Tema Tend ˆ encias em Matem ´ atica Aplicada e Computacional, 21, N. 1 (2020), 43-55 © 2020 Sociedade Brasileira de Matem´ atica Aplicada e Computacional www.scielo.br/tema doi: 10.5540/tema.2020.021.01.0043 Construction of Polygonal Color Codes from Hyperbolic Tesselations W. S. SOARES JR 1* , E. B. SILVA 2 , E. J. VIZENTIM 3 and F. P. B. SOARES 4 Received on December 3, 2018 / Accepted on September 20, 2019 ABSTRACT. 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. 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 in- teraction between the system and the surrounding environment. In theory, the problem may be solved using quantum error-correcting codes. Quantum states can be cleverly encoded so that the harmful effects of decoherence can be resisted. The classical theory of error-correcting codes was stablish by Shannon in 1948 [15]. Shor, in 1995, was the first to show an quantum error-correcting code [16], overcoming the non-cloning 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]. *Corresponding author: W. S. Soares Jr. -– E-mail: waldirjunior@utfpr.edu.br 1 Departamento de Matem´ atica, UTFPR, Universidade Tecnol´ ogica Federal de Paran´ a, Via do Conhecimento, Km 1,85503-390, Pato Branco, PR, Brasil. E-mail: waldirjunior@utfpr.edu.br https://orcid.org/0000-0001-6216-8691 2 Departamento de Matem´ atica, UEM, Universidade Estadual de Maring´ a, Av. Colombo, 5790, 87020-900, Maring´ a, PR, Brasil. E-mail: ebsilva@uem.br https://orcid.org/0000-0003-2687-5174 3 UTFPR, Universidade Tecnol´ ogica Federal de Paran´ a, Via do Conhecimento, Km 1, 85503-390, Pato Branco, PR, Brasil. E-mail: emersonjose11@hotmail.com https://orcid.org/0000-0002-1800-9834 4 Colegiado Multidisciplinar, IFPR, Instituto Federal do Paran´ a, Av. Bento Munhoz da Rocha Neto, PRT 280, 85555-000, Palmas, PR, Brasil. E-mail: franciele.soares@ifpr.edu.br https://orcid.org/0000-0001-7893-8101