Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity
$$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \geq \epsilon \big].$$
Note that if $|\psi\rangle$ is Haar random, then, by equation $2$ of this paper,
$$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \geq \epsilon \big] \leq \mathsf{exp}(-(2d-1) \epsilon).$$
Does a similar concentration bound hold for random stabilizer states too?
