Extend strong hardness from local to all stable quantum algorithms

Establish whether the strong hardness result for the Quantum Hypergraph Max-Cut problem can be extended from local algorithms defined using the quantum Wasserstein distance of order infinity to all stable algorithms defined using the quantum Wasserstein distance of order two.

Background

The paper proves a weak hardness result for stable quantum algorithms, but the required hypergraph locality parameter k depends on the algorithm’s Lipschitz constant. It proves a stronger result for local algorithms, using the quantum Wasserstein distance of order infinity, for which k can be chosen independently of the Lipschitz constant.

The authors identify the extension of this strong hardness result to all stable algorithms as the principal unresolved issue. Their obstacle is that the classical-shadows reduction introduces multiple replicas whose collective stability does not provide sufficient control over individual pairwise overlaps. They suggest developing a stronger Quantum Overlap Gap Property, potentially by studying the geometry of quantum-state space directly rather than through classical-shadow estimators.