## Reliable quantum computers (1998)

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Venue: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Citations: | 126 - 3 self |

### BibTeX

@INPROCEEDINGS{Preskill98reliablequantum,

author = {John Preskill},

title = {Reliable quantum computers},

booktitle = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},

year = {1998},

pages = {454--1969}

}

### Years of Citing Articles

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### Abstract

The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 10-6, would be a formidable factoring engine. Even a smaller less-accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence

### Citations

1950 |
Sloane The Theory of Error-Correcting Codes
- MacWilliams, A
- 1977
(Show Context)
Citation Context ...um error correction could really be possible (Unruh 1995; Landauer 1995, 1996, 1997). First of all, although very sophisticated methods have been developed to correct errors in classical information (=-=MacWilliams & Sloane 1977-=-), it is not immediately clear how to adapt these methods to correct the phase errors that plague quantum systems. Second, in a quantum computer, as in a classical analog computer, small errors can ac... |

808 | Algorithms for quantum computation: Discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...) using altogether 7-qubits (by embedding the two-dimensional Hilbert space in a space of dimension 2 7 ). Steane’s code is actually closely related to a familiar classical error-correcting code, the =-=[7,4,3]-=- Hamming code. 26 To understand why Steane’s code works, it is important to first understand the classical Hamming code. The Hamming code uses a block of 7 bits to encode 4 bits of classical informati... |

650 | Quantum theory, the Church-Turing principle, and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...rds can be characterized by a parity check matrix ⎛ ⎞ 0 0 0 1 1 1 1 H = ⎝ 0 1 1 0 0 1 1⎠ . (1) 1 0 1 0 1 0 1 Each valid codeword is a 7-bit string vcode that satisfies ∑ k Hjk (vcode) k = 0 (mod 2) ; =-=(2)-=- that is, the matrix H annihilates each codeword in mod 2 arithmetic. Since Z2 = {0, 1} is a (finite) field, familiar results of linear algebra apply here. H has three linearly independent rows and it... |

360 |
Probabilistic logics and the synthesis of reliable organisms from unreliable components
- Neumann
- 1956
(Show Context)
Citation Context ...lip errors in the rotated basis, so that after recovery the block will have undergone a bit flip in the rotated basis, or in the original basis the phase flip |0〉code → |0〉code , |1〉code → −|1〉code . =-=(13)-=- (If one qubit in the block has a phase error, and another one has a bit flip error, then recovery will be successful.) Thus we have seen that Steane’s code can enhance the reliability of stored quant... |

262 | Good quantum error-correcting codes exist
- Calderbank, Shor
- 1996
(Show Context)
Citation Context ...The source and the target of an XOR gate are interchanged if we perform a change of basis with Hadamard rotations. encoded information can be maintained with a fidelity F =1−O � ɛ t+1� (Steane 1996b; =-=Calderbank and Shor 1996-=-; Gottesman 1996; Calderbank et al. 1996, 1997). The current status of quantum coding theory is reviewed by Shor in these proceedings (Shor 1997). 4 Fault-tolerant recovery But, of course, error recov... |

232 | Quantum error-correction via codes over GF(4
- Calderbank, Rains, et al.
- 1998
(Show Context)
Citation Context ...re interchanged if we perform a change of basis with Hadamard rotations. encoded information can be maintained with a fidelity F =1−O � ɛ t+1� (Steane 1996b; Calderbank and Shor 1996; Gottesman 1996; =-=Calderbank et al. 1996-=-, 1997). The current status of quantum coding theory is reviewed by Shor in these proceedings (Shor 1997). 4 Fault-tolerant recovery But, of course, error recovery will never be flawless. Recovery is ... |

204 | Fault-tolerant quantum computation
- Shor
- 1996
(Show Context)
Citation Context ...edures that must be followed to implement fault-tolerant computation can be codified in what I will call “the laws of fault-tolerant computation.” These can be distilled from Shor’s pioneering paper (=-=Shor 1996-=-). The first law is (1) Don’t use the same bit twice. 3 A bad network of XOR gates that breaks this commandment is shown in Fig. 2 (using the notation of Fig. 1). The bit that I have called the ancill... |

192 | Fault-Tolerant Quantum Computation With Constant Error Rate,” ArXiv e-prints
- Aharonov, Ben-Or
- 1999
(Show Context)
Citation Context ...to how long a computation can proceed until errors become likely. This limitation can be overcome by using a special kind of code, a concatenated code (Knill & Laflamme 1996; Knill et al. 1996, 1997; =-=Aharonov & Ben-Or 1996-=-; Kitaev 1996ab). To understand the concept of a concatenated code, imagine that we are using Steane’s quantum error-correcting code that encodes a single qubit as a block of 7 qubits. But if we look ... |

