Abstract
This current work propose a technique to generate polygonal color codes in the hyperbolic geometry environment. The color codes were introduced by Bombin and Martin-Delgado in 2007, and the called triangular color codes have a higher degree of interest because they allow the implementation of the Clifford group, but they encode only one qubit. In 2018 Soares e Silva extended the triangular codes to the polygonal codes, which encode more qubits. Using an approach through hyperbolic tessellations we show that it is possible to generate Hyperbolic Polygonal codes, which encode more than one qubit with the capacity to implement the entire Clifford group and also having a better coding rate than the previously mentioned codes, for the color codes on surfaces with boundary with minimum distance d = 3.
Highlights
One of the great difficulties of performing quantum computing is decoherence, as Unruh warned in 1995 [20]
The color codes have been expanded, for example in the work of Soares and Silva [18], where the authors consider an approach of the color codes on compact surfaces of genus greater or equal to 2, using tools of the hyperbolic geometry, obtaining codes with parameters better than those of Kitaev [12], Bombin and Martin-Delgado [2], Albuquerque, Palazzo and Silva [1], among others
We evaluated the feasibility of this type of construction, taking into account the rigidity of the hyperbolic geometry in relation to the area of the polygons and the impossibility of decreasing the length of the side of a regular hyperbolic polygon without changing its internal angles, and we show that is it feasible as this technique can generate codes with even better parameters than those obtained until now by codes of the same nature
Summary
One of the great difficulties of performing quantum computing is decoherence, as Unruh warned in 1995 [20]. Decoherence is the decay phenomenon of superposition of states, due to the interaction between the system and the surrounding environment. The problem may be solved using quantum error-correcting codes. Quantum states can be cleverly encoded so that the harmful effects of decoherence can be resisted. The classical theory of error-correcting codes was stablish by Shannon in 1948 [15]. In 1995, was the first to show an quantum error-correcting code [16], overcoming the non-cloning theorem and achieving an analogue to the classic repeating code. Shor’s code belongs to a class of codes known as CSS codes, which was introduced by Calderbank and Shor [6] and Steane [19]
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