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arXiv · 2410.21258

Provable quantum speedups for computing persistence in topological data analysis

Casper Gyurik·Alexander Schmidhuber·Robbie King·Vedran Dunjko·Ryu Hayakawa
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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 provide an efficient quantum algorithm for a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales. Further, we prove the problem itself is $\mathsf{BQP}_1$-hard, implying that a classical solution is extremely unlikely; this stands in contrast to all previous quantum approaches to TDA, where the problems were also intractable for quantum computers, or where a rigorous proof of classical hardness still remains open. 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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Casper Gyurik, Alexander Schmidhuber, Robbie King, Vedran Dunjko, Ryu Hayakawa. 2024-10-28. Provable quantum speedups for computing persistence in topological data analysis. https://doi.org/10.1103/gvys-hl8h

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