  Matrix geometric means between two positive definite matrices can be defined
equivalently 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. This diversity already suggests the
potential for varied applications, as well as acting as a bridge between
different domains. Here we devise new quantum subroutines to efficiently
prepare quantum unitary operators that embed the standard matrix geometric mean
and its generalisations called the weighted matrix geometric mean. This enables
the construction of solutions to the algebraic Riccati equation, which is an
important class of nonlinear systems of equations that appears in machine
learning, optimal control, estimation, and filtering. Using these subroutines,
we present a new class of quantum learning algorithms called quantum geometric
mean metric learning. This has applications in efficiently finding the best
distance measure and solving classification problems in the weakly supervised
limit and for anomaly detection, for both classical and quantum problems. We
also show how our method can be generalised to a particular p^th-order system
of nonlinear equations. These quantum subroutines for matrix geometric means
are also useful in other areas of quantum information. For example, we show how
to use them in the estimation of geometric Renyi relative entropies and the
Uhlmann fidelity by means of the Fuchs-Caves observable. In particular, our
quantum algorithms for estimating the Uhlmann and Matsumoto fidelities have
optimal dependence on the precision. Finally, we provide a BQP-complete problem
based on matrix geometric means that can be solved by our subroutines, thus
characterising their computational capability.
