Quantum computing and persistence in topological data analysis

• URL: http://arxiv.org/abs/2410.21258v1
• Date: Mon, 28 Oct 2024 17:54:43 GMT
• Title: Quantum computing and persistence in topological data analysis
• Authors: Casper Gyurik, Alexander Schmidhuber, Robbie King, Vedran Dunjko, Ryu Hayakawa,
• Abstract summary: Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA is $mathsfBQP_1$-hard and contained in $mathsfBQP$. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.
• Score: 41.16650228588075
• License:
• Abstract: Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales -- is $\mathsf{BQP}_1$-hard and contained in $\mathsf{BQP}$. This result implies an exponential quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

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