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

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1270 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
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... |

1108 | Algorithms for quantum computation: discrete logarithms and factoring
- Shor
- 1994
(Show Context)
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]... |

941 | Quantum Theory: Concepts and Methods - Peres - 1995 |

854 | Quantum theory, the Church-Turing principle and the universal quantum computer
- Deutsch
- 1985
(Show Context)
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... |

599 | Simulating physics with computers - Feynman - 1982 |

571 | Quantum complexity theory
- Bernstein, Vazirani
- 1993
(Show Context)
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... |

488 | Probabilistic logics and the synthesis of reliable organisms from unreliable components - Neumann - 1956 |

441 | Rapid solution of problems by quantum computation - Deutsch, Jozsa - 1992 |

432 | Quantum mechanics helps in searching for a needle in a haystack. Phys - Grover - 1997 |

429 | On the power of quantum computation - SIMON - 1997 |

386 | Quantum Noise - Gardiner, Zoller - 2000 |

348 | Good quantum error-correcting codes exist, Phys - Calderbank, Shor - 1996 |

323 |
Scheme for reducing decoherence in quantum computer memory
- Shor
- 1995
(Show Context)
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... |

320 | Quantum circuit complexity
- Yao
- 1993
(Show Context)
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... |

304 | N.J.A.: Quantum error correction via codes over GF(4 - Calderbank, Rains, et al. |

295 | Introduction to Coding Theory - Lint - 1992 |

287 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
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... |

281 | Decoherence and the transitions from quantum to classical
- Zurek
- 1991
(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... |

271 | Introduction to Cyclotomic Fields - Washington - 1997 |

263 | Fault-tolerant quantum computation
- Shor
(Show Context)
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... |

210 | A Single Quantum Cannot Be Cloned, Nature 299 - Wooters, Zurek - 1982 |

199 |
Theory of quantum error-correcting codes Phys
- Knill, Laflamme
- 1997
(Show Context)
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 ... |

192 | Class of quantum error-correcting codes saturating the quantum Hamming bound - Gottesman - 1996 |

182 | 2-bit gates are universal for quantum computation, Phys
- DiVincenzo
- 1995
(Show Context)
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 ... |

175 | Elementary gates for quantum computation. Physical Review A, - Barenco, Bennett, et al. - 1995 |

171 | Multiple particle interference and quantum error correction.” quantph/9601029
- Steane
(Show Context)
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... |

142 | Quantum circuits with mixed states. In:
- Aharonov, Kitaev, et al.
(Show Context)
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.... |

127 | Quantum computation with quantum dots - Loss, DiVincenzo - 1998 |

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

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

98 | Quantum Computation
- Vincenzo
- 1995
(Show Context)
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... |

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

59 |
Quantum computations with cold trapped ions,” Phys
- Cirac, Zoller
- 1995
(Show Context)
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 ... |

59 | Theory of fault-tolerant quantum computation - Gottesman - 1998 |

59 | Non-binary unitary error bases and quantum codes
- Knill
- 1996
(Show Context)
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... |

55 | Fault-Tolerant Quantum Computation with Higher-Dimensional Systems,” - Gottesman - 1998 |

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

51 | 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... |

45 | Is quantum mechanics useful - Landauer |

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

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

42 | Quantum computers and dissipation,”
- Palma, Suominen, et al.
- 1946
(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... |

32 | Highly fault-tolerant parallel computation,” - Spielman - 1996 |

32 |
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... |

30 |
Threshold accuracy for quantum computation. quantph /9610011
- Knill, Laflamme, et al.
- 1996
(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... |

25 | Polynomial simulations of decohered quantum computers,
- 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... |

22 |
Completeness theorems for fault-tolerant distributed computing
- Ben-Or, Goldwasser, et al.
- 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... |

22 | 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... |

14 | 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... |

12 |
Correcting Two Dimensional Arrays
- Ga'cs, Self
(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... |

9 | Quantum Mechanics; - Cohen-Tanoudji, Diu, et al. - 1977 |

6 | Fault tolerant quantum computation, lanl e-print quant-ph/9712048 - Preskill |

5 |
Bosonic Quantum Codes for Amplitude Damping
- Chuang, Leung, et al.
(Show Context)
Citation Context ...rcuits which consist of particles with p ≥ 2 possible states to be in. We call such quantum particles qupits, as a generalization to qubits. Non binary codes where defined independently also by Chuang=-=[11]-=- and Knill[13]. The proofs that these are quantum codes[4] transform smoothly from F2 to Fp. For p = 2, we show how to convert these codes to proper quantum codes, using the set of gates and procedure... |

4 | linear algebra and its applications - Choi - 1975 |

4 |
Modern Quantum Mechanics, revised edition
- Saqurai
- 1994
(Show Context)
Citation Context ...sses of proper quantum computation code. Section 6 discusses possible extensions, conclusions and open questions. 2 Noisy Quantum Circuits In this section we recall the definitions of quantum circuits=-=[12, 8, 25]-=- with mixed states[3], We then define noisy quantum circuits, which model a physical realization of quantum circuits[1]. The faults are probabilistic, and occur in single qubits and in gates. 2.0.1 Pu... |

4 | Quantum Computation, Annual Reviews of Computational - Aharonov - 1998 |

4 | Imry: Phase Uncertainty and Loss of Interference: A General Picture, Phys. Rev. A 41 - Stern, Aharonov, et al. - 1990 |

