## Fault-tolerant quantum computation (1996)

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Venue: | In Proc. 37th FOCS |

Citations: | 215 - 4 self |

### BibTeX

@INPROCEEDINGS{Shor96fault-tolerantquantum,

author = {Peter W. Shor},

title = {Fault-tolerant quantum computation},

booktitle = {In Proc. 37th FOCS},

year = {1996},

pages = {56--65}

}

### Years of Citing Articles

### OpenURL

### Abstract

It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum error-correcting codes without decoding this data. 1.

### Citations

3231 | A Method for Obtaining Digital Signatures and Public-Key Cryptosystems - Rivest, Shamir, et al. - 1978 |

2114 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ...truction for fault-tolerant quantum circuits in detail, we first need to introduce more facts about error correcting codes. These can be found explained in more detail in coding theory books (such as =-=[17]-=-). We will be using codes with dim(C) \Gamma dim(C ? ) = 1; they are thus rather inefficient in that they code one qubit into n. The codes we use can be constructed from self-dual codes with C = C ? b... |

952 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer - Shor - 1997 |

873 | Algorithms for quantum computation: discrete logarithm and factoring
- Shor
- 1994
(Show Context)
Citation Context ...puter science. Using only polynomial resources, these quantum computers can compute certain functions which are not known to be computable on classical digital computers in less than exponential time =-=[26, 23, 25]-=-. The potentially most useful algorithms for quantum computers discovered so far include prime factorization and simulation of certain quantum mechanical systems. Given these theoretical results, a na... |

726 |
Completeness theorems for non-cryptographic fault-tolerant distributed computation
- Ben-Or, Goldwasser, et al.
- 1988
(Show Context)
Citation Context ...m computation on data that has been encoded and shared among several processors so that the data known collectively by any small subset of the processors gives no information about the unencoded data =-=[8]-=-. This is similar to the quantum mechanical requirement that measurement of the states of a small number of qubits in a fault-tolerant quantum circuit must give no information about the unencoded data... |

719 |
Quantum Theory: Concepts and Methods
- Peres
- 1998
(Show Context)
Citation Context ...e produces unreliable output, and with probability 1 \Gamma p, the gate works perfectly. This model thus assumes "fast" errors, which cannot be prevented by the quantum watchdog (quantum Zen=-=o) effect [19]. This typ-=-e of error encompasses a standard model for decoherence, where, with some small probability, the state of a gate is "measured" during its operation. Fault-tolerant circuits which can correct... |

710 | Quantum theory, the Church-Turing principle, and the universal quantum computer - Deutsch - 1985 |

656 | Quantum cryptography: Public-key distribution and coin tossing - Bennett, Brassard - 1984 |

505 | Logical reversibility of computation - Bennett - 1973 |

499 | Quantum Complexity Theory - Bernstein, Vazirani - 1997 |

469 | W.K.: Teleporting an unknown quantum state via dual classical and EPR-channels
- Bennett, Brassard, et al.
(Show Context)
Citation Context ...n to be in a certain state and then using them to perform operations on another set of qubits. This is reminiscent of techniques used in several quantum communication papers. In quantum teleportation =-=[5], if two resear-=-chers share an EPR pair, they can use this pair and classical communication to "teleport" the quantum state of a particle from one researcher to another. In [6], a small number of "USDA... |

398 |
Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components, Automata Studies
- Neumann
- 1956
(Show Context)
Citation Context ...oisy quantum circuits (although this seems plausible since noisy quantum gates might be used to simulate measurement by noisy classical gates, which can in turn perform reliable classical computation =-=[29, 20]-=-). For now, it appears to be easier to provide fault-tolerance if measurement operations are allowed during the computation. There is no fundamental physical reason for requiring that measurement be d... |

369 | On the Power of Quantum Computation
- Simon
- 1997
(Show Context)
Citation Context ...puter science. Using only polynomial resources, these quantum computers can compute certain functions which are not known to be computable on classical digital computers in less than exponential time =-=[26, 23, 25]-=-. The potentially most useful algorithms for quantum computers discovered so far include prime factorization and simulation of certain quantum mechanical systems. Given these theoretical results, a na... |

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

321 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...g about decoherence is to consider the environment to be "measuring" the state of a quantum system by interacting with it [31]. A second potential obstacle to building quantum computers is i=-=naccuracy [16, 9, 4]-=-. Quantum computers are fundamentally analog-type devices; that is, the state of a quantum superposition depends on certain continuous parameters. For example, one of the common quantum gates used in ... |

