Construction of color codes from polygons

Abstract

We propose a procedure for constructing new classes of codes, called polygonal color codes. This technique expands the triangular color codes introduced by Bombin and Martin-Delgado in 2007. Triangular color codes, built on a two-dimensional Euclidean surface with three edges, implement the entire Clifford group, but they only encode one qubit. In our case, polygonal color codes are constructed on two-dimensional surfaces with n edges, with n3 , increasing the number of encoded qubits. Considering a surface with boundary and evaluating the generators of the first homology group of this surface, we may increase the dimension of the code without losing the transversal implementation of the Clifford group. In addition, we construct two codes, one with 2 and one with 3 encoded qubits, which exemplify the method and show that we can generate codes with better parameters than the known triangular codes. For one of these constructions, we also give general parameters. OPEN ACCESS RECEIVED

Non-CSS color codes on 2D lattices : Models and Topological Properties

“…The color code has seen several interesting generalizations [10][11][12][13][14][15][16][17][18][19][20] and has also been experimentally realized [21][22][23][24][25][26][27]. Apart from direct generalizations several theoretical works have explored the physics and error correction properties of the color code and related models [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].…”