We consider the problem of testing whether an unknown $n$-qubit quantum state $|\psi\rangle$ is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using $O(n)$ copies, and conversely prove that $\Omega(\sqrt{n})$ copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.

翻译：我们考虑在仅具备单副本访问权限的条件下，测试一个未知的$n$-量子比特量子态$|\psi\rangle$是否为稳定子态的问题。我们给出了一种使用$O(n)$个副本解决该问题的算法，并反过来证明了任何算法都需要$\Omega(\sqrt{n})$个副本。我们算法背后的主要观察是：当在随机选择的稳定子基下重复测量时，在所有纯态集合中，稳定子态最有可能在测量结果中表现出线性依赖性。我们的算法旨在探测偏离这种极端行为的情况。对于下界，我们首先将稳定子态测试问题约简为区分随机稳定子态与最大混合态的任务。然后我们论证，不失一般性，只需考虑满足以下条件的测量策略即可：a) 位于克利福德群张量作用的交换子中；b) 满足正部分转置条件。通过利用这些约束，并结合关于克利福德交换子生成元的部分转置的新结果，我们推导出了样本复杂度的下界。