Construction of Polygonal Color Codes from Hyperbolic Tesselations

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.

摘要 本研究提出一种可在双曲几何环境中生成多边形着色码（color codes）的技术。着色码由邦宾（Bombin）与马丁-德尔加多（Martin-Delgado）于2007年提出，其中所谓的三角形着色码（triangular color codes）因可实现克利福德群（Clifford group）的构建而受到广泛关注，但这类编码仅能编码1个量子比特（qubit）。2018年，索阿雷斯·席尔瓦（Soares e Silva）将三角形着色码推广为多边形着色码，后者可编码多个量子比特。本研究通过双曲平铺（hyperbolic tessellations）的方法证明，可生成双曲多边形着色码（Hyperbolic Polygonal codes）：这类编码可实现多量子比特编码，能够完整实现克利福德群，且在最小距离d=3的边界曲面着色码场景中，其编码速率优于前述各类着色码。