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 give a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm for this task for every $\varepsilon_1>0$ and $\varepsilon_2\leq 2^{-\textsf{poly}(1/\varepsilon_1)}$. Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of 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：假设算法获得未知$n$量子比特量子态$|\psi\rangle$的若干副本，并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，要求判断属于哪种情况。我们针对任意$\varepsilon_1>0$和$\varepsilon_2\leq 2^{-\textsf{poly}(1/\varepsilon_1)}$的情况，给出了一个样本复杂度为$\textsf{poly}(1/\varepsilon_1)$、时间复杂度为$n\cdot \textsf{poly}(1/\varepsilon_1)$的算法。我们的证明包含以下新贡献：提出了量子态高尔斯范数的新定义，建立了量子态高尔斯-$3$范数的逆定理，并利用加性组合学的结果对泡利算子结构子集的稳定子覆盖给出了新的界。