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 |psirangle which states that: for every γ≥ 0, if the textsfGowers(|psi rangle,3)8 ≥ γ then the stabilizer fidelity of |psirangle is at least γC 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 varepsilon1-close to a stabilizer state in fidelity and rejects when |psirangle is varepsilon2 ≤ varepsilon1C-far from all stabilizer states, promised one of them is the case.