Optimal Cycle Codes Constructed From Ramanujan Graphs
SIAM Journal on Discrete Mathematics
On the Error-Correcting Capabilities of Cycle Codes of Graphs
Combinatorics, Probability and Computing
Sparse-graph codes for quantum error correction
IEEE Transactions on Information Theory
Upper bounds on the rate of low density stabilizer codes for the quantum erasure channel
Quantum Information & Computation
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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.