## FAULT-TOLERANT QUANTUM COMPUTATION WITH CONSTANT ERROR RATE (1999)

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Citations: | 195 - 12 self |

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@MISC{AHARONOV99fault-tolerantquantum,

author = {DORIT AHARONOV and MICHAEL BEN-OR},

title = {FAULT-TOLERANT QUANTUM COMPUTATION WITH CONSTANT ERROR RATE },

year = {1999}

}

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### Citations

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Citation Context ...usalem, Israel, E-mail: benor@cs.huji.ac.il 1 Introduction Quantum computation[22, 23, 74] is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm=-=[61]-=-. It is yet unclear whether and how quantum computers will be physically realizable,[49, 25, 18] but as any physical system, they in principle will be subjected to noise, such as decoherence[75, 71, 5... |

873 | Algorithms for quantum computation: discrete logarithm and factoring
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Citation Context ...sholds, at which point quantum computation becomes practical. 1 Introduction Quantum computation is believed to be more powerful than classical computation, mainly due to the celebrated Shor algorithm=-=[20]-=-. It is yet unclear whether and how quantum computers will be physically realizable,[16, 9, 7] but as any physical system, they in principle will be subjected to noise, such as decoherence[24, 23, 17]... |

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Citation Context ...rew University, Jerusalem, Israel, E-mail: doria@cs.huji.ac.il y Institute of Computer science, The Hebrew University, Jerusalem, Israel, E-mail: benor@cs.huji.ac.il 1 Introduction Quantum computation=-=[22, 23, 74]-=- is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[4... |

499 | Quantum Complexity Theory
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Citation Context ..., The Hebrew University, Jerusalem, Israel, E-mail: benor@cs.huji.ac.il 1 Introduction Quantum computation[22, 23, 74] is believed to be more powerful than classical computation, due to oracle results=-=[64, 9]-=- and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[49, 25, 18] but as any physical system, they in principle will be subjected to noise, such... |

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Citation Context ...rew University, Jerusalem, Israel, E-mail: doria@cs.huji.ac.il y Institute of Computer science, The Hebrew University, Jerusalem, Israel, E-mail: benor@cs.huji.ac.il 1 Introduction Quantum computation=-=[22, 23, 74]-=- is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[4... |

278 | Good Quantum Error-Correcting Codes Exist”, Phys - Calderbank, Shor - 1996 |

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Citation Context ...rew University, Jerusalem, Israel, E-mail: doria@cs.huji.ac.il y Institute of Computer science, The Hebrew University, Jerusalem, Israel, E-mail: benor@cs.huji.ac.il 1 Introduction Quantum computation=-=[22, 23, 74]-=- is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[4... |

238 | Quantum error correction via codes over GF(4 - Calderbank, Rains, et al. - 1997 |

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Citation Context ...ct the quantum correlations between the quantum bits (qubits). However, it was shown [12, 67] that good quantum error correcting codes exist, a result which was followed by many explicit examples(ex: =-=[62, 51]-=-). This does not trivially imply the existence of noise resistant quantum computation, since due to computation the faults propagate. One must be able to compute without allowing the errors to propaga... |

219 | Quantum Noise - Gardiner - 1991 |

215 | Algorithms for quantum computation
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Citation Context ... imply the existence of noise resistant quantum computation, since due to computation the faults propagate. One must be able to compute without allowing the errors to propagate too much. Recently Shor=-=[63]-=- showed how to use quantum codes in order to perform fault tolerant quantum computation when the noise rate, or the fault probability each time step, per qubit or gate, is polylogarithmically small. I... |

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Citation Context ...orithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[49, 25, 18] but as any physical system, they in principle will be subjected to noise, such as decoherence=-=[75, 71, 53-=-], and inaccuracies. Without error corrections, the eect of noise can accumulate and ruin the entire computation[71, 16], hence the computation must be protected. Even the simpler question of protecti... |

154 | Two-bit gates are universal for quantum computation
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Citation Context ...4] is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,=-=[49, 25, 18-=-] but as any physical system, they in principle will be subjected to noise, such as decoherence[75, 71, 53], and inaccuracies. Without error corrections, the eect of noise can accumulate and ruin the ... |

