Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This process can be used to obtain rigorous lower bounds on the ground-state energy of arbitrary local Hamiltonians by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of $n\gtrsim 100$ spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection.

中文翻译：

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通过重整化群得出局部哈密顿量的基态能量下界

给定重正化方案，我们展示了如何制定多体量子系统的可行局部密度矩阵集的易于处理的凸松弛。通过在不断增长的晶格位点组的简化状态之间引入约束层次来获得松弛。底层重整化过程的粗粒度映射用于消除大量这些约束，以便可以通过合理的计算手段强制执行剩余的约束。该过程可用于通过对所得到的约化量子态的凸松弛执行线性优化来获得任意局部哈密顿量的基态能量的严格下界。边界的质量关键取决于特定的重整化方案，该方案必须根据目标哈密顿量进行定制。我们将我们的方法应用于一维平移不变自旋模型，获得的能量界限与通过优化局部平移不变状态所获得的能量界限相当$n\gtrsim 100$旋转。除了这个演示之外，通用方法还可以应用于广泛的其他问题，例如更高空间维度的自旋系统、电子结构问题以及各种其他多体优化问题，例如纠缠和非局域性检测。