Quantum computing and persistence in topological data analysis

TL;DR

This paper demonstrates that determining the persistence of topological features in data sets can be exponentially accelerated using quantum algorithms, linking topological data analysis with quantum computational complexity.

Contribution

It establishes the quantum computational hardness of a core TDA task and introduces a quantum approach to efficiently solve it.

Findings

Determines the problem is $ extsf{BQP}_1$-hard and in $ extsf{BQP}.

Shows exponential quantum speedup for persistence detection.

Connects TDA persistence problems with quantum Hamiltonian simulation.

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.