## Fault-tolerant quantum computation by anyons (2003)

Citations: | 94 - 3 self |

### BibTeX

@MISC{Kitaev03fault-tolerantquantum,

author = {A. Yu. Kitaev},

title = { Fault-tolerant quantum computation by anyons},

year = {2003}

}

### OpenURL

### Abstract

A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.

### Citations

810 |
Quantum groups
- Drinfeld
- 1986
(Show Context)
Citation Context ...). Although Ag(s, p) does not depend on p, we retain this parameter to emphasize the duality between Ag(s, p) and Bh(s, p). 11 These operators generate an algebra D = D(G), Drinfield’s quantum double =-=[20]-=- of the group algebra C[G]. It will play a very important role below. Now we only need two symmetric combinations of Ag(s, p) and Bh(s, p), namely ∑ −1 A(s) = N Ag(s, p) B(p) = B1(s, p) (12) g∈G 11 In... |

809 | Algorithms for quantum computation: Discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...the result of fusion. Such computation is fault-tolerant by its physical nature. A quantum computer can provide fast solution for certain computational problems (e.g. factoring and discrete logarithm =-=[1]-=-) which require exponential time on an ordinary computer. Physical realization of a quantum computer is a big challenge for scientists. One important problem is decoherence and systematic errors in un... |

374 |
Quantum Groups
- Kassel
(Show Context)
Citation Context ...that the multiplication in D and the comultiplication in F are defined by the same tensor Ω ⋆ ⋆⋆. Actually, D and F are Hopf algebras dual to each other. 17(For general account on Hopf algebras, see =-=[21, 22, 23]-=-). The multiplication in F corresponds to a comultiplication in D defined as follows ∆(Dk) = Λ mn k Dm ⊗ Dn (27) (More explicitly, ∆(D(h,g)) = ∑ h1h2=h D(h1,g) ⊗D(h2,g) ). The unit element of F is 1F ... |

262 | Good quantum error-correcting codes exist
- Calderbank, Shor
- 1996
(Show Context)
Citation Context ...y possible if | Supp(E)| ≥ k. (Here Supp(E) is the set of j for which αj ̸= 0 or βj ̸= 0). One may say that the toric codes have quite poor parameters. Well, they are not “good” codes in the sense of =-=[17]-=-. However, the code TOR(k) corrects almost any multiple error of size O(k 2 ). (The constant factor in O(. . .) is related to the percolation problem). So, these codes work if the error rate is consta... |

204 | Fault-tolerant quantum computation
- Shor
- 1996
(Show Context)
Citation Context ...s in unitary transformations which occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation =-=[2]-=-, with subsequent improvements [3, 4, 5, 6]. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortu... |

192 | Fault-Tolerant Quantum Computation With Constant Error Rate,” ArXiv e-prints
- Aharonov, Ben-Or
- 1999
(Show Context)
Citation Context ... occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation [2], with subsequent improvements =-=[3, 4, 5, 6]-=-. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortunately, the threshold value of δ is rather ... |

177 |
Fractional Statistics and Anyon Superconductivity , World Scientific Singapore
- Wilczek
- 1990
(Show Context)
Citation Context ...omputation. A measurement of the final state can be performed by joining the particles in pairs and observing the result of fusion. Anyons have been studied extensively in the field-theoretic context =-=[9, 10, 11, 12, 13]-=-. So, I hardly discover any new about their algebraic properties. However, my approach differs in several respects: • The model Hamiltonians are different. • We allow a generic (but weak enough) pertu... |

155 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ... occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation [2], with subsequent improvements =-=[3, 4, 5, 6]-=-. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortunately, the threshold value of δ is rather ... |

146 | Quantum measurements and the Abelian stabilizer problem
- Kitaev
- 1995
(Show Context)
Citation Context ...ce to produce an unlimited number of copies.] The operations 3 and 4 are sufficient to perform universal classical computation. It is relatively simple to run quantum algorithms based on measurements =-=[24]-=-. Simulating a universal gate set is more subtle and requires composite qubits. That is, a usual qubit (with two distinct states) is represented by several vortex pairs. 25Concluding remarks It has b... |

80 |
Quasitriangular Hopf algebras and Yang-Baxter equations
- Majid
- 1990
(Show Context)
Citation Context ...that the multiplication in D and the comultiplication in F are defined by the same tensor Ω ⋆ ⋆⋆. Actually, D and F are Hopf algebras dual to each other. 17(For general account on Hopf algebras, see =-=[21, 22, 23]-=-). The multiplication in F corresponds to a comultiplication in D defined as follows ∆(Dk) = Λ mn k Dm ⊗ Dn (27) (More explicitly, ∆(D(h,g)) = ∑ h1h2=h D(h1,g) ⊗D(h2,g) ). The unit element of F is 1F ... |

