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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Toric code ： ウィキペディア英語版 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 〔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).〕 It is the simplest and most well studied of the quantum double models.〔A. Kitaev, Ann. Phys. 321, 2 (2006).〕 It is also the simplest example of topological order—''Z''2 topological order (first studied in the context of ''Z''2 spin liquid in 1991).〔N. Read and Subir Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991)〕〔Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gaps and Topological Orders, (Phys. Rev. B44, 2664 (1991) ).〕 The toric code can also be considered to be a ''Z''2 lattice gauge theory in a particular limit.〔E. Fradkin and S. Shenker, Phys. Rev. D 19, 3682–3697 (1979)〕 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, experimental realization requires 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. ==Error correction and computation== The toric code is defined on a two-dimensional lattice, usually chosen to be the square lattice, with a spin-½ degree of freedom located on each edge. They are chosen to be periodic. Stabilizer operators are defined on the spins around each vertex $v$ and plaquette (or face) $p$ of the lattice as follows, $$ A_v = \prod_ \sigma^x_i, \,\, B_p = \prod_ \sigma^z_i. Where here we use $i\; \backslash in\; v$ to denote the edges touching the vertex $v$, and $i\; \backslash in\; p$ to denote the edges surrounding the plaquette $p$. The stabilizer space of the code is that for which all stabilizers act trivially, hence, $$ A_v | \psi \rangle = | \psi \rangle, \,\, \forall v, \,\, B_p | \psi \rangle = | \psi \rangle, \,\, \forall p, for any state $|\; \backslash psi\; \backslash rangle$. For the toric code, this space is four-dimensional, and so can be used to store two qubits of quantum information. This can be proven by considering the number of independent stabilizer operators. The occurrence of errors will move the state out of the stabilizer space, resulting in vertices and plaquettes for which the above condition does not hold. The positions of these violations is the syndrome of the code, which can be used for error correction. The unique nature of the topological codes, such as the toric code, is that stabilizer violations can be interpreted as quasiparticles. Specifically, if the code is in a state $|\; \backslash phi\; \backslash rangle$ such that, $A\_v\; |\; \backslash phi\; \backslash rangle\; =\; -\; |\; \backslash phi\; \backslash rangle$, a quasiparticle known as an $e$ anyon can be said to exist on the vertex $v$. Similarly violations of the $B\_p$ are associated with so called $m$ anyons on the plaquettes. The stabilizer space therefore corresponds to the anyonic vacuum. Single spin errors cause pairs of anyons to be created and transported around the lattice. When errors create an anyon pair and move the anyons, one can imagine a path connecting the two composed of all links acted upon. If the anyons then meet and are annihilated, this path describes a loop. If the loop is topologically trivial, it has no effect on the stored information. The annihilation of the anyons in this case corrects all of the errors involved in their creation and transport. However, if the loop is topologically non-trivial, though reannihilation of the anyons returns the state to the stabilizer space it also implements a logical operation on the stored information. The errors in this case are therefore not corrected, but consolidated. Let us consider the noise model for which bit and phase errors occur independently on each spin, both with probability ''p''. When ''p'' is low, this will create sparsely distributed pairs of anyons which have not moved far from their point of creation. Correction can be achieved by identifying the pairs that the anyons were created in (up to an equivalence class), and then reannihilating them to remove the errors. As ''p'' increases, however, it becomes more ambiguous as to how the anyons may be paired without risking the formation of topologically non-trivial loops. This gives a threshold probability, under which the error correction will almost certainly succeed. Through a mapping to the random bond Ising model, this critical probability has been found to be around 11%.〔E. Dennis, A. Kitaev, A. Landahl, J. Preskill, J. Math. Phys. 43, 4452 (2002).〕 Other error models may also be considered, and thresholds found. In all cases studied so far, the code has been found to saturate the Hashing bound. For some error models, such as biased errors where bit errors occur more often than phase errors or vice versa, lattices other than the square lattice must be used to achieve the optimal thresholds.〔B. Roethlisberger, et al. Phys. Rev. A 85, 022313 (2012).〕〔H. Bombin, et al. Phys. Rev. X 2, 021004 (2012).〕 These thresholds are upper limits, and are useless unless efficient algorithms are found to achieve them. The most well-used algorithm is minimum weight perfect matching.〔Edmonds, Jack (1965). "Paths, trees, and flowers". Canad. J. Math. 17: 449–467.〕 When applied to the noise model with independent bit and flip errors, a threshold of around 10.5% is achieved. This falls only a little short of the 11% maximum. However, matching does not work so well when there are correlations between the bit and phase errors, such as with depolarizing noise. The means to perform quantum computation on logical information stored within the toric code has been considered, with the properties of the code providing fault-tolerance. It has been shown that extending the stabilizer space using 'holes', vertices or plaquettes on which stabilizers are not enforced, allows many qubits to be encoded into the code. However, a universal set of unitary gates can not be fault-tolerantly implemented by unitary operations and so additional techniques are required to achieve quantum computing. For example, universal quantum computing can be achieved by preparing magic states used to teleport in the required additional gates. Furthermore, preparation of magic states must be fault tolerant, which can be achieved by magic state distillation on noisy magic states. A measurement based scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two-dimensional architecture.〔R. Raussendorf, J. Harrington, Phys. Rev. Lett. 98, 190504 (2007); R. Raussendorf, J. Harrington and K. Goyal, New J. Phys. 9, 199, (2007).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Toric code」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.