1 INTRODUCTION
One of the great difficulties of performing quantum computing is decoherence, as Unruh warned in 1995 ^{20}. Decoherence is the decay phenomenon of superposition of states, due to the interaction between the system and the surrounding environment. In theory, the problem may be solved using quantum errorcorrecting codes. Quantum states can be cleverly encoded so that the harmful effects of decoherence can be resisted.
The classical theory of errorcorrecting codes was stablish by Shannon in 1948 ^{15}. Shor, in 1995, was the first to show an quantum errorcorrecting code ^{16}, overcoming the noncloning theorem and achieving an analogue to the classic repeating code. Shor’s code belongs to a class of codes known as CSS codes, which was introduced by Calderbank and Shor ^{6} and Steane ^{19}.
CSS codes, in turn, are a subclass of an even broader and richer class of codes, as shown by Gottesman ^{9}, known as stabilizer codes. One of the most important stabilizer code is Kitav toric code, introduced in ^{12}. These codes were expanded and generalized to the topological quantum codes. Important examples were obtained considering other types of tessellations of the torus ^{2}, ^{7} or considering surfaces with a higher genus, ^{1}.
Color codes were introduced in ^{4} by Bombin and MartinDelgado. These codes are also topological quantum codes, and they are also generated by surface tessellations. The color codes have a great advantage over the Kitaev codes, which is the greater number of operations that can be performed on coded qubits. In the particular case of triangular codes, Bombin and MartinDelgado proved that for the triangular codes, it is possible to implement the entire Clifford group transversally. The color codes have also been expanded, for example in the work of Soares and Silva ^{18}, where the authors consider an approach of the color codes on compact surfaces of genus greater or equal to 2, using tools of the hyperbolic geometry, obtaining codes with parameters better than those of Kitaev ^{12}, Bombin and MartinDelgado ^{2}, Albuquerque, Palazzo and Silva ^{1}, among others. In ^{17} a proposal was made to increase the number of coded qubits of the triangular codes, called polygonal codes, in order to improve the coding rate of the triangular codes, without losing the implementation capacity of the Clifford group.
Our objective in this work is to adapt polygonal codes to the environment of hyperbolic geometry because, in addition to have infinite possibilities of tessellations (the Euclidean plane has only one tessellation that satisfies the necessary conditions for the polygonal codes, as we will see in section 3), some works that used hyperbolic geometry in the generation of topological codes have achieved an improvement in the parameters, as we can see in ^{5}, ^{8}, besides those already cited ^{1} and ^{18}. We evaluated the feasibility of this type of construction, taking into account the rigidity of the hyperbolic geometry in relation to the area of the polygons and the impossibility of decreasing the length of the side of a regular hyperbolic polygon without changing its internal angles, and we show that not only is it feasible as this technique can generate codes with even better parameters than those obtained until now by codes of the same nature.
2 HYPERBOLIC GEOMETRY
Hyperbolic geometry, as well as several other nonEuclidean geometries, have arisen in response to the negation of Euclide’s fifth postulate.
Here, two models of hyperbolic geometry is cosidered: the upper halfplane,
is known as hyperbolic plane or Lobachevsky plane and this metric is known as hyperbolic metric. This application is a metric and the proof of this fact can be seen in ^{11}.
Considering the map
can be proved that f is onetoone, and that, d*, giving by
is a metric on 𝔻^{2} and has expression
If
if the integral exists.
However, to determine the area of a hyperbolic triangle the GaussBonnet Theorem can be used, which says the hyperbolic area of a hyperbolic triangle depends only on its angles. ^{11}
Theorem 2.1 (GaussBonnet).
Let ∆ be a hyperbolic triangle with inner angles α, β, γ. Then, the area of ∆ is given by
Differently from Euclidean geometry, in the hyperbolic plane two similar triangles are also congruent. This implies that, given a regular hyperbolic polygon, it is impossible to vary the length of its sides by keeping fixed the measurements of its internal angles.
2.1 Tesselations
A regular tessellation of the Euclidean or hyperbolic plane is a cover of the whole plane by regular polygons, all with the same number of sides, without superpositions of such polygons, meeting only along complete edges or at vertices. A regular tessellation is denoted by
Giving a regular tessellation
It follows that,
For this equation there are three integer solutions, that is, there are three regular tessellations in the Euclidean plane. Namely, the tessellations formed by squares
Then, it follows,
There are infinite solutions to this inequality. Therefore, there are infinite regular tessellations in the hyperbolic plane. Moreover, there are infinite tessellations of the hyperbolic plane even when we fix the number of sides of the polygon, or if we fix the amount of polygons that must meet in each vertex of the tessellation. Since we interested in colored codes, the tesselations that matter to us must be trivalent and 3colorable, which means that
3 COLOR CODES
A quantum errorcorrecting code (QECC) is an application of a complex Hilbert space ℋ^{
k
} , of dimension 2^{
k
} , to a Hilbert space of dimension 2^{
n
} where
A general class of codes, that even includes the CSSlike codes, are the socalled Stabilizer Codes ^{13}. Stabilizer codes are the quantum analog for the classical additive codes.
To define these codes, consider the set given by the Pauli matrices of a qubit
Thus, the stabilizer code 𝒞 is defined by the eigenspace associated with the eigenvalue 1 of the operators of S, that is: ^{9}
In ^{12}, Kitaev proposed a particular case of stabilizer code, which became known as Kitaev’s Toric code. In a torus
In this way, Kitaev’s Toric code is defined by:
This code has parameters
The color codes, introduced by Bombim and MartinDelgado in ^{4}, similar to the Kitaev code, is a topological code. To create this codes, the authors considered a twodimensional surface, with or without borders, and tessellated it with a tesselation satisfying 2 conditions:
