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Distance bounds for quantum color codes on compact hyperbolic surfaces

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

Published Sep 10, 2026• 0 citations

Abstract

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.

DOI: 10.1088/1402-4896/aea5f8

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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 \geq 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=4

and the encoding rate asymptotically going to 1 while

n \rightarrow \infty

.

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 \geq 2

. A performance analysis and an investigation of the asymptotic behavior of these codes are also presented.

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.

Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension

D\geq 4

. In even dimensions, the resulting type-

D/2

color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.