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Projective Plane and Planar Quantum Codes

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Abstract

Cellulations of the projective plane RP ^2 define single qubit topological quantum error correcting codes since there is a unique essential cycle in H 1 (RP 2 ;Z 2 ) . We construct three of the smallest such codes, show they are inequivalent, and identify one of them as Shor's original 9 qubit repetition code. We observe that Shor's code can be constructed in a planar domain and generalize to planar constructions of higher-genus codes for multiple qubits.

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Authors and Affiliations

  1. Microsoft Research One Microsoft Way Redmond, WA 98052, USA michaelf@microsoft.com, US

    Michael H. Freedman

  2. Project in Geometry and Physics Department of Mathematics University of California/San Diego La Jolla, CA 92093-0112, USA dmeyer@chonji.ucsd.edu, US

    David A. Meyer

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  1. Michael H. Freedman
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  2. David A. Meyer
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June 21, 2000. Final version recieved: April 27, 2001.

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Freedman, M., Meyer, D. Projective Plane and Planar Quantum Codes. Found. Comput. Math. 1, 325–332 (2001). https://doi.org/10.1007/s102080010013

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  • DOI: https://doi.org/10.1007/s102080010013

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