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"Homological Error Correction: Classical and Quantum Codes", Hector Bombin, Miguel Martin-Delgado, 2007

Reviewed August 18, 2023

Citation: Bombin, Hector, and Miguel A. Martin-Delgado. "Homological error correction: Classical and quantum codes." Journal of mathematical physics 48.5 (2007).

Web: https://arxiv.org/abs/quant-ph/0605094

Tags: Error-correcting-codes, Abelian-anyons, No-go

This paper shows that the quantum double construction for abelian groups can be adapted to give classical error correcting codes. These error correcting codes are optimal in the sense that if they are well-chosen then they can saturate the Hamming bound. They call codes of this form homological codes.

Additionally, this paper proves a no-go result: On non-orientable surfaces it is impossible to construct (quantum or classical) homological codes with coefficients in any group other than Z2. This is intuitive, because every surface is Z2 orientable but every non-orientable surface cannot be oriented with respect to any group other than Z2.

This highlights an extra point for why Z2 is special for topological error correction: you can base your codes on interesting non-orientable surfaces like to Mobius strip.