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  • Open Access

Provable Quantum Speedups for Computing Persistence in Topological Data Analysis

Casper Gyurik1, Alexander Schmidhuber2, Robbie King3, Vedran Dunjko1, and Ryu Hayakawa4,*

  • *Contact author: ryu.hayakawa@yukawa.kyoto-u.ac.jp
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PRX Quantum 7, 020361 – Published 16 June, 2026

DOI: https://doi.org/10.1103/gvys-hl8h

Abstract

Topological data analysis (TDA) aims to extract noise-robust features from a dataset 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 BQP1-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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Physics Subject Headings (PhySH)

Popular Summary

Topological data analysis (TDA) extracts the ”shape” of high-dimensional data by tracking holes—loops, cavities, and their higher-dimensional analogs—that appear and disappear as the resolution of observation changes.

Determining whether a particular hole survives this process is a core computational task in TDA; we prove, using tools from computational complexity theory, that this task is efficiently solvable on a quantum computer yet lies beyond the reach of any efficient classical algorithm.

We achieve this by encoding the persistence question into quantum circuits and showing that it captures the full power of quantum computation.

Unlike previous claims of quantum advantage in TDA, our result is backed by a rigorous complexity-theoretic argument rather than heuristic or empirical evidence.

While our result is theoretical in nature and currently applies to worst-case instances rather than practical datasets, it lays a rigorous foundation for the future development of practically useful quantum algorithms for shape-based analysis of complex real-world data.

Article Text

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Outline

Information

PRX Quantum 7, 020361– Published 16 June, 2026
  • Received 14 December 2025
  • Revised 29 March 2026
  • Accepted 5 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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