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

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 959
Author(s):  
Edson Donizete de Carvalho ◽  
Waldir Silva Soares ◽  
Eduardo Brandani da Silva
Keyword(s):  
Hexagonal Lattice ◽  
Voronoi Cell ◽  
Quantum Codes ◽  
Flat Torus ◽  
Dimensional Lattice ◽  
Topological Quantum ◽  
Flat Tori ◽  
Toric Codes

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.

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices

1992 ◽  
Vol 436 (1896) ◽  
pp. 55-68 ◽  
Keyword(s):  
Simple Formula ◽  
Three Dimensional ◽  
Voronoi Cell ◽  
Dimensional Lattice ◽  
Usual Treatment ◽  
Low Dimensional

The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n ≼ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi’s theorem that every lattice of dimension n ≼ 3 is of the first kind, and of Fedorov’s classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

THE EUCLIDEAN DISTORTION OF FLAT TORI

2013 ◽  
Vol 05 (02) ◽  
pp. 205-223 ◽  
Author(s):  
ISHAY HAVIV ◽  
ODED REGEV
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Upper Bound ◽  
Gaussian Function ◽  
Smoothing Parameter ◽  
Gaussian Measures ◽  
Dimensional Lattice ◽  
Flat Tori ◽  
Euclidean Distortion

We show that for every n-dimensional lattice [Formula: see text] the torus [Formula: see text] can be embedded with distortion [Formula: see text] into a Hilbert space. This improves the exponential upper bound of O(n3n/2) due to Khot and Naor (FOCS 2005, Math. Ann. 2006) and gets close to their lower bound of [Formula: see text]. We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point [Formula: see text] to a Gaussian function centered at u in the Hilbert space [Formula: see text]. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine–Zolotarev bases.

scholarly journals Topological quantum codes from self-complementary self-dual graphs

2015 ◽  
Vol 14 (11) ◽  
pp. 4057-4066 ◽  
Author(s):  
Avaz Naghipour ◽  
Mohammad Ali Jafarizadeh ◽  
Sedaghat Shahmorad
Keyword(s):  
Quantum Codes ◽  
Dual Graphs ◽  
Topological Quantum

Families of codes of topological quantum codes from tessellations tessellations {4i+2,2i+1}, {4i,4i}, {8i-4,4} and ${12i-6,3}

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1424-1440
Author(s):  
Clarice Dias de Albuquerque ◽  
Reginaldo Palazzo Jr. ◽  
Eduardo Brandani da Silva
Keyword(s):  
Specific Property ◽  
Bipartite Graphs ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Topological Quantum ◽  
Complete Bipartite Graphs

In this paper we present some classes of topological quantum codes on surfaces with genus $g \geq 2$ derived from hyperbolic tessellations with a specific property. We find classes of codes with distance $d = 3$ and encoding rates asymptotically going to 1, $\frac{1}{2}$ and $\frac{1}{3}$, depending on the considered tessellation. Furthermore, these codes are associated with embedding of complete bipartite graphs. We also analyze the parameters of these codes, mainly its distance, in addition to show a class of codes with distance 4. We also present a class of codes achieving the quantum Singleton bound, possibly the only one existing under this construction.

New classes of TQC associated with self-dual, quasi self-dual and denser tessellations

2010 ◽  
Vol 10 (11&12) ◽  
pp. 956-970
Author(s):  
C. D. Albuquerque ◽  
R. Palazzo Jr. ◽  
E. B. Silva
Keyword(s):  
Minimum Distance ◽  
Bipartite Graphs ◽  
The Self ◽  
Quantum Codes ◽  
Complete Graphs ◽  
Mathematical Structures ◽  
Topological Quantum ◽  
Complete Bipartite Graphs ◽  
Compact Surfaces ◽  
The One

In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

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