Toric Codes

Clarice Dias de Albuquerque, Eduardo Brandani da Silva, Waldir Silva Soares

Anton Betten, Harald Fripertinger, Adalbert Kerber +2 more

Arun B. Aloshious, Pradeep Kiran Sarvepalli

In this paper, we study nonbinary toric codes and color codes. While the relationships between the qubit color codes and toric codes have been studied extensively from various perspectives, similar studies for qudit toric codes and color codes are missing. We show that just as in the binary case, qudit color codes can be mapped to qudit toric codes. Qudit color codes are known to have the capability for implementing the generalized Clifford group transversally. The proposed mapping of qudit color codes to toric codes would allow us to port the computational capabilities of the qudit color codes to qudit toric codes. Another application would be to decode qudit color codes via qudit toric codes. Along the way, we show an important structural result for higher-dimensional qudit toric codes. Specifically, we show that the toric codes over prime power dimension can be viewed as a direct sum of copies of codes over the prime dimension. This result simplifies the study of codes over prime power dimension by reducing them to the study of codes over the prime dimension. Furthermore, we provide explicit circuits for the transformation between qudit color codes and toric codes.

Arun B. Aloshious, Pradeep Kiran Sarvepalli

Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).

Olaf Manz

Johan P. Hansen

Toric varieties and their associated toric codes, as well as determination of their parameters with intersection theory, are presented in the two dimensional case. Linear Secret Sharing Schemes with strong multiplication are constructed from toric varieties and codes by the J. L. Massey construction. Asymmetric Quantum Codes are obtained from toric codes by the A.R. Calderbank, P.W. Shor and A.M. Steane construction of stabilizer codes from linear codes containing their dual codes. Decoding of a class of toric codes is presented.