163 |
Decoherence and the transition from quantum to classical. Phys. Today
- Zurek
- 1991
(Show Context)
Citation Context ...phase errors as well as bit flip errors. To accomplish this, we observe (following Steane11,12 ) that we can change the basis for each qubit by applying the Hadamard rotation R = 1 √ 2 ( 1 1 1 −1 ) . =-=(9)-=- Then phase errors in the |0〉, |1〉 basis become bit flip errors in the rotated basis |˜0〉 ≡ 1 √ (|0〉 + |1〉) , |˜1〉 ≡ 2 1 √ (|0〉 − |1〉) . (10) 2 It will therefore be sufficient if our code is able to d... |

159 | Mixed state entanglement and quantum error correction
- Bennett, DiVincenzo, et al.
- 1996
(Show Context)
Citation Context ...mputation. What then is the meaning of the fifth law? While any code can be used in principle, some codes are better than others. For example, there is a 5-qubit code that can recover from one error (=-=Bennett et al. 1996-=-, Laflamme et al. 1996), and Gottesman has exhibited a universal set of fault-tolerant gates for this code. But the gate implementation is quite complex. The 7-qubit Steane code requires a larger bloc... |

155 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ...n can proceed until errors become likely. This limitation can be overcome by using a special kind of code, a concatenated code (Knill & Laflamme 1996; Knill et al. 1996, 1997; Aharonov & Ben-Or 1996; =-=Kitaev 1996-=-a, b). To understand the concept of a concatenated code, imagine that we are using Steane's quantum error-correcting code that encodes a single qubit as Proc. R. Soc. Lond. A (1998)Reliable quantum c... |

153 |
A class of quantum error-correcting codes saturating the quantum hamming bound
- Gottesman
- 1996
(Show Context)
Citation Context ...of an XOR gate are interchanged if we perform a change of basis with Hadamard rotations. encoded information can be maintained with a fidelity F =1−O � ɛ t+1� (Steane 1996b; Calderbank and Shor 1996; =-=Gottesman 1996-=-; Calderbank et al. 1996, 1997). The current status of quantum coding theory is reviewed by Shor in these proceedings (Shor 1997). 4 Fault-tolerant recovery But, of course, error recovery will never b... |

146 | Quantum measurements and the Abelian stabilizer problem - Kitaev - 1995 |

143 |
Stabilizer codes and quantum error correction
- Gottesman
(Show Context)
Citation Context ...e flow equations are too complicated to solve exactly, we can obtain approximate solutions by making conservative assumptions. Crudely speaking, the conclusions are as follows (Gottesman et al. 1998; =-=Gottesman 1997-=-b). If storage errors are negligible, then the threshold rate for gate errors is about 10-4. If storage errors dominate, then the threshold error rate is about 10-5 per time step (where the unit of ti... |

142 |
A single quantum cannot be cloned. Nature 299, 802. Discussion P. Marcer (BCS Cybernetic Machine Group
- Wootters, Zurek
- 1982
(Show Context)
Citation Context ...the measurement will destroy the delicate quantum information that is encoded in the device. Finally, to protect against errors we must encode information in a redundant manner. But a famous theorem (=-=Wootters & Zurek 1982-=-; Dieks 1982) says that quantum information cannot be copied, so it is not obvious how to store information with the required redundancy. But by now all of these apparent obstacles have been overcome—... |

142 |
Perfect quantum error correction code
- Laflamme, Miquel, et al.
- 1996
(Show Context)
Citation Context ...2 ǫ ; (36) ǫ (L) ∼ ǫ0 thus, to be reasonably confident that we can complete a computation with T gates without making an error we must choose the block size 7L to be block size ∼ [ ] log2 7 log ǫ0T . =-=(37)-=- log ǫ0/ǫ If the code that is concatenated has block size n and can correct t + 1 errors, the power log 2 7 ∼ 2.8 in Eq. (37) is replaced by log n/ log(t + 1); this power approaches 2 for the family o... |

136 | Multiple particle interference and quantum error correction
- Steane
- 1996
(Show Context)
Citation Context ...rror occurs, (2) the bit flip |0〉 ↔|1〉occurs, (3) the relative phase of |0〉 and |1〉 flips, (4) both a bit flip and a phase flip occur. Now it is clear how a quantum error-correcting code should work (=-=Steane 1996-=-b; Knill & Laflamme 1997). By making a suitable measurement, we wish to diagnose which of these four possibilities actually occurred. Of course, in general, the state of the qubit will be a linear com... |

