The Quantum Overlap Gap Property and Algorithmic Hardness for the Quantum Hypergraph Max-Cut Problem

2026-09-09

AI Summary

本文使用量子重叠间隙属性分析了量子超图最大割问题的平均情况近似难度，证明了对于稳定量子算法和局部量子算法的硬度结果，并为流行量子算法设定了具体深度下界。

Abstract

In this work, we analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem using the theoretical framework of the Quantum Overlap Gap Property (QOGP). We establish two main results. Our first result applies to a wide class of stable quantum algorithms, satisfying a Lipschitz property with respect to the quantum Wasserstein distance of order $2$. We show a weak hardness result, demonstrating that for any Lipschitz constant $L$, there is some $k$ such that $L$-stable algorithms cannot approximate the optimal solution to Quantum Hypergraph Max-Cut on $k$-uniform hypergraphs in the average case. Additionally, we establish a strong hardness result where $k$ is independent of $L$, but only for a more restricted class of local quantum algorithms defined using the quantum Wasserstein distance of order $\infty$. We apply these results to establish concrete depth lower bounds for popular quantum algorithms for preparing near-optimal states for this problem.

Problem

• Quantum Hypergraph Max-Cut
• Quantum Overlap Gap Property
• Algorithmic Hardness

Innovation

• Quantum Overlap Gap Property
• Lipschitz property
• quantum Wasserstein distance
• algorithmic hardness