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We introduce a new type of sparse CSS quantum error correcting code based onthe homology of hypermaps. Sparse quantum error correcting codes are ofinterest in the building of quantum computers due to their ease ofimplementation and the possibility of developing fast decoders for them. Codesbased on the homology of embeddings of graphs, such as Kitaev's toric code,have been discussed widely in the literature and our class of codes generalizethese. We use embedded hypergraphs, which are a generalization of graphs thatcan have edges connected to more than two vertices. We develop theorems andexamples of our hypermap-homology codes, especially in the case that we choosea special type of basis in our homology chain complex. In particular the moststraightforward generalization of the $m \times m$ toric code tohypermap-homology codes gives us a $[(3/2)m^2,2,m]$ code as compared to thetoric code which is a $[2m^2,2,m]$ code. Thus we can protect the same amount ofquantum information, with the same error-correcting capability, using lessphysical qubits.

Hypermap-Homology Quantum Codes (Ph.D. thesis)