Construction of Polygonal Color Codes from Hyperbolic Tesselations

This current work propose a technique to generate polygonal color codes in the hyperbolic geometry environment. The color codes were introduced by Bombin and Martin-Delgado in 2007, and the called triangular color codes have a higher degree of interest because they allow the implementation of the Clifford group, but they encode only one qubit. In 2018 Soares e Silva extended the triangular codes to the polygonal codes, which encode more qubits. Using an approach through hyperbolic tessellations we show that it is possible to generate Hyperbolic Polygonal codes, which encode more than one qubit with the capacity to implement the entire Clifford group and also having a better coding rate than the previously mentioned codes, for the color codes on surfaces with boundary with minimum distance $d = 3$ .

Keywords:
color codes; topological quantum codes; hyperbolic geometry

Authorship

W.S. SOARES JR^**Corresponding author: W. S. Soares Jr. -- E-mail: waldirjunior@utfpr.edu.br

Departamento de Matemática, UTFPR, Universidade Tecnológica Federal de Paraná, Via do Conhecimento, Km 1,85503-390, Pato Branco, PR, Brasil. E-mail: waldirjunior@utfpr.edu.brUniversidade Tecnológica Federal de ParanáBrasilPato Branco, PR, BrasilDepartamento de Matemática, UTFPR, Universidade Tecnológica Federal de Paraná, Via do Conhecimento, Km 1,85503-390, Pato Branco, PR, Brasil. E-mail: waldirjunior@utfpr.edu.br

http://orcid.org/0000-0001-6216-8691

E.B. SILVA

Departamento de Matemática, UEM, Universidade Estadual de Maringá, Av. Colombo, 5790, 87020-900, Maringá, PR, Brasil. E-mail: ebsilva@uem.brUniversidade Estadual de MaringáBrasilMaringá, PR, BrasilDepartamento de Matemática, UEM, Universidade Estadual de Maringá, Av. Colombo, 5790, 87020-900, Maringá, PR, Brasil. E-mail: ebsilva@uem.br

http://orcid.org/0000-0003-2687-5174

E.J. VIZENTIM

UTFPR, Universidade Tecnológica Federal de Paraná, Via do Conhecimento, Km 1, 85503-390, Pato Branco, PR, Brasil. E-mail: emersonjose11@hotmail.comUniversidade Tecnológica Federal de ParanáBrasilPato Branco, PR, BrasilUTFPR, Universidade Tecnológica Federal de Paraná, Via do Conhecimento, Km 1, 85503-390, Pato Branco, PR, Brasil. E-mail: emersonjose11@hotmail.com

http://orcid.org/0000-0002-1800-9834

F.P.B. SOARES

Colegiado Multidisciplinar, IFPR, Instituto Federal do Paraná, Av. Bento Munhoz da Rocha Neto, PRT 280, 85555-000, Palmas, PR, Brasil. E-mail: franciele.soares@ifpr.edu.brInstituto Federal do ParanáBrasilPalmas, PR, BrasilColegiado Multidisciplinar, IFPR, Instituto Federal do Paraná, Av. Bento Munhoz da Rocha Neto, PRT 280, 85555-000, Palmas, PR, Brasil. E-mail: franciele.soares@ifpr.edu.br

http://orcid.org/0000-0001-7893-8101

*Corresponding author: W. S. Soares Jr. -- E-mail: waldirjunior@utfpr.edu.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (6)
Tables (1)
Formulas (16)

Thumbnail

Figure 1:
Support of face (f) e vertices (v) operators of Kitaev’s Toric Code.

Thumbnail

Figure 2:
An example of a t-string (or string-net). Note that a string of a certain color branches into two parts, of two other colors. The two parts combine and return to a string with original color. This string-net is equivalent a whole original color string and to see this is just to do the product of this t-string with the face operators generated by the marked faces ¹⁸18. W.S. Soares Jr & E.B. Silva. Hyperbolic Quantum Color Codes. Quantum Information and Computation, 18(3 and 4) (2018), 308-320..

Thumbnail

Figure 3:
A triangular code encodes only one qubit, a t-string with three endpoints, each one on one edge of its respective color.

Thumbnail

Figure 4:
Homologically non-trivial strings in a polygon with 5 sides.

