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- Title
Hyperbolic quantum color codes with normal subgroup structure derived from the Reidemeister–Schreier method.
- Authors
Albuquerque, Clarice Dias; Lazari, Henrique; Palazzo Jr., Reginaldo; Campos, Daniel Silva
- 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 { 6 j , 3 } tessellation with encoding rate going asymptotically to 1. The regular tessellation { 6 j , 3 } includes the two types of tessellations: (1) the densest tessellation { 12 i - 6 , 3 } when j = 2 i - 1 and (2) the tessellation { 12 i , 3 } when j = 2 i , for i ∈ N . An analysis of the minimum distance achieved by this class of regular normal hyperbolic quantum color codes is performed.
- Subjects
COLOR codes; HYPERBOLIC groups; QUANTUM groups; SYMMETRY groups; HYPERBOLIC geometry
- Publication
Computational & Applied Mathematics, 2024, Vol 43, Issue 4, p1
- ISSN
0101-8205
- Publication type
Article
- DOI
10.1007/s40314-024-02710-w