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

Kunal Marwaha

Berkeley Center for Quantum Information and Computation, University of California, Berkeley, California 94720, USA

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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 $\textit{maximum cut problem}$ (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 $\textit{classical}$ 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.

For further reading go to our presentation Local Competitors to QAOA

 

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[3] Kunal Marwaha and Stuart Hadfield, "Bounds on approximating Max kXOR with quantum and classical local algorithms", Quantum 6, 757 (2022).

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[5] Zeqiao Zhou, Yuxuan Du, Xinmei Tian, and Dacheng Tao, "QAOA-in-QAOA: Solving Large-Scale MaxCut Problems on Small Quantum Machines", Physical Review Applied 19 2, 024027 (2023).

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The above citations are from Crossref's cited-by service (last updated successfully 2023-05-29 20:37:24) and SAO/NASA ADS (last updated successfully 2023-05-29 20:37:25). The list may be incomplete as not all publishers provide suitable and complete citation data.