Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs
Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs
复制标题
非 k 超有限复形上的量子系统：扩展图上经典统计力学的推广
DOI：
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发表时间：
2013
影响因子：
1
通讯作者：
M. Hastings
中科院分区：
文献类型：
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作者：
M. Freedman;M. Hastings
我们构造了推广扩展图的胞复数族。这些图族称为非k-超限图族，推广了非超限(NH)图族的概念。粗略地说，这样的复合体具有这样的性质，即人们不能移除一小部分点，而只留下一个在大尺度上看起来是k-1维的对象。然后我们考虑这些络合物上的某些量子系统。作为证明量子PCP猜想的尝试的一部分，未来的目标是构建一族哈密顿量，使得每个低能态都有拓扑序。这一目标是通过构造一个环码哈密顿量来实现的，该环码哈密顿量具有这样的性质，即每个没有顶点缺陷的低能态都有拓扑序，这一性质不适用于任何格子ZD中的任何局部系统，甚至不适用于任何1-超有限复数。此外，这种NH络合物在量子编码理论中也有应用。利用NH复形推广了Tillich和Zemor的超图乘积码[1]。
We construct families of cell complexes that generalize expander graphs. These families are called non-k-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the property that one cannot remove a small fraction of points and be left with an object that looks k - 1-dimensional at large scales. We then consider certain quantum systems on these complexes. A future goal is to construct a family of Hamiltonians such that every low energy state has topological order as part of an attempt to prove the quantum PCP conjecture. This goal is approached by constructing a toric code Hamiltonian with the property that every low energy state without vertex defects has topological order, a property that would not hold for any local system in any lattice Zd or indeed on any 1-hyperfinite complex. Further, such NH complexes find application in quantum coding theory. The hypergraph product codes[1] of Tillich and Zemor are generalized using NH complexes.
关于 $p$-adic 分析塔中 Betti 数的增长
DOI：
10.4171/ggd/227
发表时间：
2014
期刊：
arXiv: Geometric Topology
影响因子：
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作者：
N. Bergeron;P. Linnell;W. Lück;R. Sauer
通讯作者：
R. Sauer