The Fubini–Study metric and one natural-gradient step

The model state is prepared by the circuit $R_Y(\theta)$ followed by $R_Z(\phi)$ on one qubit,

$\lvert\psi(\theta,\phi)\rangle=R_Z(\phi)\,R_Y(\theta)\lvert0\rangle,\qquad R_P(t)=e^{-itP/2}.$

Its Fubini–Study metric is the real symmetric $2\times2$ matrix

$g_{ij}=\mathrm{Re}\Big[\langle\partial_i\psi\vert\partial_j\psi\rangle-\langle\partial_i\psi\vert\psi\rangle\langle\psi\vert\partial_j\psi\rangle\Big],\qquad i,j\in\{\theta,\phi\}.$

Enter $g_{\theta\theta}$, $g_{\theta\phi}$ and $g_{\phi\phi}$ at $\theta=\pi/3$ (the metric does not depend on $\phi$). Each value is graded with absolute tolerance $10^{-3}$; exact expressions such as sin(pi/3)^2/4 are accepted.

Take the loss $L(\theta,\phi)=\langle\psi(\theta,\phi)\vert Z\vert\psi(\theta,\phi)\rangle=\cos\theta$ for the same model. Starting from $(\theta,\phi)=(\pi/3,\,0)$, perform one quantum-natural-gradient step with learning rate $\eta=0.1$ and no regularisation,

$\begin{pmatrix}\theta'\\\phi'\end{pmatrix}=\begin{pmatrix}\theta\\\phi\end{pmatrix}-\eta\,g^{+}\,\nabla L,$

where $g$ is the metric of the previous step and $g^{+}$ its (pseudo-)inverse. Enter $\theta'$ and $\phi'$. Each value is graded with absolute tolerance $10^{-3}$; exact expressions such as pi/3+0.2*sqrt(3) are accepted.