  Parameterized quantum circuits (PQCs) are a central component of many
variational quantum algorithms, yet there is a lack of understanding of how
their parameterization impacts algorithm performance. We initiate this
discussion by using principal bundles to geometrically characterize two-qubit
PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a
simple equation relating the Ricci scalar (geometry) and concurrence
(entanglement). By calculating the Ricci scalar during a variational quantum
eigensolver (VQE) optimization process, this offers us a new perspective to how
and why Quantum Natural Gradient outperforms the standard gradient descent. We
argue that the key to the Quantum Natural Gradient's superior performance is
its ability to find regions of high negative curvature early in the
optimization process. These regions of high negative curvature appear to be
important in accelerating the optimization process.
