We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|\psi\rangle$ promised $(i)$ $|\psi\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|\psi\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which is the case. We show that for every $\varepsilon_1>0$ and $\varepsilon_2\leq \varepsilon_1^C$, there is a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm that decides which is the case (where $C>1$ is a universal constant). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of quantum states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics.

翻译：我们考虑以下任务：假设某算法获得未知 $n$ 量子比特量子态 $|\psi\rangle$ 的若干副本，且已知以下两种情况之一成立：$(i)$ $|\psi\rangle$ 在保真度意义下 $\varepsilon_1$-接近某个稳定子态；$(ii)$ $|\psi\rangle$ 与所有稳定子态 $\varepsilon_2$-远离。本文证明对于任意 $\varepsilon_1>0$ 及满足 $\varepsilon_2\leq \varepsilon_1^C$ 的参数（其中 $C>1$ 为普适常数），存在样本复杂度为 $\textsf{poly}(1/\varepsilon_1)$、时间复杂度为 $n\cdot \textsf{poly}(1/\varepsilon_1)$ 的算法可判定具体情形。我们的证明过程包含：提出量子态高尔斯范数的新定义，建立量子态高尔斯-$3$ 范数的逆定理，并利用加性组合学结论对泡利算符结构子集的稳定子覆盖给出新的界。