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arXiv 1602.08828quant-phcond-mat.str-el
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一维低能本征态的严格RG算法和面积律

Rigorous RG algorithms and area laws for low energy eigenstates in 1D

Itai Arad, Zeph Landau, Umesh Vazirani, Thomas Vidick

备注：
Multiple improvements in presentation; results unchanged. To appear in Comm. Math. Phys
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摘要：

量子多体系统研究的核心挑战之一是在经典计算机上模拟它们的复杂性。Landau等人最近的一个进展给出了一个多项式时间算法来计算带隙一维量子系统的唯一基态的简洁经典描述。尽管取得了这一进展，许多问题仍然没有得到解决，包括是否存在严格有效的算法时，地面空间是退化（和聚（$n$）维），或聚（$n$）的最低能量状态的一维系统，甚至是否承认这些国家的简洁的经典描述或面积定律。在本文中，我们给出了一个新的算法，找到低能量状态的一维系统，严格合理的RG型变换的基础上。在这个过程中，我们解决了上述一些开放的问题，包括多项式时间算法的聚（$n$）退化的地面空间和$n^{O(\log n)}$算法的聚（$n$）的最低能量状态的一维系统（在一个温和的密度条件下）。我们注意到，对于这些类的系统的存在一个简洁的经典描述和面积法没有严格证明这项工作之前。该算法是自然和有效的，并且对于寻找无阻挫哈密顿量的唯一基态的情况，运行时间是$\tilde{O}(nM(n))$，其中$M(n)$是乘以两个$n\times n$矩阵所需的时间。

英文摘要：

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance of Landau et al. gave a polynomial time algorithm to actually compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly($n$) dimensional), or for the poly($n$) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified RG type transformation. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly($n$) degenerate ground spaces and an $n^{O(\log n)}$ algorithm for the poly($n$) lowest energy states for 1D systems (under a mild density condition). We note that for these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is $\tilde{O}(nM(n))$, where $M(n)$ is the time required to multiply two $n\times n$ matrices.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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