Toric code

The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice.[1] It is the simplest and most well studied of the quantum double models.[2] It is also the simplest example of topological order—Z2 topological order (first studied in the context of Z2 spin liquid in 1991).[3][4] The toric code can also be considered to be a Z2 lattice gauge theory in a particular limit.[5] It was introduced by Alexei Kitaev.

The toric code gets its name from its periodic boundary conditions, giving it the shape of a torus. These conditions give the model translational invariance, which is useful for analytic study. However, some experimental realizations require open boundary conditions, allowing the system to be embedded on a 2D surface. The resulting code is typically known as the planar code. This has identical behaviour to the toric code in most, but not all, cases.

  1. ^ A. Y. Kitaev, Proceedings of the 3rd International Conference of Quantum Communication and Measurement, Ed. O. Hirota, A. S. Holevo, and C. M. Caves (New York, Plenum, 1997)
  2. ^ Kitaev, Alexei (2006). "Anyons in an exactly solved model and beyond". Annals of Physics. 321 (1): 2–111. arXiv:cond-mat/0506438. Bibcode:2006AnPhy.321....2K. doi:10.1016/j.aop.2005.10.005. ISSN 0003-4916. S2CID 118948929.
  3. ^ Read, N.; Sachdev, Subir (1 March 1991). "Large-Nexpansion for frustrated quantum antiferromagnets". Physical Review Letters. 66 (13): 1773–1776. Bibcode:1991PhRvL..66.1773R. doi:10.1103/physrevlett.66.1773. ISSN 0031-9007. PMID 10043303.
  4. ^ Wen, X. G. (1 July 1991). "Mean-field theory of spin-liquid states with finite energy gap and topological orders". Physical Review B. 44 (6): 2664–2672. Bibcode:1991PhRvB..44.2664W. doi:10.1103/physrevb.44.2664. ISSN 0163-1829. PMID 9999836.
  5. ^ Fradkin, Eduardo; Shenker, Stephen H. (15 June 1979). "Phase diagrams of lattice gauge theories with Higgs fields". Physical Review D. 19 (12): 3682–3697. Bibcode:1979PhRvD..19.3682F. doi:10.1103/physrevd.19.3682. ISSN 0556-2821.