局部矩阵乘积算子：标准形、压缩与控制理论
Local Matrix Product Operators: Canonical Form, Compression, & Control Theory
摘要：
我们提出了一种新的方法来压缩矩阵乘积算子（MPO），表示局部项的总和，如哈密尔顿算子。正如区域法律所规定的那样，这些本地运营商可能会被完全指定，每个站点的信息量很小。标准的矩阵乘积状态（MPS）工具不适合这种情况下，由于广泛的施密特值共存的密集的，和约旦块的转移矩阵。我们改善这些问题，通过引入一个“几乎施密特分解”，尊重地方。我们的方法是“$\varepsilon$ -接近”的精度MPS为基础的方法有限的MPO，并无缝扩展到热力学极限，MPS技术是不适用的。在控制理论的框架下，我们的方法推广了Kung的模型降阶算法。提供了几个例子，包括所有MPO版本的运营商递归方法（Lanczos算法）直接在热力学极限。所有的结果都伴随着实用的算法，非常适合在DMRG中出现的大MPO的远程或准二维模型。
英文摘要：
We present a new method for compressing matrix product operators (MPOs) which represent sums of local terms, such as Hamiltonians. Just as with area law states, such local operators may be fully specified with a small amount of information per site. Standard matrix product state (MPS) tools are ill-suited to this case, due to extensive Schmidt values that coexist with intensive ones, and Jordan blocks in the transfer matrix. We ameliorate these issues by introducing an "almost Schmidt decomposition" that respects locality. Our method is "$\varepsilon$-close" to the accuracy of MPS-based methods for finite MPOs, and extends seamlessly to the thermodynamic limit, where MPS techniques are inapplicable. In the framework of control theory, our method generalizes Kung's algorithm for model order reduction. Several examples are provided, including an all-MPO version of the operator recursion method (Lanczos algorithm) directly in the thermodynamic limit. All results are accompanied by practical algorithms, well-suited for the large MPOs that arise in DMRG for long-range or quasi-2D models.
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