133 |
Quantum Computations with Cold Trapped Ions
- Cirac, Zoller
- 1995
(Show Context)
Citation Context ...& Knight 1996). For example, in an ion trap computer we might store quantum information in a two-dimensional space spanned by the ground state of the ion and a particular long-lived metastable state (=-=Cirac & Zoller 1995-=-). But during the operation of the device, the ion might make an unexpected transition to another state. If that state decays quickly to the ground state, then the error can be detected and corrected ... |

128 |
Error correcting codes in quantum theory
- Steane
- 1996
(Show Context)
Citation Context ... occurs; (2) the bit flip 10) <+- I1) occurs; (3) the relative phase of ]0) and I1) flips; (4) both a bit flip and a phase flip occur. Now it is clear how a quantum error-correcting code should work (=-=Steane 1996-=-b; Knill & Laflamme 1997). By making a suitable measurement, we wish to diagnose which of these four possibilities actually occurred. Of course, in general, the state of the qubit will be a linear com... |

92 |
Simulating physics with computers,” Int
- Feynman
- 1982
(Show Context)
Citation Context ...assical information; that is, there are 16 = 2 4 strings of length 7 that are the valid codewords. The codewords can be characterized by a parity check matrix ⎛ ⎞ 0 0 0 1 1 1 1 H = ⎝ 0 1 1 0 0 1 1⎠ . =-=(1)-=- 1 0 1 0 1 0 1 Each valid codeword is a 7-bit string vcode that satisfies ∑ k Hjk (vcode) k = 0 (mod 2) ; (2) that is, the matrix H annihilates each codeword in mod 2 arithmetic. Since Z2 = {0, 1} is ... |

85 | Bulk spin resonance quantum computation
- Gershenfeld, Chuang
- 1997
(Show Context)
Citation Context ... be a powerful and valuable device (assuming a reasonable processing speed). From the perspective of the current state of the technology (Monroe et 1al. 1995; Turchette et al. 1995; Cory et al. 1996; =-=Gershenfeld & Chuang 1997-=-), these numbers seem daunting. But in fact a machine that meets far less demanding specifications may still be very useful (Preskill 1997). First of all, quantum computers can do other things besides... |

85 | Quantum networks for elementary arithmetic operations - Vedral, Barenco, et al. - 1995 |

84 |
Scheme for reducing decoherence in quantum memory
- Shor
- 1995
(Show Context)
Citation Context ... by applying the Hadamard rotation R = 1 √ 2 ( 1 1 1 −1 ) . (9) Then phase errors in the |0〉, |1〉 basis become bit flip errors in the rotated basis |˜0〉 ≡ 1 √ (|0〉 + |1〉) , |˜1〉 ≡ 2 1 √ (|0〉 − |1〉) . =-=(10)-=- 2 It will therefore be sufficient if our code is able to diagnose bit flip errors in this rotated basis. But if we apply the Hadamard rotation to each of the 7 qubits, then Steane’s logical 0 and log... |

80 |
Universal quantum simulator
- Lloyd
- 1996
(Show Context)
Citation Context ...manding specifications may still be very useful (Preskill 1997). First of all, quantum computers can do other things besides factoring, and some of these other tasks (in particular quantum simulation-=-=Lloyd 1996-=-) might be accomplished with a less reliable or smaller device. Furthermore, our estimate of the accuracy threshold might be too conservative for a number of reasons. For example, the estimate was obt... |

78 | Theory of quantum error-correcting codes
- Knill, Laflamme
- 1997
(Show Context)
Citation Context ...he bit flip 10) <+- I1) occurs; (3) the relative phase of ]0) and I1) flips; (4) both a bit flip and a phase flip occur. Now it is clear how a quantum error-correcting code should work (Steane 1996b; =-=Knill & Laflamme 1997-=-). By making a suitable measurement, we wish to diagnose which of these four possibilities actually occurred. Of course, in general, the state of the qubit will be a linear combination of these four s... |

63 | Resilient quantum computation: error models and thresholds
- Knill, Laflamme, et al.
- 1998
(Show Context)
Citation Context ...qubit so that M in the rotated basis is a product of I’s and Z’s acting on the individual qubits. (We rotate by R = 1 ( ) 1 1 √ (19) 2 1 −1 for each qubit acted on by X in M, and by R ′ = 1 ( ) 1 i √ =-=(20)-=- 2 i 1 for each qubit acted on by Y .) In this basis, the value of M is just the parity of the bits for which Z’s appear. We can measure the parity (much as we did e Actually, it is also acceptable if... |