154 | A single quantum cannot be cloned, Nature 299 - Wootters, Zurek - 1982 |

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Citation Context ...ect the quantum correlations between the quantum bits (qubits). However, it was shown [4, 22] that good quantum error correcting codes exist, a result which was followed by many explicit examples(ex: =-=[18, 15]-=-). This does not immediately imply the existence of noise resistant quantum computation, since the computation causes the effect of faults to spread. Recently Shor[19] showed how to use quantum codes ... |

145 | Elementary gates for quantum computation, Phys. Rev. A 52 - Barenco, Bennet, et al. - 1995 |

138 | Multiple particle interference and quantum error correction
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Citation Context ... simpler question of protecting quantum information is harder than the classical analog because one must also protect the quantum correlations between the quantum bits (qubits). However, it was shown =-=[12, 67]-=- that good quantum error correcting codes exist, a result which was followed by many explicit examples(ex: [62, 51]). This does not trivially imply the existence of noise resistant quantum computation... |

115 | Quantum circuits with mixed states
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Citation Context ...of a noisy quantum system is in general a probability distribution over pure states, i.e. a mixed state, and not merely a pure state as in the standard model, we use quantum circuits with mixed states=-=[3]-=-. Since noise is a dynamic process which depends on time, the circuit will be divided to levels, or time steps. Unlike Yao[25] we allow qubits to be input and output at different times to the circuit.... |

114 | Oracle quantum computing - Berthiaume, Brassard - 1994 |

94 | Bulk spin resonance quantum computation - Gershenfeld, Chuang - 1997 |

72 | Topics in quantum computers
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Citation Context ...putation is believed to be more powerful than classical computation, mainly due to the celebrated Shor algorithm[20]. It is yet unclear whether and how quantum computers will be physically realizable,=-=[16, 9, 7]-=- but as any physical system, they in principle will be subjected to noise, such as decoherence[24, 23, 17], and inaccuracies. Without error corrections, the effect of noise can ruin the entire computa... |

54 | A theory of fault-tolerant quantum computation - Gottesman - 1998 |

51 | Non-binary unitary error bases and quantum codes
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Citation Context ...t for systematic errors. Similar results to those of this paper where independently discovered by Knill, La amme and Zurek[41]. Non binary codes where dened independently also by Chuang[17] and Knill[=-=4-=-3]. 2 Noisy Quantum Circuits In this section we recall the denitions of quantum circuits[59, 23, 74] with mixed states[3], and dene noisy quantum circuits[3]. 2.1 Pure states We deal with systems of n... |

51 | DiVincenzo: Quantum computation with quantum dots, Phys - Loss, P - 1998 |

47 | Is quantum mechanics useful - Landauer - 1995 |

46 |
Quantum computations with cold trapped ions,” Phys
- Cirac, Zoller
- 1995
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Citation Context ...4] is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor's algorithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,=-=[49, 25, 18-=-] but as any physical system, they in principle will be subjected to noise, such as decoherence[75, 71, 53], and inaccuracies. Without error corrections, the eect of noise can accumulate and ruin the ... |

46 | Maintaining coherence in quantum computers
- Unruh
- 1995
(Show Context)
Citation Context ...lgorithm[20]. It is yet unclear whether and how quantum computers will be physically realizable,[16, 9, 7] but as any physical system, they in principle will be subjected to noise, such as decoherence=-=[24, 23, 17]-=-, and inaccuracies. Without error corrections, the effect of noise can ruin the entire computation[23, 6], so we need to protect the computation against quantum noise. Already the question of protecti... |

43 | Implementation of a quantum search algorithm on a quantum computer, Nature 393 - Jones, Mosca, et al. - 1998 |

42 | Fault-tolerant quantum computation with higher-dimensional systems - Gottesman - 1999 |

40 | Quantum data processing and error correction”, Phys - Schumacher, Nielsen - 1996 |

