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Designs Codes and Cryptography

Surface and color codes from semi-regular tessellations on non-orientable surfaces

Eduardo Brandani da Silva, Douglas Fernando Copatti, Waldir Silva Soares, Evandro Mazetto Brizola

Published Sep 15, 2025• 0 citations
Publisher
DOI
10.1007/s10623-025-01732-7

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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.

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Construction of binary quantum codes on closed orientable surfaces

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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.
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 $ \geq 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.
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.