287 | Quantum circuit complexity
- Yao
(Show Context)
Citation Context ... that calculate on encoded qubits [32]. Some of their ideas might be useful for simplifying our constructions. 2. Quantum circuits The model of quantum computation we use is the quantum circuit model =-=[30]-=-. Our quantum computation will be done in the quantum state space of n two-state quantum systems (e.g., spin1 2 particles). Each of these two-state quantum systems can be in a superposition of two qua... |

278 | Good Quantum Error-Correcting Codes Exist”, Phys
- Calderbank, Shor
- 1996
(Show Context)
Citation Context ...with it called the "environment." Thus, decoherence-reduction methods can often be used to correct inaccuracy, and vice versa. It has already been shown that the use of quantum error correct=-=ing codes [24, 11, 27, 15, 7]-=- can reduce both decoherence and inaccuracy dramatically during transmission and storage of quantum data. We build upon these techniques to show that the use of these codes can also reduce decoherence... |

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

238 | Conservative logic - Fredkin, Toffoli - 1982 |

231 |
Scheme for reducing decoherence in quantum computer memory”, Phys
- Shor
- 1995
(Show Context)
Citation Context ...with it called the "environment." Thus, decoherence-reduction methods can often be used to correct inaccuracy, and vice versa. It has already been shown that the use of quantum error correct=-=ing codes [24, 11, 27, 15, 7]-=- can reduce both decoherence and inaccuracy dramatically during transmission and storage of quantum data. We build upon these techniques to show that the use of these codes can also reduce decoherence... |

220 | Elementary gates for quantum computation
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...not depend on k. Such a set of quantum gates powerful enough to realize quantum computation is called a universalsset of quantum gates; It has been shown that most sets of quantum gates are universal =-=[3]-=-. In a measurement operation, we measure the value (0 or 1) of one of the qubits. This will project the system into a superposition of states where this qubit has a definite value of either 0 or 1. If... |

179 |
Decoherence and the transition from quantum to classical
- Zurek
(Show Context)
Citation Context ... decay easily; this decay phenomenon is called decoherence. One way of thinking about decoherence is to consider the environment to be "measuring" the state of a quantum system by interactin=-=g with it [31]-=-. A second potential obstacle to building quantum computers is inaccuracy [16, 9, 4]. Quantum computers are fundamentally analog-type devices; that is, the state of a quantum superposition depends on ... |

165 | Mixed State Entanglement and Quantum Error Correction”, Phys
- Bennett, DiVincenzo, et al.
- 1996
(Show Context)
Citation Context ...with it called the "environment." Thus, decoherence-reduction methods can often be used to correct inaccuracy, and vice versa. It has already been shown that the use of quantum error correct=-=ing codes [24, 11, 27, 15, 7]-=- can reduce both decoherence and inaccuracy dramatically during transmission and storage of quantum data. We build upon these techniques to show that the use of these codes can also reduce decoherence... |

154 | Stabilizer codes and quantum error correction - Gottesman - 1997 |

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

147 | Quantum computations with cold trapped ions - Cirac, Zoller - 1995 |

145 |
Elementary gates for quantum computation, Phys. Rev. A 52
- Barenco, Bennet, et al.
- 1995
(Show Context)
Citation Context ...not depend on k. Such a set of quantum gates powerful enough to realize quantum computation is called a universal set of quantum gates; It has been shown that most sets of quantum gates are universal =-=[3]-=-. In a measurement operation, we measure the value (0 or 1) of one of the qubits. This will project the system into a superposition of states where this qubit has a definite value of either 0 or 1. If... |

138 | Multiple particle interference and quantum error correction
- Steane
- 1996
(Show Context)
Citation Context |

134 | Quantum vs. classical communication and computation - Buhrman, Cleve, et al. - 1998 |

115 | Quantum circuits with mixed states
- Aharonov, Kitaev, et al.
- 1998
(Show Context)
Citation Context ...odel of computation, augmented with measurement operations during the computation. For noise-free quantum circuits, these measurement operations can always be delayed until the end of the computation =-=[2]-=-; thus previous definitions of quantum circuits have sometimes only permitted measurement steps at the end, as this model is easier to work with and was believed to be equivalent. It has not yet been ... |

115 |
W.K.: Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
- Bennett, Brassard, et al.
- 1996
(Show Context)
Citation Context ...In quantum teleportation [5], if two researchers share an EPR pair, they can use this pair and classical communication to "teleport" the quantum state of a particle from one researcher to an=-=other. In [6], a small -=-number of "USDA" pairs of qubits known to be in perfect EPR states can used to purify a set of noisy EPR states, sacrificing some of them but yielding a large set of good EPR pairs. This par... |

