Depth-free Frobenius map computation in $\mathbb{F}_{p^l}$

Question

I read it somewhere that evaluation of Frobenius maps on elements of $\mathbb{F}_{p^l}$ is multiplicative depth-free. This boils down to the claim that computation of $a^p$ is depth-free inside $\mathbb{F}_{p^l}$.
This is what I understood,
Since elements of $\mathbb{F}_{p^l}$ are polynomials of the form $a_0 + a_1*x + ... + a_{l-1}*x^{l-1}$, let us consider an element $a$.
Now when we compute $a^p$, all the coefficients remain same by Fermat's little theorem, but the positions of these coefficients get shuffled. For e.g. $a_2*x^2 \to a_2*x^{2*p \mod l }$. This means that the exponentiation with $p$ has been reduced to some shifting operations inside the polynomial.
But, how can we say that this exponentiation is depth-free meaning that we can do this with a constant number of mult-gates? Any help is appreciated. Thanks!