Quantum algorithm limits revealed for hypergraph max cut problems
The Quantum Overlap Gap Property and Algorithmic Hardness for the Quantum Hypergraph Max-Cut Problem
Computational Complexity
Summary
Solving the Quantum Hypergraph Max-Cut problem means finding the best way to break a complex quantum system into parts. The authors studied why some popular quantum algorithms struggle to get close to the best solution, showing there are inherent limits depending on the algorithm’s properties. They used a mathematical tool called the Quantum Overlap Gap Property to prove that for many algorithms, especially those that are stable or local, it’s impossible to approximate the best answer well. This helps explain why some quantum methods can't easily solve these problems even on average.
What this means in practice
- •For quantum algorithm developers: Determine limits on how close certain quantum algorithms can get to optimal solutions for Quantum Hypergraph Max-Cut problems based on algorithm stability and locality.
- •For quantum hardware engineers: Guide the design of quantum hardware by understanding depth requirements needed for algorithms to prepare near-optimal states for complex quantum optimization tasks.
A theory result. No direct application yet.
Authors
Mikhail Mints, Eric R. Anschuetz
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.