We present a novel classical algorithm designed to learn the stabilizer group—namely, the group of Pauli strings for which a state is a $\pm 1$ eigenvector—of a given matrix product state (MPS). The algorithm is based on a clever and theoretically grounded biased sampling in the Pauli (or Bell) basis. Its output is a set of independent stabilizer generators whose total number is directly associated with the stabilizer nullity, notably a well-established nonstabilizer monotone. We benchmark our method on $T$-doped states randomly scrambled via Clifford unitary dynamics, demonstrating very accurate estimates up to highly entangled MPS with bond dimension $\chi \sim {10}^3$. Our method, thanks to a very favorable scaling $\mathcal{O}({\chi }^3)$, represents the first effective approach to obtain a genuine magic monotone for MPS, enabling systematic investigations of quantum many-body physics out of equilibrium.

中文翻译：

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揭开矩阵产品状态的稳定剂组的面纱


我们提出了一种新颖的经典算法，旨在学习给定矩阵乘积状态 (MPS) 的稳定组，即状态为 $\pm 1$ 特征向量的泡利弦组。该算法基于泡利（或贝尔）基础上的巧妙且有理论依据的偏置采样。其输出是一组独立的稳定器生成器，其总数与稳定器无效性直接相关，尤其是公认的非稳定器单调性。我们在通过 Clifford 酉动力学随机扰乱的 $T$ 掺杂态上对我们的方法进行了基准测试，证明了对具有键尺寸 $\chi \sim {10}^3$ 的高度纠缠 MPS 的非常准确的估计。我们的方法，由于非常有利的缩放 $\mathcal{O}({\chi }^3)$ ，代表了第一个获得 MPS 真正神奇单调的有效方法，从而能够系统地研究不平衡的量子多体物理。