Hyperbolic quantum color codes with normal subgroup structure derived from the Reidemeister–Schreier method

Abstract

Given the importance of hyperbolic quantum color codes and Euclidean quantum color codes, this paper considers the study of the former codes on compact surfaces with genus g≥2 from the mathematical point of view. Identifying the normal subgroup in the decomposition of the full symmetry group of the {p,3} tessellation is relevant because it provides the algebraic structure for identifying and constructing a class of linear shrunk hyperbolic quantum color codes. Under this assumption, the normal subgroup’s presentation, the whole process’s kernel, is derived from the Reidemeister–Schreier method. As a result, we present a class of regular normal hyperbolic quantum color codes derived from the {6j,3} tessellation with encoding rate going asymptotically to 1. The regular tessellation {6j,3} includes the two types of tessellations: (1) the densest tessellation {12i-6,3} when j=2i-1 and (2) the tessellation {12i,3} when j=2i, for i∈N. An analysis of the minimum distance achieved by this class of regular normal hyperbolic quantum color codes is performed.