Quantum algorithms for matrix geometric means

Liu, N; Wang, Q; Wilde, MM; Zhang, Z

Quantum algorithms for matrix geometric means

Liu, N Wang, Q Wilde, MM Zhang, Z

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Publisher:: NATURE PORTFOLIO
Publication Type:: Journal Article
Citation:: Npj Quantum Information, 2025, 11, (1)
Issue Date:: 2025-12-01

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Field	Value	Language
dc.contributor.author	Liu, N
dc.contributor.author	Wang, Q
dc.contributor.author	Wilde, MM
dc.contributor.author	Zhang, Z https://orcid.org/0000-0002-7436-0426
dc.date.accessioned	2025-07-19T07:07:58Z
dc.date.available	2025-07-19T07:07:58Z
dc.date.issued	2025-12-01
dc.identifier.citation	Npj Quantum Information, 2025, 11, (1)
dc.identifier.issn	2056-6387
dc.identifier.issn	2056-6387
dc.identifier.uri	http://hdl.handle.net/10453/188552
dc.description.abstract	Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.
dc.language	English
dc.publisher	NATURE PORTFOLIO
dc.relation.ispartof	Npj Quantum Information
dc.relation.isbasedon	10.1038/s41534-025-00973-7
dc.rights	info:eu-repo/semantics/openAccess
dc.subject.classification	4613 Theory of computation
dc.subject.classification	4902 Mathematical physics
dc.subject.classification	5108 Quantum physics
dc.title	Quantum algorithms for matrix geometric means
dc.type	Journal Article
utslib.citation.volume	11
pubs.organisational-group	University of Technology Sydney
pubs.organisational-group	University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	University of Technology Sydney/Faculty of Engineering and Information Technology/School of Computer Science
utslib.copyright.status	open_access	*
dc.rights.license	This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.date.updated	2025-07-19T07:07:57Z
pubs.issue	1
pubs.publication-status	Accepted
pubs.volume	11
utslib.citation.issue	1

Abstract:

Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.

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http://hdl.handle.net/10453/188552