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 two results: (i) Assuming $|\psi\rangle$ is a phase state, i.e., $|\psi\rangle=\frac{1}{\sqrt{2^n}}\sum \limits_{x \in \{0,1\}^n} {f(x)}|x\rangle$ where $f:\{0,1\}^n\rightarrow \{-1,1\}$, then we give a $\textsf{poly}(1/\varepsilon_1)$ sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$ time algorithm for every $\varepsilon_1 > 0$ and $\varepsilon_2 \leq \textsf{poly}(\varepsilon_1)$, for tolerant testing stabilizer states. (ii) For arbitrary quantum states $|\psi\rangle$, assuming a conjecture in additive combinatorics, 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；或 (ii) $|\psi\rangle$ 与所有稳定子态 $\varepsilon_2$-远离，需判定属于何种情形。我们证明了两项结果：(i) 若 $|\psi\rangle$ 为相位态，即 $|\psi\rangle=\frac{1}{\sqrt{2^n}}\sum \limits_{x \in \{0,1\}^n} {f(x)}|x\rangle$，其中 $f:\{0,1\}^n\rightarrow \{-1,1\}$，则对于任意 $\varepsilon_1 > 0$ 与 $\varepsilon_2 \leq \textsf{poly}(\varepsilon_1)$，我们给出一种样本复杂度为 $\textsf{poly}(1/\varepsilon_1)$、时间复杂度为 $n\cdot \textsf{poly}(1/\varepsilon_1)$ 的算法，用于稳定子态的容忍性测试。(ii) 对于任意量子态 $|\psi\rangle$，在加法组合学中某猜想成立的假设下，我们针对任意 $\varepsilon_1>0$ 与 $\varepsilon_2\leq 2^{-\textsf{poly}(1/\varepsilon_1)}$ 给出一种样本复杂度为 $\textsf{poly}(1/\varepsilon_1)$、时间复杂度为 $n\cdot \textsf{poly}(1/\varepsilon_1)$ 的算法。我们的证明包含：量子态高尔斯范数的新定义、量子态高尔斯-$3$ 范数的逆定理，以及利用加法组合学结论得到的保罗算子结构子集稳定子覆盖新边界。