On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction

Gilles Zémor (1)

1 IMB - Institut de Mathématiques de Bordeaux

We review constructions of quantum surface codes and give an alternative, algebraic, construction of the known classes of surface codes that have fixed rate and growing minimum distance. This construction borrows from Margulis's family of Cayley graphs with large girths, and highlights the analogy between quantum surface codes and cycle codes of graphs in the classical case. We also attempt a brief foray into the class of quantum topological codes arising from higher dimensional manifolds and find these examples to have the same constraint on the rate and minimum distance as in the 2-dimensional case.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Gilles Zémor. On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction. International Workshop on Coding and Cryptology, Jun 2009, Zhangjiajie, China. pp.259-273. ⟨hal-00387084⟩