61 |
Measurement of conditional phase shifts for quantum logic
- Turchette, Hood, et al.
- 1995
(Show Context)
Citation Context ... per gate of about one in a million would be a powerful and valuable device (assuming a reasonable processing speed). From the perspective of the current state of the technology (Monroe et 1al. 1995; =-=Turchette et al. 1995-=-; Cory et al. 1996; Gershenfeld & Chuang 1997), these numbers seem daunting. But in fact a machine that meets far less demanding specifications may still be very useful (Preskill 1997). First of all, ... |

54 | Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing
- Cory, Price, et al.
- 1998
(Show Context)
Citation Context ...in a million would be a powerful and valuable device (assuming a reasonable processing speed). From the perspective of the current state of the technology (Monroe et 1al. 1995; Turchette et al. 1995; =-=Cory et al. 1996-=-; Gershenfeld & Chuang 1997), these numbers seem daunting. But in fact a machine that meets far less demanding specifications may still be very useful (Preskill 1997). First of all, quantum computers ... |

51 | The Theory of Error Correcting Codes - Macillams, Sloane - 1977 |

47 |
Quantum error correction with imperfect gates
- Kitaev
- 1997
(Show Context)
Citation Context ...n can proceed until errors become likely. This limitation can be overcome by using a special kind of code, a concatenated code (Knill & Laflamme 1996; Knill et al. 1996, 1997; Aharonov & Ben-Or 1996; =-=Kitaev 1996-=-ab). To understand the concept of a concatenated code, imagine that we are using Steane’s quantum error-correcting code that encodes a single qubit as a block of 7 qubits. But if we look more closely ... |

46 |
Is quantum mechanics useful
- Landauer
- 1995
(Show Context)
Citation Context ...ea how quantum error correction would work, or whether it would work. Indeed, there were a number of reasons for pessimism about whether quantum error correction could really be possible (Unruh 1995; =-=Landauer 1995-=-, 1996, 1997). First of all, although very sophisticated methods have been developed to correct errors in classical information (MacWilliams & Sloane 1977), it is not immediately clear how to adapt th... |

44 |
Quantum error correction and orthogonal
- Calderbank, Rains, et al.
- 1997
(Show Context)
Citation Context ...syndrome measurement increases with t like a power t b . Then the probability that t + 1 errors accumulate before the measurement is complete will behave like Block Error Probability ∼ ( t b ǫ )t+1 , =-=(30)-=- where ǫ is the probability of error per step. We may then choose t to minimize the error probability (t ∼ e−1ǫ−1/b , assuming t is large), obtaining ( Minimum Block Error Probability ∼ exp −e −1 bǫ −... |

44 | Reliable computation with cellular automata
- Gacs
- 1986
(Show Context)
Citation Context ...hat we want to store one qubit in an unknown pure state |ψ〉. Due to imperfections in our storage device, the state ρout that we recover will have suffered a loss of fidelity: F ≡ 〈ψ|ρout|ψ〉 = 1 − ǫ . =-=(14)-=- But if we store the qubit using Steane’s 7-qubit block code, if each of the 7qubits is maintained with fidelity F = 1 − ǫ, if the errors on the qubits are uncorrelated, and if we can perform error re... |

43 | Semiclassical Fourier transform for quantum computation. Physical Review Letters 76:3228–3231 - Griffiths, Niu - 1996 |

38 | Fault-tolerant error correction with efficient quantum codes
- DiVincenzo, Shor
- 1996
(Show Context)
Citation Context ... scheme for fault-tolerant syndrome measurement have been described here only for the 7-qubit code, but they can be adapted to more complex codes that have the capability to recover from many errors (=-=DiVincenzo & Shor 1996-=-; Steane 1997). As the complexity of the code increases, Steane’s scheme becomes substantially more efficient than Shor’s. 5 Fault-tolerant quantum gates We have seen that coding can protect quantum i... |

38 | Breakdown of predictability in gravitational collapse, Phys - Hawking - 1976 |

