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Projective plane and planar quantum codes. (English) Zbl 0995.94037
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)].

94B60 Other types of codes
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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