Freedman, Michael H.; Meyer, David A. Projective plane and planar quantum codes. (English) Zbl 0995.94037 Found. Comput. Math. 1, No. 3, 325-332 (2001). The idea of topological or anyonic quantum computation arose independently in papers of A. Yu. Kitaev [Fault-tolerant quantum computation by anyons, quant-ph/9707021] and M. Freedman [Proc. Natl. Acad. Sci. 95, 98-101 (1998; Zbl 0895.68053)]. Topological properties of quantum systems might play a crucial role in stabilizing large-scale quantum computers. In this short note, the authors study a very beautiful example of this approach. Using celluations of the projective plane, the authors construct three inequivalent quantum error-correcting codes for a single qubit. They also identify one of the codes with Shor’s original 9 qubit code. The idea is based on Kitaev’s above-mentioned paper. Another equivalent construction is given by A. Yu. Kitaev and S. B. Bravyi [Quantum codes on a lattice with boundary, quant-ph/9811052]. A unified approach based on topological quantum field theories is given by M. Freedman et al. [Topological quantum computation, Bull. Am. Math. Soc. (to appear)]. Reviewer: Zhenghan Wang (Bloomington) Cited in 23 Documents MSC: 94B60 Other types of codes 81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy 57M20 Two-dimensional complexes (manifolds) (MSC2010) Keywords:planar quantum codes; quantum error correcting codes Citations:Zbl 0895.68053 PDFBibTeX XMLCite \textit{M. H. Freedman} and \textit{D. A. Meyer}, Found. Comput. Math. 1, No. 3, 325--332 (2001; Zbl 0995.94037) Full Text: DOI arXiv