SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A164-A192, January 2021.
Tensor network operators, such as the matrix product operator (MPO) and the projected entangled-pair operator (PEPO), can provide efficient representation of certain linear operators in high-dimensional spaces. This paper focuses on the efficient representation of tensor network operators with long-range pairwise interactions such as the Coulomb interaction. For MPOs, we find that all existing efficient methods exploit an upper-triangular low-rank (UTLR) property, i.e., the upper-triangular part of the matrix can be well approximated by a low-rank matrix, while the matrix itself can be full-rank. This allows us to convert the problem of finding the efficient MPO representation into a matrix completion problem. We develop a modified incremental singular value decomposition method (ISVD) to solve this ill-conditioned matrix completion problem. This algorithm yields equivalent MPO representation to that developed in [E. M. Stoudenmire and S. R. White, Phys. Rev. Lett., 119 (2017), 046401]. In order to efficiently treat more general tensor network operators, we develop another strategy for compressing tensor network operators based on hierarchical low-rank matrix formats, such as the hierarchical off-diagonal low-rank (HODLR) format and the $\mathcal{H}$-matrix format. Though the preconstant in the complexity is larger, the advantage of using the hierarchical low-rank matrix format is that it is applicable to both MPOs and PEPOs. For the Coulomb interaction, the operator can be represented by a linear combination of $\mathcal{O}(\log(N)\log(N/\epsilon))$ MPOs/PEPOs, each with a constant bond dimension, where $N$ is the system size and $\epsilon$ is the accuracy of the low-rank truncation. Neither the modified ISVD nor the hierarchical low-rank algorithm assumes that the long-range interaction takes a translation-invariant form.

中文翻译：

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具有长距离成对交互作用的张量网络算子的低秩表示

SIAM科学计算杂志，第43卷，第1期，A164-A192页，2021年1月。
张量网络算子，例如矩阵乘积算子（MPO）和投影纠缠对算子（PEPO），可以提供高维空间中某些线性算子的有效表示。本文着重于具有远程成对相互作用（例如库仑相互作用）的张量网络算子的有效表示。对于MPO，我们发现所有现有的有效方法都利用了上三角低秩（UTLR）属性，即矩阵的上三角部分可以由低阶矩阵很好地近似，而矩阵本身可以是全职。这使我们可以将寻找有效MPO表示的问题转换为矩阵完成问题。我们开发了一种改进的增量式奇异值分解方法（ISVD），以解决此病态矩阵完成问题。该算法产生的等效MPO表示形式与[EM Stoudenmire和SR White，Phys。Rev. Lett。，119（2017），046401]。为了有效地处理更一般的张量网络运营商，我们开发了另一种策略，用于基于分层低秩矩阵格式（例如，分层对角低秩（HODLR）格式和$ \ mathcal {H } $-矩阵格式。尽管复杂性的先决条件是较大的，但使用分层低秩矩阵格式的优点是它适用于MPO和PEPO。对于库仑相互作用，运算符可以用$ \ mathcal {O}（\ log（N）\ log（N / \ epsilon））$ MPO / PEPO的线性组合表示，每个MPO / PEPO具有恒定的键维，其中$ N $是系统大小，$ \ epsilon $是低阶截断的精度。