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

TL;DR

This paper explores the application of Quantum Approximate Optimization Algorithms (QAOA) to MaxCut problems on low-girth graphs, including expander and planar graphs, demonstrating quantum advantage over classical algorithms in several cases.

Contribution

The paper introduces a novel application of QAOA and multi-angle QAOA to low-girth graphs, deriving an iterative formula for expected cut fraction and demonstrating quantum advantage through numerical experiments.

Findings

QAOA outperforms classical algorithms by up to 5.2% on additive product graphs.

ma-QAOA further improves performance by up to 2.5%.

QAOA shows advantage on planar graphs like tiling grid graphs.

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.