Code and Data for "Anticoncentration and state design of random tensor networks"

We investigate quantum random tensor network states where the bond dimensions scale polynomially with the system size, N. Specifically, we examine the delocalization properties of random Matrix Product States (RMPS) in the computational basis by deriving an exact analytical expression for the Inverse Participation Ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions χ∼γN, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as γ increases. Additionally, we provide numerical evidence for the frame potential, measuring the 2-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for χ≫\sqrt{N}. We extend this analysis to two-dimensional systems using random Projected Entangled Pair States (PEPS), where we similarly observe the convergence of IPRs to their Haar values for χ≫\sqrt{N}. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar-anticoncentrated and approximate unitary designs, regardless of the spatial dimension.本研究聚焦于键维度随系统尺寸N呈多项式缩放的量子随机张量网络态。具体而言，我们通过推导适用于开放边界条件与闭合边界条件的任意阶逆参与比（Inverse Participation Ratio，IPR）的精确解析表达式，研究了计算基矢下随机矩阵乘积态（random Matrix Product States，RMPS）的退局域化特性。当键维度χ~γN时，我们确定了对应重叠概率分布的首项阶次，并证明当γ增大时，该分布会收敛至哈尔随机态（Haar-random states）的特征分布——波特-托马斯分布（Porter-Thomas distribution）。此外，我们通过可衡量与哈尔系综（Haar ensemble）间2-距离的框架势（frame potential）提供了数值证据，证实了当χ≫√N时，随机MPS会收敛至类哈尔行为。我们将该分析拓展至采用随机投影纠缠对态（random Projected Entangled Pair States，PEPS）的二维系统，并同样观察到当χ≫√N时，逆参与比会收敛至其哈尔值。上述研究结果表明，键维度随系统尺寸呈多项式缩放的随机张量网络，无论空间维度如何，均具备完全的哈尔反聚集（Haar-anticoncentrated）特性，且属于近似幺正设计。