A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

Sam McArdle; András Gilyén,; Mario Berta

doi:10.22331/q-2026-04-10-2058

A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

Sam McArdle¹, András Gilyén,², and Mario Berta^3,4

¹AWS Center for Quantum Computing, Pasadena, CA 91125, USA
²Alfréd Rényi Institute of Mathematics, Budapest, Hungary
³Institute for Quantum Information, RWTH Aachen University, Aachen, Germany
⁴Department of Computing, Imperial College London, London, UK

Published:	2026-04-10, volume 10, page 2058
Editor:	Jin-Peng Liu
Eprint:	arXiv:2209.12887v3
Doi:	https://doi.org/10.22331/q-2026-04-10-2058
Citation:	Quantum 10, 2058 (2026).

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Abstract

Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyze and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over previously known rigorous classical algorithms for computing the persistent Betti numbers to constant additive error – the salient task for applications. However, we also introduce a quantum-inspired classical power method with provable scaling only quadratically worse than the quantum algorithm. This gives a provable classical algorithm with scaling comparable to existing classical heuristics. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence for this being the case.

► BibTeX data

@article{McArdle2026streamlinedquantum,
  doi = {10.22331/q-2026-04-10-2058},
  url = {https://doi.org/10.22331/q-2026-04-10-2058},
  title = {A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits},
  author = {McArdle, Sam and Gily{\'{e}}n,, Andr{\'{a}}s and Berta, Mario},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2058},
  month = apr,
  year = {2026}
}

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 17:17:16) and SAO/NASA ADS (last updated successfully 2026-08-08 17:17:17). The list may be incomplete as not all publishers provide suitable and complete citation data.

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