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

• URL: http://arxiv.org/abs/2101.05513v2
• Date: Thu, 15 Apr 2021 05:08:13 GMT
• Title: Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs
• Authors: Kunal Marwaha
• Abstract summary: We show that for all degrees $D ge 2$ of girth $> 5$, QAOA$ has a larger expected cut fraction than QAOA$$ on $G$. We conjecture that for every constant $p$, there exists a local classical MAX-CUT algorithm that performs as well as QAOA$_p$ on all graphs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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