Quantum algorithms for matrix geometric means

TL;DR

This paper introduces quantum algorithms for matrix geometric means, enabling efficient solutions to nonlinear systems, geodesic computations, and optimization problems with applications in machine learning and quantum information.

Contribution

It develops quantum subroutines for matrix geometric means and applies them to solve nonlinear equations, construct quantum learning algorithms, and estimate quantum information measures.

Findings

Quantum subroutines for matrix geometric means are effective.

Applications include quantum learning algorithms and fidelity estimation.

Achieves optimal precision dependence in quantum entropy measures.

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 R\'enyi 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.