  Variational quantum algorithms are a promising class of algorithms that can
be performed on currently available quantum computers. In most settings, the
free parameters of a variational circuit are optimized using a classical
optimizer that updates parameters in Euclidean geometry. Since quantum circuits
are elements of the special unitary group, we can consider an alternative
optimization perspective that depends on the structure of this group. In this
work, we investigate a Riemannian optimization scheme over the special unitary
group and we discuss its implementation on a quantum computer. We illustrate
that the resulting Riemannian gradient-flow algorithm has favorable
optimization properties for deep circuits and that an approximate version of
this algorithm can be performed on near-term hardware.
