Quantum TDA hardness–tractability conjecture

Establish that topological data analysis features computed from point clouds (such as persistent homology or related invariants) are classically hard to compute while being tractable for quantum computers, thereby resolving the conjecture cited by Berry et al. and supporting quantum-advantage claims for such feature extraction.

Background

The authors compare their feature-extraction setting to quantum topological data analysis (TDA), where features are extracted from unlabeled point clouds. They note that the conjectured classical hardness vs. quantum tractability of these features—referenced in Berry et al. (2024)—has not yet been rigorously established.

A proof of this conjecture would provide a foundation for quantum-advantageous feature extraction in TDA-like settings and clarify whether quantum devices offer provable benefits for computing such features.

References

However, even in this case, and even if there was a rigorous proof that these features are hard to compute classically but are tractable for quantum computers (at present this is still a conjecture), this would not suffice for the type of separation claim we desire.

— Machine learning with minimal use of quantum computers: Provable advantages in Learning Under Quantum Privileged Information (LUQPI)  (2601.22006 - Bokov et al., 29 Jan 2026) in Section 3, Taxonomy of scenarios – Other (Topological Data Analysis)