Grover在单纯复形上的上下行走
Up and down grover walks on simplicial complexes
摘要：
提出了单纯复形上上下Grover游动的概念，并研究了它们的性质.这些是抽象的Szegedy游动，它是希尔伯特空间上一种特殊的酉算子。本文所引入的算子是利用单纯复形的组合结构定义的图上的Grover游动。但对移位算子进行了修改，使其包含了单纯复形中每个单形的方向信息。众所周知，这类酉算子的谱结构完全由其判别算子决定。它与单纯复形上的组合拉普拉斯算子以及单纯复形的几何甚至拓扑有着密切的联系。特别地，给出了上下判别式的谱与可定向性的关系，判别式的谱的对称性与单纯复形的组合结构的关系等定理.文中还给出了有限和无限单纯复形的一些例子。最后，一些方面的发现概率和平稳措施进行了讨论。
英文摘要：
A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatorial structures of simplicial complexes. But the shift operators are modified so that it can contain information of orientations of each simplex in the simplicial complex. It is well-known that the spectral structures of this kind of unitary operators are completely determined by its discriminant operators. It has strong relationship with combinatorial Laplacian on simplicial complexes and geometry, even topology, of simplicial complexes. In particular, theorems on a relation between spectrum of up and down discriminants and orientability, on a relation between symmetry of spectrum of discriminants and combinatorial structure of simplicial complex are given. Some examples, both of finite and infinite simplicial complexes, are also given. Finally, some aspects of finding probability and stationary measures are discussed.
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