Euclidean and Hyperbolic Asymmetric Topological Quantum Codes

In the last three decades, several constructions of quantum error-correcting codes were presented in the literature. Among these codes, there are the asymmetric ones, i.e., quantum codes whose $Z$-distance $d_z$ is different from its $X$-distance $d_x$. The topological quantum codes form an important class of quantum codes, where the toric code, introduced by Kitaev, was the first family of this type. After Kitaev's toric code, several authors focused attention on investigating its structure and the constructions of new families of topological quantum codes over Euclidean and hyperbolic surfaces. As a consequence of establishing the existence and the construction of asymmetric topological quantum codes in Theorem \ref{main}, the main result of this paper, we introduce the class of hyperbolic asymmetric codes. Hence, families of Euclidean and hyperbolic asymmetric topological quantum codes are presented. An analysis regarding the asymptotic behavior of their distances $d_x$ and $d_z$ and encoding rates $k/n$ versus the compact orientable surface's genus is provided due to the significant difference between the asymmetric distances $d_x$ and $d_z$ when compared with the corresponding parameters of topological codes generated by other tessellations. This inherent unequal error-protection is associated with the nontrivial homological cycle of the $\{p,q\}$ tessellation and its dual, which may be appropriately explored depending on the application, where $p\neq q$ and $(p-2)(q-2)\ge 4$. Three families of codes derived from the $\{7,3\}$, $\{5,4\}$, and $\{10,5\}$ tessellations are highlighted.