109 | A silicon-based nuclear spin quantum computer. Nature 393 - Kane - 1998 |

91 | A framework for fast quantum mechanical algorithms - Grover - 1998 |

86 | Scheme for reducing decoherence in quantum memory - Shor - 1995 |

85 | Quantum Information Theory - Shor |

71 |
Polynomial time algorithms for prime factorisation and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...puter science. Using only polynomial resources, these quantum computers can compute certain functions which are not known to be computable on classical digital computers in less than exponential time =-=[26, 23, 25]-=-. The potentially most useful algorithms for quantum computers discovered so far include prime factorization and simulation of certain quantum mechanical systems. Given these theoretical results, a na... |

64 | Z4-Kerdock codes, orthogonal spreads, and extremal line-sets, preprint
- Calderbank, Cameron, et al.
(Show Context)
Citation Context ... computation. They only generate unitary matrices in the Clifford group, which is a finite group of unitary transformations in 2n-dimensional complex space that arises in several areas of mathematics =-=[10]-=-. 7The transformations in this group corresponding to classical computation are the linear Boolean transformations, which can be built out of xor and not gates. To obtain a set of gates universal for... |

62 | Efficient Simulation of Quantum Systems by Quantum Computers - Zalka - 1998 |

47 |
Is quantum mechanics useful
- Landauer
- 1995
(Show Context)
Citation Context ...m computers can successfully be built, scaling these up to computers that are large enough to yield useful computations could present formidable difficulties. One of these difficulties is decoherence =-=[16, 28, 12]-=-. Quantum computation involves manipulating the quantum states of objects that are in coherent quantum superpositions. These superpositions, however, tend to be quite fragile and decay easily; this de... |

46 | Maintaining coherence in quantum computers
- Unruh
- 1995
(Show Context)
Citation Context ...m computers can successfully be built, scaling these up to computers that are large enough to yield useful computations could present formidable difficulties. One of these difficulties is decoherence =-=[16, 28, 12]-=-. Quantum computation involves manipulating the quantum states of objects that are in coherent quantum superpositions. These superpositions, however, tend to be quite fragile and decay easily; this de... |

46 |
Quantum computations with cold trapped ions,” Phys
- Cirac, Zoller
- 1995
(Show Context)
Citation Context ...certain quantum mechanical systems. Given these theoretical results, a natural question is whether such computers could ever be built. Ingenious designs for such computers have recently been proposed =-=[13]-=-, and currently several experiments are underway in attempts to build small working prototypes [18]. Even if small quantum computers can successfully be built, scaling these up to computers that are l... |

45 |
Upper bound on the redundancy of self-correcting arrangements of unreliable elements
- Dobrushin, I
- 1977
(Show Context)
Citation Context ...to reduce error in classical computers at relatively low cost, general techniques for performing reliable computation with noisy gates require a logarithmic increase in the size of classical circuits =-=[14, 21]-=-. These techniques involve keeping several copies of every bit, and periodically reconciling them by setting them to the majority value [29, 20, 22]. Digital circuitry is reliable enough that these te... |

43 | Simulation of Many-Body Fermi Systems on a Universal Quantum Computer - Abrams, Lloyd - 1997 |

37 | Is Quantum Mechanically Coherent Computation Useful - Landauer - 1995 |

34 |
Maintaining coherence in quantum computers, Phys
- Unruh
- 1995
(Show Context)
Citation Context ...m computers can successfully be built, scaling these up to computers that are large enough to yield useful computations could present formidable difficulties. One of these difficulties is decoherence =-=[16, 28, 12]-=-. Quantum computation involves manipulating the quantum states of objects that are in coherent quantum superpositions. These superpositions, however, tend to be quite fragile and decay easily; this de... |

28 | Ensemble quantum computing by nuclear magnetic resonance spectroscopy - Cory, Fahmy, et al. - 1997 |

28 | Bulk spin-resonance quantum computation Science - Gershenfeld, Chuang - 1997 |

26 |
Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets
- Calderbank, Cameron, et al.
- 1997
(Show Context)
Citation Context ...omputation. They only generate unitary matrices in the Clifford group, which is a finite group of unitary transformations in 2 n -dimensional complex space that arises in several areas of mathematics =-=[10]-=-. The transformations in this group corresponding to classical computation are the linear Boolean transformations, which can be built out of XOR and NOT gates. To obtain a set of gates universal for q... |

25 | Fast version of Shor’s quantum factoring algorithm - Zalka - 1998 |

25 | On a lower bound for the redundancy of reliable networks with noisy gates
- Pippenger, Stamoulis, et al.
- 1991
(Show Context)
Citation Context ...to reduce error in classical computers at relatively low cost, general techniques for performing reliable computation with noisy gates require a logarithmic increase in the size of classical circuits =-=[14, 21]-=-. These techniques involve keeping several copies of every bit, and periodically reconciling them by setting them to the majority value [29, 20, 22]. Digital circuitry is reliable enough that these te... |