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Quantum Information Processing

New quantum surface codes from semi-regular tessellations

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

Published Nov 3, 2023• 3 citations
Publisher
DOI
10.1007/s11128-023-04147-2

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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.
We utilize the symmetry groups of regular tessellations on two-dimensional surfaces of different constant curvatures, including spheres, Euclidean planes, and hyperbolic planes to encode a qubit or qudit into the physical degrees of freedom on these surfaces, which we call tessellation codes. We show that tessellation codes exhibit decent error correction properties by analysis via geometric considerations and the representation theory of the isometry groups on the corresponding surfaces. Interestingly, we demonstrate how this formalism enables the implementation of certain logical operations through geometric rotations of surfaces in real space, opening a new approach to logical quantum computation. We provide a variety of concrete constructions of such codes associated with different tessellations, which give rise to different logical groups. This formalism sheds a new light on quantum code and logical operation construction.
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.