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arXiv · 2510.07014

Computational complexity of the homology problem with orientable filtration: MA-completeness

Ryu Hayakawa·Casper Gyurik·Mahtab Yaghubi Rad·Vedran Dunjko
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Abstract

We show the existence of an MA-complete homology problem for a certain subclass of simplicial complexes. The problem is defined through a new concept of orientability of simplicial complexes that we call a "uniform orientable filtration", which is related to sign-problem freeness in homology. The containment in MA is achieved through the design of new, higher-order random walks on simplicial complexes associated with the filtration. For the MA-hardness, we design a new gadget with which we can reduce from an MA-hard stoquastic satisfiability problem. Therefore, our result provides the first natural MA-complete problem for higher-order random walks on simplicial complexes, combining the concepts of topology, persistent homology, and quantum computing.

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Ryu Hayakawa, Casper Gyurik, Mahtab Yaghubi Rad, Vedran Dunjko. 2025-10-08. Computational complexity of the homology problem with orientable filtration: MA-completeness. https://arxiv.org/abs/2510.07014

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