Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs

TL;DR

This paper compares the performance of a quantum algorithm (QAOA$_2$) with classical algorithms on high-girth regular graphs for MAX-CUT, showing classical algorithms can outperform the quantum approach.

Contribution

It provides a graph-size-independent analysis of QAOA$_2$ on high-girth graphs and demonstrates classical algorithms can surpass QAOA$_2$ in expected cut fraction.

Findings

QAOA$_2$ outperforms QAOA$_1$ on high-girth regular graphs.

A classical 2-local randomized algorithm outperforms QAOA$_2$ on these graphs.

Supports the conjecture that classical algorithms can match QAOA$_p$ performance.

Abstract

The $p$-stage Quantum Approximate Optimization Algorithm (QAOA$_p$) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond $p=1$. We analyze QAOA$_2$ for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any $D$-regular graph of girth $> 5$ (i.e. without triangles, squares, or pentagons). We show that for all degrees $D \ge 2$ and every $D$-regular graph $G$ of girth $> 5$, QAOA$_2$ has a larger expected cut fraction than QAOA$_1$ on $G$. However, we also show that there exists a $2$-local randomized classical algorithm $A$ such that $A$ has a larger expected cut fraction than QAOA$_2$ on all $G$. This supports our conjecture that for every constant $p$, there exists a local classical MAX-CUT algorithm that performs as well as QAOA$_p$ on all graphs.