Clique Homology is QMA1-hard

Abstract

We tackle the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch 20 years ago. We show that this decision problem is QMA1-hard. Moreover, we show that a version of the problem satisfying a suitable promise and certain constraints is contained in QMA. This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we are able to significantly strengthen this by showing that the problem remains QMA1-hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and homological algebra. We discuss potential implications for the problem of quantum advantage in topological data analysis.

arXiv comment50 pages, 20 figures

References in corpus (7)

• Lattice Models with N=2 Supersymmetry
• Towards quantum advantage via topological data analysis
• Analyzing Prospects for Quantum Advantage in Topological Data Analysis
• On the complexity of quantum partition functions
• Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis
• Complexity of Supersymmetric Systems and the Cohomology Problem
• Representation of the Fermionic Boundary Operator

Cited by in corpus (5)

• Quantum Algorithm for Estimating Betti Numbers Using Cohomology Approach
• Provable quantum speedups for computing persistence in topological data analysis
• Comparing quantum and classical Monte Carlo algorithms for estimating Betti numbers of clique complexes
• Holey graphs: very large Betti numbers are testable
• Quantum Merlin-Arthur proof systems for synthesizing quantum states