Thumbnail

Figure 5:
Code

P (1, 2, 1)

with parameters

[[12, 2, 3]]

generated by tessellation

\{4, 8, 8\}

in a quadrilateral. Highlighted we have a t-string with the minimum distance of the code, with 3 qubits of its support.

Thumbnail

Figure 6:
Code

P (1, 2, 2)

with parameters

[[15, 3, 3]]

generated by tessellation

\{8, 3\}

in a pentagon

\{5, 15\}

. Highlighted we have a t-string with the minimum distance of the code, with the 3 qubits of its support.

Thumbnail

Table 1:
Comparison between the color codes with borders with minimum distance

d = 3

that implement the Clifford group.

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(3.1)

Figure 1: Support of face (f) e vertices (v) operators of Kitaev’s Toric Code.

Figure 2: An example of a t-string (or string-net). Note that a string of a certain color branches into two parts, of two other colors. The two parts combine and return to a string with original color. This string-net is equivalent a whole original color string and to see this is just to do the product of this t-string with the face operators generated by the marked faces ¹⁸18. W.S. Soares Jr & E.B. Silva. Hyperbolic Quantum Color Codes. Quantum Information and Computation, 18(3 and 4) (2018), 308-320..

Figure 3: A triangular code encodes only one qubit, a t-string with three endpoints, each one on one edge of its respective color.

Figure 4: Homologically non-trivial strings in a polygon with 5 sides.

Figure 5: Code $P (1, 2, 1)$ with parameters $[[12, 2, 3]]$ generated by tessellation $\{4, 8, 8\}$ in a quadrilateral. Highlighted we have a t-string with the minimum distance of the code, with 3 qubits of its support.

Figure 6: Code $P (1, 2, 2)$ with parameters $[[15, 3, 3]]$ generated by tessellation $\{8, 3\}$ in a pentagon $\{5, 15\}$ . Highlighted we have a t-string with the minimum distance of the code, with the 3 qubits of its support.

Table 1: Comparison between the color codes with borders with minimum distance $d = 3$ that implement the Clifford group.

Code	$[[n, k, d]]$	k/n
Triangular	$[[7, 1, 3]]$	1/7
Polygonal	$[[12, 2, 3]]$	1/6
Hyperbolic Polygonal	$[[15, 3, 3]]$	1/5

(2.1)

d s = \frac{\sqrt{d x^{2} + d y^{2}}}{y},

(2.2)

f (z) = \frac{z i + 1}{z + i},

(2.3)

d^{*} (z, w) = d (f^{- 1} (z), f^{- 1} (w)); (z, w \in 𝔻^{2})

(2.4)

μ (R) = \int \int_{R} \frac{d x d y}{y^{2}}

\frac{2}{p} + \frac{2}{q} = π .

p q - 2 p - 2 q = 0 \Rightarrow (p - 2) (q - 2) = 4 .

\frac{2 π}{p} + \frac{2 π}{q} < π .

(2.5)

p q - 2 p - 2 q > 0 \Rightarrow (p - 2) (q - 2) > 4

𝒞 = {| ψ 〉; mml:m | ψ 〉 = | ψ 〉, \forall mml:m \in S} .

𝒞 = {| ψ 〉; X_{v} | ψ 〉 = | ψ 〉; Z_{f} | ψ 〉 = | ψ 〉; \forall v, f} .

\prod_{p \in R} B_{p}^{σ} = \prod_{p \in G} B_{p}^{σ} = \prod_{p \in B} B_{p}^{σ}

(3.1)

S_{μ}^{R σ} S_{μ}^{G σ} S_{μ}^{B σ} ~ 1

ℙ_{l + 1}^{n} = \{U | U P U^{†} P^{†} \in ℙ_{l}^{n}, \forall P \in ℙ_{1}^{n}\},

H = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}), K = (\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}), Λ (X) = (\begin{matrix} I_{2} & 0 \\ 0 & X \end{matrix})

H \cdot X_{f} \cdot \bar{H} = Z_{f}, H \cdot Z_{f} \cdot H = X_{f}, H \cdot X \cdot H = Z e H \cdot Z \cdot H = X .

K \cdot Z_{f} \cdot K = Z_{f} a n d K \cdot X_{f} \cdot K = {(- 1)}^{t / 2} X_{f} \cdot Z_{f},

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Construction of Polygonal Color Codes from Hyperbolic Tesselations

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[1] *Corresponding author: W. S. Soares Jr. -- E-mail: waldirjunior@utfpr.edu.br