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Journal of Mathematical Physics

On geometrically uniform codes and topological quantum MDS codes

Vandenberg Lopes Vieira, S. O. Juriaans

Published Sep 1, 2022• 1 citation
Publisher

Abstract

In this paper, we present generalized edge-pairings for the family of hyperbolic tessellations {4λ, 4} with the purpose of obtaining the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations {4λ, 4λ}, implying that the associated codes achieve the least error probability or, equivalently, that these codes are optimum codes. We also consider the topological quantum error-correcting codes on surfaces with genus g ≥ 2, as proposed in other papers. From that, we focus on the class of topological quantum maximum distance separable codes, showing that the only such codes are precisely the ones with minimum distance d ≤ 2.

DOI
10.1063/5.0052815

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Abstract We consider the minimum distance of quantum color codes associated to trivalent face-3-colorable cellular tessellations of closed orientable surfaces of genus g ⩾ 2 . One important point is that a short cycle in a shrunken lattice does not necessarily correspond to a nontrivial logical operator, as it may belong to the boundary space. We prove directly that the face vertex incidence matrix satisfies rank ⁡ H = n f − 2 , recovering the standard value k = 4 g . We derive an algebraic criterion that determines the exact minimum distance d = 2 . For cases not covered by this criterion, we propose a short-cycle dimension method which compares the space generated by short cycles with the corresponding boundary space and gives rigorous upper bounds on the minimum distance. The method extends to regular and semi-regular families. The criterion of existence leads to exactly 38 admissible pairs ( g , p ) for regular hyperbolic tessellations of type { p , 3 } with 2 ⩽ g ⩽ 10 . In twenty cases the exact value d = 2 is attained, while for the remaining eighteen cases rigorous upper bounds of 4 , 6 or 8 are obtained.
Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4 and the encoding rate asymptotically going to 1 while n\rightarrow\infty.
Current work presents a new approach to quantum color codes on compact surfaces with genus g≥2g \geq 2g≥2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4d=4d=4 and the encoding rate asymptotically going to 1 while n→∞n \rightarrow \inftyn→∞.
In this paper, we construct new families of asymmetric quantum surface codes (AQSCs) over non-orientable surfaces of genus g ≥ 2 by applying tools of hyperbolic geometry. More precisely, we prove that all asymmetric quantum surface codes constructed over orientable surfaces with genus h can be reproduced, with exactly the same parameters, over a non-orientable surface with genus g = 2h. Additionally, if S is a non-orientable surface of genus g, we show that the new surface code constructed on a {p, q} tessellation of S has ratio k/n greater than the ratio of an AQSC constructed on the same {p, q} tessellation over an orientable surface of the same genus g.
In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.