Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.

中文翻译：

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广义 Rokhsar-Kivelson 波函数中的魔法

魔力是量子态的一个属性，它表征了它与稳定态的偏差，作为实现通用量子计算的有用资源，例如在使用 Clifford 运算的方案中。在这项工作中，我们在一类称为广义 Rokhsar-Kivelson 系统的模型（即允许随机矩阵形式 (SMF) 分解的哈密顿量）中研究由稳定器 Renyi 熵量化的魔法。这些系统的基态波函数可以在其相图中明确地写出，并且它们的性质可以与相关的经典统计力学问题相关，从而允许强大的分析和数值方法，这在传统的量子多体设置中通常不可用。因此，我们能够用波函数系数来表达 SRE，波函数系数可以理解为相关经典问题的自由能差。我们将这一见解应用于一系列量子多体 SMF 哈密顿量，这使我们能够对大型高维系统的 SRE 进行数值研究，并在某些情况下获得分析结果。我们观察到，SRE 的行为在这些系统中的量子相变中相对没有特征，尽管它确实是奇异的（在其一阶或高阶导数中，取决于转变的性质）。相反，我们发现 SRE 的最大值通常出现在远离量子临界点的尖点处，此时导数突然改变符号。此外，我们将 SRE 和重叠对数与特定稳定器状态进行比较，这些稳定器状态在这些系统的基态相图中渐近实现。我们发现它们表现出惊人相似的行为，这反过来又对魔法的最小相对熵建立了严格的界限。