We show an improved inverse theorem for the Gowers-$3$ norm of $n$-qubit quantum states $|\psi\rangle$ which states that: for every $\gamma\geq 0$, if the $\textsf{Gowers}(|\psi \rangle,3)^8 \geq \gamma$ then the stabilizer fidelity of $|\psi\rangle$ is at least $\gamma^C$ 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 $|\psi\rangle$ is $\varepsilon_2 \leq \varepsilon_1^C$-far from all stabilizer states, promised one of them is the case.

翻译：我们针对$n$量子比特量子态$|\psi\rangle$的Gowers-$3$范数给出了一个改进的逆定理：对于任意$\gamma\geq 0$，若$\textsf{Gowers}(|\psi \rangle,3)^8 \geq \gamma$，则$|\psi\rangle$的稳定子保真度至少为$\gamma^C$，其中$C>1$为常数。该结论导出了一个常数样本量的多项式时间容错测试算法，用于检测稳定子态：当未知量子态在保真度意义下$\varepsilon_1$接近某个稳定子态时算法接受；当$|\psi\rangle$与所有稳定子态的距离满足$\varepsilon_2 \leq \varepsilon_1^C$时算法拒绝（假设两种情况必居其一）。