66 |
Quasi-Hopf algebras, group cohomology, and orbifold models, Integrable systems and quantum groups
- Dijkgraaf, Pasquier, et al.
- 1990
(Show Context)
Citation Context ...omputation. A measurement of the final state can be performed by joining the particles in pairs and observing the result of fusion. Anyons have been studied extensively in the field-theoretic context =-=[9, 10, 11, 12, 13]-=-. So, I hardly discover any new about their algebraic properties. However, my approach differs in several respects: • The model Hamiltonians are different. • We allow a generic (but weak enough) pertu... |

47 |
Quantum error correction with imperfect gates
- Kitaev
- 1997
(Show Context)
Citation Context ...t is not so simple. First of all, we need a quantum code with local stabilizer operators. I start with a class of stabilizer quantum codes associated with lattices on the torus and other 2-D surfaces =-=[6, 8]-=-. Qubits live on the edges of the lattice whereas the stabilizer operators correspond to the vertices and the faces. These operators can be put together to make up a 1 Actually, the threshold is not k... |

44 |
Quantum error correction and orthogonal
- Calderbank, Rains, et al.
- 1997
(Show Context)
Citation Context ...re two relations between the stabilizer operators, ∏ s As = 1 and ∏ p Bp = 1. So, there are m = 2k 2 − 2 independent stabilizer operators. It follows from the general theory of additive quantum codes =-=[15, 16]-=- that dim L = 2 n−m = 4. However, there is a more instructive way of computing dim L. Let us find the algebra L(L) of all linear operators on the space L — this will give us full information about thi... |

34 | Accuracy threshold for quantum computation, Technical report, Quantum Physics e-Print archive
- Knill, Laflamme, et al.
- 1996
(Show Context)
Citation Context ... occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation [2], with subsequent improvements =-=[3, 4, 5, 6]-=-. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortunately, the threshold value of δ is rather ... |

25 | Threshold Estimate for Fault Tolerant Quantum Computation - Zalka - 1997 |

18 | Concatenated quantum codes - Knill, Laflamme - 1996 |

14 |
Fractional statistics and the quantum hall effect
- Arovas, Schrieffer, et al.
- 1984
(Show Context)
Citation Context ...ffect. It does not occur if both particles are of the same type. Note that abelian anyons exist in real solid state systems, namely, they are intrinsicly related to the fractional quantum Hall effect =-=[18]-=-. However, these anyons have different braiding properties. In the fractional quantum Hall system with filling factor p/q, there is only one basic type of anyonic particles with (real) electric charge... |

9 |
Non-abelian vortices and nonabelian statistics, Phys
- Lo, Preskill
- 1993
(Show Context)
Citation Context ...omputation. A measurement of the final state can be performed by joining the particles in pairs and observing the result of fusion. Anyons have been studied extensively in the field-theoretic context =-=[9, 10, 11, 12, 13]-=-. So, I hardly discover any new about their algebraic properties. However, my approach differs in several respects: • The model Hamiltonians are different. • We allow a generic (but weak enough) pertu... |

8 | de Wild Propitius, Quantum symmetries in discrete gauge theories, Phys - Bais, Driel, et al. - 1992 |

4 |
Fractional statistics on a torus. Phys
- Einarsson
- 1990
(Show Context)
Citation Context ...rrors). It seems that the anyons are more fundamental and can be used as a universal probe for this hidden order. Indeed, the ground state degeneracy on the torus follows from the existence of anyons =-=[19]-=-. Here is the original Einarsson’s proof applied to our two types of particles. We derived the ground state degeneracy from the commutation relations between the operators Z1, Z2, X1, X2. These operat... |

1 | de Wild Propitius, Discrete gauge theories - Bais, M - 1995 |

1 |
Concatenated quantum codes, e-print quant-ph/9608012
- Knill, Laflamme
- 1996
(Show Context)
Citation Context |

1 |
Wild Propitius, Discrete gauge theories, e-print hep-th/9511201
- Bais, de
- 1995
(Show Context)
Citation Context |

1 |
The notion of symmetry and computational feedback in the paradigm of steady, simultaneous quantum computation, Int
- Castagnoli, Rasetti
- 1993
(Show Context)
Citation Context ...s unified description of anyonic excitations and long range entanglement in the ground state. An attempt to use one-dimensional anyons for quantum computation was made by G. Castagnoli and M. Rasetti =-=[14]-=-, but the question of fault-tolerance was not considered. 1 Toric codes and the corresponding Hamiltonians Consider a k × k square lattice on the torus (see fig. 1). Let us attach a spin, or qubit, to... |