I'm studying the quantum mechanics and reading a paper named "Barren plateaus in quantum neural network training landscape." In the paper, the author calculates the gradient of the objective function. I'm a newbie in quantum mechanics and want to know the process to get the gradient of the objective function, which is in Dirac notation.
Here is the information from the paper (Eq. 1~3)
$U(\theta) = U(\theta_1, ..., \theta_L) =\prod_{l=1}^L U_l(\theta_l)W_l$
where
$ U(\theta) =exp(-i\theta_l V_l)$
and the objective function $E(\theta)$ expressed as the expectation value over Hermitian operator $H$
$E(\theta) =i\langle{0|U(\theta)^\dagger H U(\theta)|0\rangle}$
In the last equation, the paper says that the gradient of the objective function takes the simple form as
$\partial_k E = i\langle0|U^\dagger_{\_}[V_k, U^\dagger_{+}HU_{+}]U_{-}|0\rangle$
where $U_{-} = \prod^{k-1}_{l=o}U_l(\theta_l)W_l$
and
$U_{+} = \prod^{L}_{l=k}U_l(\theta_l)W_l$
In the equation, I can't catch the process of calculating the partial derivatives
Is there any reference for calculating the gradient of dirac notation process for me?