A note on polynomial-time tolerant testing stabilizer states

We show an improved inverse theorem for the Gowers-$3$ norm of $n$-qubit quantum states $|ψ\rangle$ which states that: for every $γ\geq 0$, if the $\textsf{Gowers}(|ψ\rangle,3)^8 \geq γ$ then the stabilizer fidelity of $|ψ\rangle$ is at least $γ^C$ for some constant $C>1$. This implies a constant-sample polynomial-time tolerant testing algorithm for stabilizer states which accepts if an unknown state is $\varepsilon_1$-close to a stabilizer state in fidelity and rejects when $|ψ\rangle$ is $\varepsilon_2 \leq \varepsilon_1^C$-far from all stabilizer states, promised one of them is the case.