具有长程成对相互作用的张量网络算子的低秩表示
Low-rank representation of tensor network operators with long-range pairwise interactions
摘要：
张量网络算子，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。本文主要研究具有长程成对相互作用（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一个独特的“上三角低秩”（UTLR）属性，即矩阵的上三角部分可以很好地近似为低秩矩阵，而矩阵本身可以是满秩的。这使我们能够将找到有效的MPO表示的问题转换为矩阵完成问题。我们提出了一种改进的增量奇异值分解方法（ISVD）来解决这个病态矩阵完备化问题。该算法产生与[Stoudenmire和White，Phys. Rev. Lett. 2017年]。为了有效地处理更一般的张量网络算子，我们开发了另一种策略，用于基于分层低秩矩阵格式压缩张量网络算子，例如分层非对角低秩（HODLR）格式和$\mathcal{H}$ -矩阵格式。虽然复杂度中的前常数较大，但使用分层低秩矩阵格式的优点在于其适用于MPO和PEO两者。对于库仑相互作用，算子可以由$\mathcal{O}(\log(N)\log(N/ε))$ MPO/PEO的线性组合表示，每个MPO/PEO具有恒定的键尺寸，其中$N$是系统尺寸，$ε$是低秩截断的准确度。无论是修改后的ISVD，也不是分层低秩算法假设的长程相互作用采取的抑制不变的形式。
英文摘要：
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 a peculiar "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 [Stoudenmire and White, Phys. Rev. Lett. 2017]. 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 pre-constant 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 $\mathcal{O}(\log(N)\log(N/ε))$ MPOs/PEPOs, each with a constant bond dimension, where $N$ is the system size and $ε$ 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.
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