Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains
Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains
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利用矩阵乘积状态和张量序列中的矩阵对称性和物理对称性
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发表时间：
2013
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通讯作者：
T. Schulte
中科院分区：
文献类型：
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作者：
T. Huckle;K. Waldherr;T. Schulte
我们关注与量子多体系统模拟中出现的矩阵和向量相关的对称性。自旋哈密顿量具有特殊的矩阵对称性质，例如全对称性。此外，系统可以表现出物理对称性，转化为感兴趣的特征向量的对称性。这两种类型的对称性都可以在稀疏表示格式中利用，例如所需特征向量的矩阵积状态 (MPS)。本文总结了典型物理系统（例如伊辛模型）的哈密顿量的对称性，并列出了相关特征向量的所得属性。基于 MPS（张量链或张量链）及其规范范式的概述，我们展示了矢量的对称性如何转化为 MPS 矩阵之间的关系，以及反过来，哪些对称性是由 MPS 矩阵内的关系产生的。在这种情况下，我们分析了不同类型的对称性，并导出了代表这些对称性的 MPS 的适当范式。通过使用这些范式来利用这种对称性将导致 MPS 矩阵中自由度的数量减少。本文为从（多）线性代数角度提出的众所周知的和新的结果提供了一个统一的平台。
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This article summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of MPS (Tensor Trains or Tensor Chains) and their canonical normal forms, we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context, we analyse different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This article provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.