Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

Entropy

Eduardo Brandani da Silva¹  Edson Donizete de Carvalho²  Waldir Silva Soares³ 

[1] Departament of Mathematics, UEM, Av. Colombo 5790, Maringa, PR 87020-900, Brazil;Department of Mathematics, UNESP, Ilha Solteira, SP 15385-000, Brazil;Department of Mathematics, UTFPR, Pato Branco, PR 85503-390, Brazil;

关键词: color codes;    surface codes;    toric codes;    flat torus;    lattice partition;   

DOI  :  10.3390/e23080959

来源: DOAJ

【 摘 要 】

In this work, we show that an n-dimensional sublattice ${\Lambda }^{\prime }=m\Lambda$ of an n-dimensional lattice $\Lambda$ induces a $G={\mathbb{Z}}_m^n$ tessellation in the flat torus $T_{{\beta }^{\prime }}={\mathbb{R}}^n/{\Lambda }^{\prime }$, where the group G is isomorphic to the lattice partition $\Lambda /{\Lambda }^{\prime }$. As a consequence, we obtain, via this technique, toric codes of parameters $\left[[2m^2,2,m]\right]$, $\left[[3m^3,3,m]\right]$ and $\left[[6m^4,6,m^2]\right]$ from the lattices ${\mathbb{Z}}^2$, ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^4$, respectively. In particular, for $n=2$, if ${\Lambda }_1$ is either the lattice ${\mathbb{Z}}^2$ or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell ${\mathcal{P}}_0^{\prime }$ of each hexagonal sublattice ${\Lambda }^{\prime }$ of hexagonal lattices $\Lambda$, using either the fundamental cell ${\mathcal{P}}_0$ or the Voronoi cell ${\mathcal{V}}_0$. These partitions allow us to present new classes of toric codes with parameters $\left[[3m^2,2,m]\right]$ and color codes with parameters $\left[[18m^2,4,4m]\right]$ in the flat torus from families of hexagonal lattices in ${\mathbb{R}}^2$.

【 授权许可】

Unknown