37 |
Is Quantum Mechanically Coherent Computation Useful
- Landauer
- 1995
(Show Context)
Citation Context ...ng code (those with an even number of 1’s), |0〉code = = 1 √ 8 ⎛ 1 √ ⎝ 8 ∑ even v ∈ Hamming ⎞ |v〉 ⎠ ( |0000000〉 + |0001111〉 + |0110011〉 + |0111100〉 ) +|1010101〉 + |1011010〉 + |1100110〉 + |1101001〉 , 7 =-=(6)-=-� � � � � � � � � � � � |0〉 ❣ ❣ ❣ ❣ Measure |0〉 ❣ ❣ ❣ ❣ Measure |0〉 ❣ ❣ ❣ ❣ Measure Figure 2: Computation of the bit-flip syndrome for Steane’s 7-qubit code. Repeating the computation in the rotated ... |

34 | Accuracy threshold for quantum computation, Technical report, Quantum Physics e-Print archive
- Knill, Laflamme, et al.
- 1996
(Show Context)
Citation Context ...curacy, there is a limit to how long a computation can proceed until errors become likely. This limitation can be overcome by using a special kind of code, a concatenated code (Knill & Laflamme 1996; =-=Knill et al. 1996-=-, 1997; Aharonov & Ben-Or 1996; Kitaev 1996ab). To understand the concept of a concatenated code, imagine that we are using Steane’s quantum error-correcting code that encodes a single qubit as a bloc... |

34 | Operator Quantum Error Correction
- Kribs, Laflamme, et al.
- 2006
(Show Context)
Citation Context ...the meaning of the fifth law? While any code can be used in principle, some codes are better than others. For example, there is a five-qubit code that can recover from one error (Bennett et al. 1996; =-=Laflamme et al. 1996-=-), and Gottesman has exhibited a universal set of fault-tolerant gates for this code. But the gate implementation is quite complex. The seven-qubit Steane code requires a larger block, but it is much ... |

31 |
Demonstration of a fundamental quantum logic
- Monroe, Meekhof, et al.
- 1995
(Show Context)
Citation Context ...its and an error rate per gate of about one in a million would be a powerful and valuable device (assuming a reasonable processing speed). From the perspective of the current state of the technology (=-=Monroe et al. 1995-=-; Turchette et al. 1995; Cory et al. 1996; Gershenfeld & Chuang 1997), these numbers seem daunting. But in fact a machine that meets far less demanding specifications may still be very useful (Preskil... |

30 | The Physical Nature of Information - Landauer - 1996 |

28 | Difficulties for the evolution of pure states into mixed states - Banks, Peskin, et al. - 1984 |

27 | Quantum error correcting codes need not completely reveal the error syndrome - Shor, Smolin - 1996 |

27 | Efficient networks for quantum factoring - Beckman, Chari, et al. - 1996 |

25 | Threshold Estimate for Fault Tolerant Quantum Computation
- Zalka
- 1997
(Show Context)
Citation Context ...es (that is, by using the ancilla as the source and the data as the target), we can avoid applying the R gates to the data, and hence can reduce the likelihood of damaging the data with faulty gates (=-=Zalka 1996-=-; Steane 1997).) Due to error propagation, a single error that occurs during the preparation of the Shor state can result in two phase errors in the state, and these can both propagate to the data if ... |

24 | The capacity of a noisy quantum channel - Lloyd - 1997 |

24 | Time, Parallelism and Noise Requirements for Reliable Quantum Computing. Fortschritte der Physik - Space - 1999 |

19 | 2n-quasihole states realize 2n-1-dimensional spinor braiding statistics in paired quantum hall states - Nayak, Wilczek - 1996 |

18 | Decoherence limits to quantum computation using trapped ions
- Plenio, Knight
- 1997
(Show Context)
Citation Context ...nvironment, or rotates in the two-dimensional space in an unpredictable way. But there is another possible type of error, in which the qubit leaks out of the twodimensional space into a larger space (=-=Plenio & Knight 1996-=-). For example, in an ion-trap computer we might store quantum information in a two-dimensional space spanned by the ground state of the ion and a particular long-lived metastable state (Cirac & Zolle... |

17 |
Active Stabilization, Quantum Computation, and Quantum State Synthesis
- Steane
- 1997
(Show Context)
Citation Context ... by using the ancilla as the source and the data as the target), we can avoid applying the R gates to the data, and hence can reduce the likelihood of damaging the data with faulty gates (Zalka 1996; =-=Steane 1997-=-).) 10 = � �s|0〉 R � |0〉 |0〉 |0〉 |0〉 � � � � � � � � � R R R R Measure Figure 11: Construction and verification of the Shor state. If the measurement outcome is 1, then the state is discarded and a ne... |