Gapped Clique Homology is QMA1-Hard and Contained in QMA

Explore a conference talk from TQC 2024 that delves into the complexity analysis of the homology problem in computational topology, specifically focusing on clique complexes. Learn how researchers demonstrate that determining the existence of holes in clique complexes of weighted graphs, under specific gap promises, is both QMA1-hard and contained within QMA. Discover innovative techniques for establishing lower bounds on combinatorial Laplacian operator eigenvalues through spectral sequences and their connection to Hodge theory. Understand the implications of these findings for quantum topological data analysis (TDA), including insights into why quantum TDA algorithms may resist dequantization. Examine the intersection of classical topology and quantum complexity classes, building on connections between homology and supersymmetric quantum mechanics. Presented by Robbie King and Tamara Kohler at the 19th Conference on the Theory of Quantum Computation, Communication and Cryptography, this 24-minute presentation offers valuable insights into potential quantum advantages in topological data analysis.