Abstract
In this work, we show that an n-dimensional sublattice of an n-dimensional lattice induces a tessellation in the flat torus , where the group G is isomorphic to the lattice partition . As a consequence, we obtain, via this technique, toric codes of parameters , and from the lattices , and , respectively. In particular, for , if is either the lattice or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell of each hexagonal sublattice of hexagonal lattices , using either the fundamental cell or the Voronoi cell . These partitions allow us to present new classes of toric codes with parameters and color codes with parameters in the flat torus from families of hexagonal lattices in .
Highlights
The concepts and ideas of classical error-correcting codes theories were an inspiration and model to build quantum error-correcting codes
Kitaev [4] proposed a particular class of stabilizer codes, which is associated with a Z2 lattice. These codes depend on the topology of a surface, and they belong to the general class of topological quantum codes (TQC), which, in turn, belong to the class of stabilizer quantum codes
In the toric codes obtained from three-dimensional flat torus, the stabilizer operators X and Z are attached to the cells and edges which tessellate the torus surface, and the encoded qubits are related to the homological non-trivial faces on the torus
Summary
The concepts and ideas of classical error-correcting codes theories were an inspiration and model to build quantum error-correcting codes. In the toric codes obtained from three-dimensional flat torus, the stabilizer operators X and Z are attached to the cells and edges which tessellate the torus surface, and the encoded qubits are related to the homological non-trivial faces on the torus. In a similar way to the toric codes obtained from four-dimensional flat torus, the stabilizer operators X and Z are attached to the cells and edges of the paralellepiped, which tessellates the torus surface, and the encoded qubits are related to the homological non-trivial faces on each two-dimensional flat torus. Color codes obtained from two-dimensional flat torus were proposed on hexagon lattices [10], i.e., lattices in which the Voronoi cells are given by regular hexagons (three-valent tessellations), and where the qubits are attached to each vertex of the regular hexagons that tessellate the torus These two-dimensional lattices are known in quantum coding theory by honeycomb lattices (because of the shape of its Voronoi cells).
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