighting open questions about broader graph families and comparisons to classical approximation bounds.

Abstract

Maximum cut (MaxCut) on graphs is a classic NP-hard problem. In quantum computing, Farhi, Gutmann, and Goldstone proposed the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Its guarantee on cut fraction (the fraction of edges in the output cut over all edges) was mainly studied for high-girth graphs, i.e., graphs with only long cycles. On the other hand, low-girth graphs are ubiquitous in theoretical computer science, including expander graphs being outstanding examples with wide applications in theory and beyond. In this paper, we apply QAOA to MaxCut on a set of expander graphs proposed by Mohanty and O'Donnell known as additive product graphs. Additionally, we apply multi-angle QAOA (ma-QAOA) to better utilize the graph structure of additive product graphs in ansatz design. In theory, we derive an iterative formula to calculate the expected cut fraction of such graphs. This formula also extends to the quantum MaxCut problem. On the other hand, we conduct numerical experiments to compare between best-known classical local algorithms and QAOA with constant depth. Our results demonstrate that QAOA outperforms the best-known classical algorithms by 0.3% to 5.2% on several additive product graphs, while ma-QAOA further enhances this advantage by an additional 0.6% to 2.5%. In particular, we observe cases that ma-QAOA exhibits superiority over best-known classical algorithms but QAOA does not. Furthermore, we extend our experiments to planar graphs such as tiling grid graphs, where QAOA also demonstrates an advantage.
Paper Structure (17 sections, 6 theorems, 40 equations, 6 figures, 6 tables, 3 algorithms)

This paper contains 17 sections, 6 theorems, 40 equations, 6 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main results.
  3. Preliminaries
  4. The quantum approximate optimization algorithm and MaxCut
  5. X-Ramanujan graphs
  6. The multi-angle quantum approximate optimization algorithm
  7. Classical local algorithms for MaxCut
  8. Theoretical Guarantee of QAOA for Additive Product Graphs
  9. Classical MaxCut
  10. Notations.
  11. Iterations for infinite additive product graph X.
  12. Generalizing classical MaxCut to quantum MaxCut
  13. Experiments
  14. Comparison with classical local algorithms
  15. Numerical evaluation on QAOA
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $X$ is an additive product graph defined in defn:additive-graph. Then for any $p$ and any parameters $(\boldsymbol{\gamma},\boldsymbol{\beta})\in [0,2\pi]^{2p}$, the expected cut fraction of the additive product graph satisfies where the expectations $\mathbb{E}[\underline{A}_C(a,b)]$ are defined on different subgraphs and follow the recursive formulas in (eq:EA) and (eq:G).

Figures (6)

  • Figure 1: The expected cut fractions of QAOA and the best-known classical local algorithms are illustrated for the graphs shown in \ref{['fig:graph-set1']}. The results in \ref{['fig:a']}, \ref{['fig:b']}, and \ref{['fig:c']} correspond to the colors blue, red, and green, respectively. The solid lines, dash-dotted lines, and dashed lines represent the performance of ma-QAOA, QAOA, and the best-known classical local algorithms discussed in \ref{['sec:experiments']}.
  • Figure 2: The infinite ($3,4$)-biregular tree constructed by the single-edge graphs on $K_{3,4}$.
  • Figure 3: One example of the additive product graph. A portion of vertices on the graph are labeled with the corresponding string.
  • Figure 4: The subgraph in additive product graph corresponding to the edge $\{v,w\}$ in $\underline{A}_1$ at $p=2$. The labels of vertices and edges come from the generation of additive product graph.
  • Figure 5: The additive product graphs used in the QAOA and classical algorithm experiments.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1: Main Theorem
  • Definition 1
  • Definition 2: mohanty2020x
  • Theorem 2: hirvonen2014large
  • Theorem 3: barak2022classical
  • Proposition 1
  • Theorem 4: expected cut fraction on additive product graph
  • Definition 3
  • Corollary 1
  • proof : Proof sketch