35 | Fault-tolerant quantum computation with local gates - Gottesman - 2000 |

34 |
Maintaining coherence in quantum computers, Phys
- Unruh
- 1995
(Show Context)
Citation Context ...orithm[61]. It is yet unclear whether and how quantum computers will be physically realizable,[49, 25, 18] but as any physical system, they in principle will be subjected to noise, such as decoherence=-=[75, 71, 53-=-], and inaccuracies. Without error corrections, the eect of noise can accumulate and ruin the entire computation[71, 16], hence the computation must be protected. Even the simpler question of protecti... |

32 | Quantum computation and dissipation
- Palma, Suominen, et al.
- 1995
(Show Context)
Citation Context ...lgorithm[20]. It is yet unclear whether and how quantum computers will be physically realizable,[16, 9, 7] but as any physical system, they in principle will be subjected to noise, such as decoherence=-=[24, 23, 17]-=-, and inaccuracies. Without error corrections, the effect of noise can ruin the entire computation[23, 6], so we need to protect the computation against quantum noise. Already the question of protecti... |

30 | How to share a quantum secret - Cleve, Gottesman, et al. - 1999 |

29 | Highly fault-tolerant parallel computation - Spielman - 1996 |

28 |
Threshold Accuracy for Quantum Computation LANL e-print
- Knill, Laflamme, et al.
(Show Context)
Citation Context ... are allowed to operate only on nearest neighbor qubits (In this case the threshold will be smaller.) Similar results to those of this paper where independently discovered by Knill, Laflamme and Zurek=-=[14]-=-. Organization of paper: In section 2 we define the model of noisy quantum circuits. Section 3 is devoted to describe one step of the simulation, given any quantum computation code. In section 4 we pr... |

23 | Polynomial Simulations of Decohered Quantum Computers”, (Online preprint quantph/9611029
- Aharonov, Ben-Or
- 1996
(Show Context)
Citation Context ...l, E-mail: benor@cs.huji.ac.il 1sThe error corrections which where described so far used a combination of classical and quantum operations. We would like to define a model of noisy quantum computation=-=[1]-=-, such that errors and error corrections can be described entirely inside this model. Working in the correct model will enable to prove the result of this paper. Sequential quantum computation can not... |

21 |
Completeness theorems for faulttolerant distributed computing
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- 1988
(Show Context)
Citation Context ...ata from the damaged qubits. Hence there is a strong connection between quantum error correction codes to secret sharing schemes that are used to perform secure fault-tolerant distributed computation =-=[8]-=-. The second class of quantum codes is thus the quantum analog of random polynomial codes[8]. To adopt the techniques of [8] to the quantum setting one can use the same encoding but instead of selecti... |

21 | Quantum computers, factoring and decoherence
- Chuang, Laflamme, et al.
- 1995
(Show Context)
Citation Context ... as any physical system, they in principle will be subjected to noise, such as decoherence[24, 23, 17], and inaccuracies. Without error corrections, the effect of noise can ruin the entire computation=-=[23, 6]-=-, so we need to protect the computation against quantum noise. Already the question of protecting quantum information is harder than the classical analog because one should also protect the quantum co... |

13 |
Self-correcting two-dimensional arrays
- Gács
- 1989
(Show Context)
Citation Context ... above scheme to achieve fault tolerance with polylogarithmically small noise rate, . To improve this result, we use concatenated simulations, which generalizes the works of Tsirelson [19] and Gac's [=-=27-=-] to the quantum case. The idea is that the eective noise rate of the simulating circuit can be decreased by simulating it again, and so on for several levels. It will suce that each level of simulati... |

11 | Limitations of noisy reversible computation
- Aharonov, Ben-Or, et al.
- 1996
(Show Context)
Citation Context ... circuit. This is crucial, since with the restriction that all qubits are initialized at time 0, it is not possible to compute fault tolerantly without an exponential blowup in the size of the circuit=-=[2]-=-. Between the time steps, we add the noise process, which is a probabilistic process: each qubit or gate undergoes a fault with independent probability η per step, and η is referred to as the noise ra... |