Magic, or nonstabilizerness, characterizes how far away a state is from the stabilizer states, making it an important resource in quantum computing, under the formalism of the Gotteman-Knill theorem. In this paper, we study the magic of the one-dimensional (1D) random matrix product states (RMPSs) using the $L_1$-norm measure. We first relate the $L_1$ norm to the $L_4$ norm. We then employ a unitary four-design to map the $L_4$ norm to a 24-component statistical physics model. By evaluating partition functions of the model, we obtain a lower bound on the expectation values of the $L_1$ norm. This bound grows exponentially with respect to the qudit number $n$, indicating that the $1\mathrm{D}$ RMPS is highly magical. Our numerical results confirm that the magic grows exponentially in the qubit case.

中文翻译：

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随机矩阵乘积状态的魔力

魔力或非稳定态描述了一个态与稳定态的距离，使其成为戈特曼-尼尔定理形式下量子计算的重要资源。在本文中，我们使用以下公式研究一维 (1D) 随机矩阵乘积状态 (RMPS) 的魔力$L_1$-标准测量。我们首先关联$L_1$规范为$L_4$规范。然后，我们采用统一的四设计来绘制$L_4$24 分量统计物理模型的范数。通过评估模型的配分函数，我们获得了期望值的下界$L_1$规范。这个界限相对于 Qudit 数量呈指数增长$n$，表明$1\mathrm{D}$RMPS 非常神奇。我们的数值结果证实，在量子比特的情况下，魔力呈指数级增长。