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Foundations of Computational Mathematics

, Volume 1, Issue 3, pp 325–332 | Cite as

Projective Plane and Planar Quantum Codes

  • Michael H. Freedman
  • David A. Meyer
Article

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.

Key words. Topological code, Quantum error correction, Planar code. AMS Classification. 94Bxx, 81P99, 57M20. 

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Copyright information

© Society for the Foundation of Computational Mathematics 2000

Authors and Affiliations

  • Michael H. Freedman
    • 1
  • David A. Meyer
    • 2
  1. 1.Microsoft Research One Microsoft Way Redmond, WA 98052, USA michaelf@microsoft.comUS
  2. 2.Project in Geometry and Physics Department of Mathematics University of California/San Diego La Jolla, CA 92093-0112, USA dmeyer@chonji.ucsd.eduUS

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