3 | Smolin J A and Wootters W K Mixed state entanglement and quantum error correction, Phys - Bennett, DiVincenzo - 1996 |

3 | Bulk spin-resonance quantum computation - NA, IL - 1997 |

3 | A 1997 Quantum error correction with imperfect gates Quantum - Yu |

3 |
Resilient quantum computation, Science, vol 279, p.342
- Knill, Laflamme, et al.
- 1998
(Show Context)
Citation Context ...aller.) The scheme applies also for corrections of random inaccuracies, but not for systematic errors. Similar results to those of this paper where independently discovered by Knill, La amme and Zurek=-=[4-=-1]. Non binary codes where dened independently also by Chuang[17] and Knill[43]. 2 Noisy Quantum Circuits In this section we recall the denitions of quantum circuits[59, 23, 74] with mixed states[3], ... |

3 |
A potentially realisable quantum computer, Science 261 1569; see also Science 263
- Lloyd
- 1993
(Show Context)
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 ... |

3 |
K-A & Ekert A K 1996 Quantum computers and dissipation
- Palma, Suominen
(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... |

3 |
A 1996 Quantum data processing and error correction Phys
- Schumacher, Nielsen
(Show Context)
Citation Context ...ith p = 2. In this example the set of gates and procedures described in [63] is used, modied tost the denition of proper quantum codes. For the second class of proper codes, we recall the result in [6=-=0]-=-, showing that a quantum error correction exists if and only if the environment gains no information about the unencoded data from the damaged qubits. Hence there is a strong connection between quantu... |

3 |
Quantum networks
- Deutch
- 1989
(Show Context)
Citation Context ...d entirely inside this model. Working in the correct model will enable to prove the result of this paper. Sequential quantum computation can not be noise resistant[1], so we work with quantum circuits=-=[8, 25]-=-, and since the state 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 circ... |

2 |
Kitaev A Yu and Nisan N, Quantum Circuits with Mixed States
- Aharonov
- 1998
(Show Context)
Citation Context ...ithmic cost in time and size. The error corrections which have been described so far use a combination of classical and quantum operations. We would like to dene a model of noisy quantum computation [=-=3]-=-, such that errors and error corrections can be described entirely inside this model. In this paper we are able to prove the main result due to working in such a formal framework. Sequential quantum c... |

2 | Ekert A, Suominen K A and Torma P, Approximate quantum Fourier transform and decoherence, Phys - Barenco - 1996 |

2 | Vwani Roychowdhury, and Farrokh Vatan, "On Universal and FaultTolerant Quantum Computing", in preparation - Boykin, Mor, et al. |

2 | error correction and orthogonal - Lett - 1997 |

2 |
Bosonic Quantum Codes for Amplitude Damping, in Phys
- Chuang, Leung, et al.
- 1997
(Show Context)
Citation Context ...racies, but not 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[=-=1-=-7] and Knill[43]. 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 wit... |

2 |
Reliable storage of information in a system of unreliable components with local interactions
- Cirel'son
(Show Context)
Citation Context ...ow to apply the 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 le... |

2 | Fahmy A F, and Havel T F, Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing - Cory - 1996 |

2 | Cirac J I and Zoller P 1995 Decoherence, continuous observation, and quantum computing: A cavity QED - Pellizzari, Gardiner |

2 | Personal communication - Prasad - 2014 |

2 | Vagner I D and Kventsel G 1997 Quantum computation in quantum-Hall systems - Privman - 1998 |

2 | Aharonov Y and Imry Y, "Phase uncertainty and loss of interference: a general picture" Phys - Stern - 1990 |

2 |
quantum error correcting codes exist
- Good
- 1995
(Show Context)
Citation Context ...dy the question of protecting quantum information is harder than the classical analog because one should also protect 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 computati... |

2 |
A potentially realizable computer
- Lloyd
- 1993
(Show Context)
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... |

2 | A and Tsirelson - Khalfin - 1992 |

1 | A and Tsirelson - Khal - 1992 |

1 | La R 1997 A theory of quantum error-correcting codes, Phys - Knill |

1 | Kraus K, states, eects and operations: Foundamental notions of Quantum Theory - Hellwig, Kraus - 1970 |

1 |
Factoring in a dissipative quantum computer Phys. Rev. A 54 2605-2613, and
- Miquel, Paz, et al.
- 1996
(Show Context)
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... |

1 |
correcting two dimentional arrays
- Self
- 1989
(Show Context)
Citation Context ...t tolerance when the noise rate η is taken to be logarithmically small. Improvement to constant noise rate To improve this result, we use concatenated simulations, which generalizes the works of Gacs =-=[10]-=- to the quantum case. The idea is that since the simulating circuit is also a circuit, it’s effective noise rate can be improved by simulating it again, and so on for several levels. Even a very small... |

1 | Kraus K, states, effects and operations: Foundamental notions of Quantum Theory - Hellwig, Kraus - 1970 |

1 | 62 van Lint J H Coding Theory - Lett - 1998 |

1 |
11 D. Deutch. Quantum networks
- Lett
- 1995
(Show Context)
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... |

1 | Reliable stomge of information in a system of unreliable components with local interactions, volume 653 - Cirel’son - 1978 |

1 | Non-bhary unitary error bases and quantum codes. quant-ph/9608048 - Knin - 1996 |