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Quantum Information and Computation

Surface codes and color codes associated with non-orientable surfaces

Debashis Bhowmik, Dipendu Maity, Eduardo Brandani da Silva

Published Sep 1, 2021• 1 citation
Publisher

Abstract

Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

DOI
10.26421/qic21.13-14-4

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Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus ≥1\ge 1≥1 and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus ≥2n+1\geq 2n+1≥2n+1 for n≥1n\geq 1n≥1. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios k/nk/nk/n and d/nd/nd/n better than the codes obtained from orientable surfaces.
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→∞.
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.
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 construct several new quantum Floquet codes on compact, orientable, as well as non-orientable surfaces. In order to obtain such codes, we identify these surfaces with hyperbolic polygons and examine hyperbolic semi-regular tessellations on such surfaces. The method of construction presented here generalizes similar constructions concerning hyperbolic Floquet codes on connected and compact surfaces with genus g≥2g \geq 2g≥2. A performance analysis and an investigation of the asymptotic behavior of these codes are also presented.