it must be trivalent, that is, three edges meet at each vertex
have 3colorable faces, wich means it is possible to color all the faces of the tessellation using only 3 colors (red, green and blue, for instance), such that two faces sharing an edge do not have the same color.
Considering a coloration of faces, it also induces a coloration of the edges, so that an edge of a certain color does not belong to the border of a face with the same color.
Once the surface is properly tessellated, the authors now associated the qubits onetoone with the vertices of the tessellation, diferently of Kitaev’s Toric code where the qubits were associated to edges.
The generators of the stabilizers are the face operators, known by “plaquette” operators, with both X and Z operators on each face, acting on the vertices of the face in question. For each face p of the tessellation, such operators are denoted by
Separating the faces according to their color, in the sets R (red), G (green) and B (blue), it follows:
with
For every colored tesselation the shrunk lattices can be defined. There exists three of them, one for each color, which are auxiliary lattices. For instance in the red shrunk lattice considers as vertex each red face. The edges of this new lattice conect the red faces (wich are vertices now) and note each new edge contain two vertices of the original lattice, which means that each edge corresponds to two qubits. Still, the green and blue faces of the colored tessellation are the faces of that new auxiliary lattice.
The string operators acting on color codes may be green, blue or red, depending on which net is being considered. Regardless of the color, they can be of type X or Z. These string operators are denoted by
In general, one has ^{4}:
Equation (3.1) shows that there are only two independent colors.
A fundamental property of color codes is that, in addition to a string operator of one color being able to be deformed to another homologous string operator, it is also possible to combine two strings of different colors to produce a third equivalent operator of different color from the two original ones.
3.1 Triangular and Polygonal Codes
The most interesting particular case of the color codes is obtained when tessellations on surfaces with boundary are considered. This construction was firstly considered in ^{4}. A color code was created on a planar region bounded by a triangle in which each edge of the triangle is in the border, and each edge has one color. This code encodes only one qubit, and its nontrivial homology string is a tstring, that is, a string that departs from one edge of a certain color, branches into two other colors, and each part of its branching ends in the border which has its color.
The tstrings in the triangular codes are the key to implement the Clifford group. The Clifford group is the second level of the socalled Clifford hierarchy (CH), introduced by Gottesman and Chuang ^{10} for quantum operations, and defined by
where U is an unitary operator acting on n qubits. In this definition of CH, the first level of the hierarchy is
Although the implementation of Clifford group does not imply the universality of quantum computation (which we would have at the third level of CH), there are important tasks performed by the operations this group, such as quantum distillation, quantum teleportation and dense coding, for instance.
The difference in relation to the surface codes is that, geometrically, the tstrings X and the tstrings Z are exactly the same. Thus, these two such tstrings are denoted by T
^{
X
} and T
^{
Z
} , and consequently
Thus, given the encoded operators
Further,
where t is the number of vertices of the face. Thus, using a semiregular tessellation where all faces have a number of vertices, multiple of 4, such as
In 2018, Soares and Silva introduced the so called Polygonal Codes, whose rules are similar to those of triangular codes, but with a polygon as the border. Then, with more edges it was obtained more encoded qubits. The main property of the polygonal codes is to increase the number of encoded qubits, but keeping the implementation of the Clifford group, wich means that all homology group should be generated by tstrings.
In the Figure 4 an example of the structure of a polygonal code with 5 edges is showed. Here all strings with nontrivial homology (except for homeomorphisms) are explicit. It must be observed that strings 1, 2, and 3 (as labeled in the figure) generate all other strings, which means that such code encodes 3 qubits, and in addition, since these strings are tstrings, this ensures that the polygonal codes of Soares and Silva are in the same conditions as the triangular codes of Bombin and MartinDelgado, which means that these codes also have the capacity of implementation of the whole group of Clifford. What has been exemplified for the particular case of a 5sided polygon has been shown in the general case in ^{17}, that is, if a polygon has n sides, then the polygonal code generated by it encodes
The smallest triangular code presented by Bombin and MartinDelgado has parameters
In the works ^{1}^{), (}^{17} and ^{8}, using properties of the hyperbolic geometry in connection with topological codes, it was achieved good results by improving the parameters of the codes that preceded them. Therefore, it is natural to investigate whether it is possible to use hyperbolic tesselations to try to reproduce triangular and polygonal codes in the hyperbolic geometry environment, improving its parameters and maintaining its ability to implement the Clifford group.
Code 

k/n 
Triangular 

1/7 
Polygonal 

1/6 
Hyperbolic Polygonal 

1/5 
The main difficulty in adapting triangular codes to the hyperbolic geometry environment is the area. Given a hyperbolic triangle ABC, as already quoted in the section 2, its area S is always less than π. Thus, when considering a tessellation
Applying the hyperbolic tessellation techniques by ruler and compass described in ^{14}, the Poincaré disc can be tesselated with any tessellation
With the polygon and tessellation presented we were able to generate a code with the best parameters of the category, in the direction of encoding rate, in comparison with others in the literature that have the same minimum distance. Table 1 presents some topological codes with minimum distance