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Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori
Entropy ( IF 2.1 ) Pub Date : 2021-07-27 , DOI: 10.3390/e23080959
Edson Donizete de Carvalho 1, 2 , Waldir Silva Soares 3 , Eduardo Brandani da Silva 2
Affiliation  

In this work, we show that an n-dimensional sublattice Λ=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ=Rn/Λ, where the group G is isomorphic to the lattice partition Λ/Λ. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0 of each hexagonal sublattice Λ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

中文翻译:

n维平面圆环上晶格划分的拓扑量子码

在这项工作中,我们证明了一个n维亚晶格Λ=Λ一个的Ñ维晶格Λ 诱导一个 G=Zn 平面圆环中的镶嵌 β=电阻n/Λ,其中群G与格子划分同构Λ/Λ. 因此,我们通过这种技术获得了参数的复曲面代码[[22,2,]], [[33,3,]][[64,6,2]] 从格子 Z2, Z3Z4, 分别。特别是,对于n=2, 如果 Λ1 是格子 Z2 或六边形点阵,通过点阵划分,我们得到两种等价的方式来覆盖基本单元格 0 每个六边形亚晶格 Λ 六边形格子 Λ, 使用基本单元格 0 或 Voronoi 细胞 0. 这些分区允许我们用参数呈现新的复曲面代码类别[[32,2,]] 和带参数的颜色代码 [[182,4,4]] 在来自六边形格子家族的扁平圆环中 电阻2.
更新日期:2021-07-27
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